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This article is cited in 1 scientific paper (total in 1 paper) Renormalization in one-dimensional dynamics A. S. Skripchenkoab a HSE University b Skolkovo Institute of Science and Technology, Russia
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     1. IntroductionAn interval exchange transformation (IET) is a one-to-one semicontinuous map from a half-open interval to itself with a finite number of discontinuity points, such that on each interval of continuity this map is a shift. This kind of maps appear in the theory of dynamical systems and low-dimensional topology in different contexts. For example, interval exchange transformations can be seen as the first return maps on transversals to the billiard flows in rational polygons. In addition, interval exchange transformations are closely related to measured foliations with singularities that are defined by closed 1-forms on orientable surfaces. Namely, taking any interval exchange transformation, one can construct an oriented surface for which this IET is the first return map for a foliation of fixed direction on some interval transversal to this direction. And vice versa, for a measured foliation defined by some 1-form the first return map on every transversal is an IET. IETs are also interesting from the point of view of other branches of dynamical systems: certain classes of IETs appear in symbolic dynamics (for instance, as a geometric model for Sturmian and epi-Sturmian sequences) or as a natural extension of circle rotations in the KAM theory. Therefore, the ergodic properties of IETs, such as minimality, ergodicity, the number of ergodic measures, weak and strong mixing, questions of solvability of cohomological equations, and so on, as well as topological and geometric interpretations of the results obtained in terms of the properties of foliations and moduli spaces play a valuable role in several branches of mathematics. For the first time IETs were defined by Arnold and Oseledts in the 1960s; slightly later Katok and Stepin described the connection between interval exchange transformations and billiards in rational polygons. At the same time some serious progress on the research on measured foliations on surfaces was achieved; measured foliations were introduced by Thurston. In the 1970s it was proved that typical IETs are minimal (Keane) and not strongly mixing (Katok). In the 1980s, due to efforts of H. Masur and Veech several breakthrough results on the ergodic properties of IETs were established, namely, it was shown that a typical IET is uniquely ergodic. At the same time the question of whether a typical IET is weakly mixing remained open till 2007, when a key result was proved by Avila and Forni. In the same period of time some observations on cohomological equations for IETs were made by Forni (on the one hand) and Yoccoz, Marmi, and Moussa (on the other). This work became a base for the further study of the theory of small divisors and its connections with the KAM theory. In parallel the research on the applications of the ergodic and spectral properties of IETs to the geometry of moduli spaces of abelian and quadratic differentials was conducted. In this context the main emphasis should be made on results on the Lyapunov spectrum of the Kontsevich–Zorich cocycle and its connection with the number of connected components of the moduli spaces obtained by A. Zorich and Kontsevich, as well as on a theorem due to Avila and Viana on the simplicity of the Lyapunov spectrum. Finally, the most valuable result in this direction was due to Eskin and Mirzakhani, who obtained a full classification of actions of the group $\operatorname{SL}(2,\mathbb R)$ on the moduli space. The key tool, used in all the above works to study the ergodic, topological, and even symbolic properties of IETs, is a renormalization algorithm, which in this case is called Rauzy induction. One step of Rauzy induction is the transformation that builds another IET from the original one so that the resulting IET is equivalent to the original one in terms of the behaviour of orbits, but is defined on a smaller support interval; the renormalization algorithm defines an infinite sequence of such transformations. Each step of Rauzy induction is described by a matrix used to express the lengths of the resulting IETs in terms of the lengths of the original IET; adding the renormalization condition on the sum of the lengths of segments allows us to determine a sequence of projective transformations. Thus, in the case of IETs the renormalization procedure is given by a multidimensional fraction algorithm. In the case of two intervals this algorithm coincides with the Gauss map. The ergodic and combinatorial properties of the renormalization algorithm as an independent dynamical system determine many important ergodic properties of the original IET. At the same time, for many topologically meaningful problems one needs to study the ergodic properties and characteristics of orbits, as well as the geometric properties of moduli spaces not for IETs themselves but for their natural generalizations, like IETs with flips, linear involutions, interval translation mappings, systems of isometries, and so on. In comparison to classical IETs, all these generalizations were much less extensively investigated. IETs with flips were introduced by Nogueira and studied subsequently by Skripchenko and Troubetzkoy. Linear involutions were defined by Nogueira and Danthony, and some important results on their dynamics were obtained by Lanneau and Boissy, and by Lanneau, Marmi, and Skripchenko. An important contribution to the research on interval translation mappings was made by Bruin, Troubetzkoy, and Clack, while several recent observations are due to Artigiani, Fougeron, Skripchenko, and Hubert. Systems of isometries were defined for the first time in geometric group theory by Levitt, Gaboriau, and Paulin. Connections between systems of isometries and the Novikov problem of the asymptotic behaviour of plane sections of a triply periodic surface was discovered by Dynnikov; subsequently, Dynnikov, in collaboration with Hubert and Skripchenko, achieved a significant progress in the understanding of the ergodic properties of measured foliations associated with systems of isometries; their results widely used some theorems proved by Avila, Hubert, and Skripchenko. In all these cases the research was focused on the construction of an efficient renormalization process and on the study of the ergodic properties of such an algorithm and applications of these observations to the description of the original dynamical system. In the current paper we provide a review of the known results in this context, as well as their topological and geometric applications, and formulate challenging open questions. The survey contains the following sections. In § 2 we introduce IETs and formulate their most valuable dynamical properties. This section also contains a topological interpretation of the results described in terms of measured foliations the surfaces and a geometric interpretation in terms of the key features of the moduli spaces of abelian differentials. In § 3 we define Markovian multidimensional fraction algorithms and discuss their ergodic and spectral properties. All renormalization algorithms that we consider here fit in this class. Section 4 is dedicated to interval exchange transformations with flips and to the particularities of the renormalization process for IETs with flips. In § 5 we discuss foliations on surfaces defined by quadratic differentials, and also some properties of the dynamics of the linear involutions associated with these foliations. Section 6 contains a review of the known results concerning renormalization and the ergodic properties of interval translation mappings, and also the proofs of a series of new statements about Bruin–Troubetzkoy interval translation mappings. In § 7 we describe systems of isometries and two-dimensional band complexes with vertical foliations on them. We provide two approaches toward the renormalization problem and the result of the application of these two approaches to dynamics, topology, and geometric group theory. Section 8 is devoted to applications of the results described above to the Novikov problem of the asymptotic behaviour of plane sections of 3-periodic surfaces. The author is grateful to I. Dynnikov for reading the previous version of this text attentively and critically and to V. M. Buchstaber for his attention to this work. 2. Interval exchange transformations: definition and key resultsIn this section we consider permutations of the n elements $1,\dots,n$ that are irreducible in the following sense: for any $1\leqslant k<n$. Given a vector $\lambda\in\mathbb{R}_{+}^n$ and a permutation $\pi$, an interval exchange transformation (IET) $T(x)=T_{\lambda,\pi}(x)$ is a map that maps the half-open interval $I=[0,\lambda_1+\cdots+\lambda_n)$ to itself in the following way:
$$
\begin{equation}
T(x)=x+\sum_{\pi(j)<\pi(i)} \lambda_j-\sum_{j<i} \lambda_j
\end{equation}
\tag{1}
$$
if $x\in I_i=\bigl[\sum_{j<i} \lambda_j,\sum_{j\leqslant i} \lambda_j\bigr)$. The orbit of the point $x$ under the action of $T$ is the set of images of $x$ under the sequence of iterations of $T$: We will be interested in the ergodic and topological properties of orbits of the map $T(x)$. For interval exchange transformations defined by an irreducible permutation Keane [43] presented a sufficient condition of minimality. Definition 2. Let $T$ be an IET defined by an irreducible permutation. We say that $T$ satisfies Keane’s condition if any discontinuity point of $T$ has an infinite orbit and for any two discontinuity points of $T$ their orbits do not intersect. In particular, if the lengths of intervals of continuity of the map $T$ are rationally independent, then Keane’s condition is satisfied. Keane showed that the following statement holds. Corollary 2. Given an irreducible permutation, almost all (with respect to the Lebesgue measure) sets of lengths $\{\lambda_i\}_{i=1}^n$ define minimal IETs. Keane’s theorem can be seen as a natural generalization of the statement about the density of orbits of an irrational rotation of the circle. Definition 3. The map $F\colon X\to X$ is called uniquely ergodic if it admits exactly one ergodic Borel invariant measure. In [43] Keane posed a conjecture that was later proved independently by Masur [55] and Veech [69]. Theorem 3 ( Masur and Veech). Almost all (with respect to the Lebesgue measure) IETs are uniquely ergodic. As in Corollary 2, by ‘almost all’ IETs we mean ‘for a fixed irreducible permutation and almost all vectors of lengths’. At the same time, one cannot replace ‘almost all’ by ‘all’ since there exist examples of minimal IETs that admit more than one ergodic measure (see, for example, [44] or [46]). More precisely, an upper bound for the number of ergodic measures admitted by an IET on $n$ intervals is based on results by Katok [41] and Veech [69] and is described by the following statement. Actually, this is an estimate for the genus of a surface that carries a vertical foliation such that the IET in question is the first return map on a transversal to this foliation (this surface is called a suspension surface). The efficiency of this boundary was shown by Sataev in [61], in terms of flows on the surface, and by Fickensher in [33], in terms of interval exchange transformations. In both cases explicit examples of a surface of genus $g$ with a linear flow or an IET with exactly $g$ invariant measures were constructed. The ergodic properties of IETs have an explicit topological interpretation in terms of the properties of measured foliations on smooth surfaces and holomorphic differential forms on closed Riemann surfaces. The notion of a measured foliation was introduced by Thurston in connection with the study of classes of diffeomorphisms of a surface; this notion means that there exists a foliation that is smooth (of class $\mathcal C^{\infty}$) everywhere except at a finite number of isolated points; these points are saddles with $k\geqslant 3$ separatrices, and the foliation is equipped with a finite transversal measure that is $\mathcal C^{\infty}$-equivalent to the Lebesgue measure. In order to describe the ergodic properties of measured foliations on Riemann surfaces we need to introduce some notation. Let $M_{g,b}$ be an oriented surface of genus $g$ with $b$ boundary components, and let
$$
\begin{equation*}
\operatorname{MCG}(M_{g,b})= \operatorname{Diff}^{+}(M_{g,b})/\operatorname{Diff}_0(M_{g,b})
\end{equation*}
\notag
$$
be its modular group. Let $\mathcal{M}\mathcal{F}(M_{g,b})$ be the space of equivalence classes of the measured foliations on $M_{g,b}$ (the equivalence relation is generated by the diffeomorphisms isotopic to the identity and by the Whitehead moves). Let $\mathcal{P}\mathcal{M}\mathcal{F}(M_{g,b})$ be the associated projective space. It is known that $\mathcal{M}\mathcal{F}(M_{g,b})$ has the structure of a topological piecewise linear manifold (and, moreover, the set of equivalence classes is homeomorphic to $\mathbb{R}^{6g-6}\setminus\{0\}$ if $g\geqslant 2$). Thus, the Lebesgue measure is naturally defined in every chart (see details in [32]). In the same paper [55] where Theorem 3 was proved, Masur proved the following statement. Theorem 5 (Masur). The action of $\operatorname{MCG}(M_{g,b})$ on $\mathcal{P}\mathcal{M}\mathcal{F}(M_{g,b})$ is ergodic (with respect to the Lebesgue measure), and almost all (with respect to the Lebesgue measure) measured foliations in the set $\mathcal{M}\mathcal{F}(M_{g,b})$ are uniquely ergodic. The mixing properties of IETs were also extensively studied. Definition 4. Let $F\colon X\to X$ be a map defined on the probability space $X$ with measure $\mu$ and $\sigma$-algebra $\mathcal{X}$. If $F$ preserves $\mu$, then we say that $F$ is strongly mixing if for any measured $A,B \in \mathcal{X}$. The map $F$ is called weakly mixing if for any measurable $A,B \in \mathcal{X}$ we have Katok showed that IETs cannot be strongly mixing maps ([42]). At the same time, as shown in [9], weak mixing is rather a typical situation. Another aspect of the research on IETs consists in the study of the solvability of cohomological equations. In dynamical systems a cohomological equation is an equation of the following type: where $T\colon X\to X$ and $g\colon X\to \mathbb{R}$ are known ($T$ is usually a bijection of some set $X$) and one needs to find when the equation (2) is solvable for a given map $f\colon X\to\mathbb{R}$.For the first time cohomological equations for IETs were considered by Forni [34], who studied linear area-preseving flows on surfaces of high genera and time changes for these flows. Using advanced analytical tools Forni showed the existence of a solution of the cohomological equation for almost all directions of the linear flow on a fixed surface with an abelian differential. His result was slightly refined for cohomological equations for IETs by Marmi, Moussa, and Yoccoz [53]. Their work provided a foundation for the further study of the rigidity properties of affine IETs and, more generally, for the development of methods of the KAM theory in one-dimensional dynamics (some further results were published in [54]). In particular, we talk about the extension of the notion of a Roth number to IETs. An irrational number $\alpha$ is called a Roth number if for any $\varepsilon>0$ there exists a positive constant $C_\varepsilon$ such that $|q\alpha-p| \geqslant C_{\varepsilon}q^{1+\varepsilon}$ for all rational $p/q$. Roth numbers admit several equivalent definitions, including the following one: $\alpha$ is a Roth number if and only if for all $r,s\in\mathbb{R}$ such that $r>s+1\geqslant 1$ and all functions $\Phi\in\mathcal{C}^r(\mathbb{T})$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ with zero mean $\displaystyle\int_{\mathbb{T}}\Phi\,dx=0$ there exists a unique function $\Psi\in\mathcal{C}^s(\mathbb{T})$ with zero mean such that where $R_{\alpha}$ is a rotation of the circle $\mathbb T$: $R_{\alpha}(x)=x+\alpha$. Roth’s theorem implies that all algebraic irrational numbers are Roth numbers; moreover, the Roth numbers form a set of full measure, which is invariant under the action of $\operatorname{SL}(2,\mathbb Z)$. In [56] this result was generalized to the case of IETs: the authors defined IETs of Roth type (see § 1.3 in [56] and Definition 12 in § 5.3 below) and proved the following statement. Theorem 7 (Marmi, Moussa, and Yoccoz). Let $T$ be an IET of Roth type such that Keane’s condition is satisfied. Denote intervals of continuity of this map by $(I_\alpha)$ and consider the following classes of functions: Then there exists a function $\chi$ that is constant on each $I_\alpha$ and a bounded function $\Psi$ such that Theorem 8 (Marmi, Moussa, and Yoccoz). The set of parameters that give rise to IETs of Roth type has a full Lebesgue measure. The key instrument that is used in the proofs of all results listed above is the renormalization algorihtm. In the case of IETs it is called Rauzy induction. Let $T$ be an IET. We denote its singular points by $u^{\mathrm{t}}_i$ and the singular points of $T^{-1}$ by $u^\mathrm{b}_i$. We consider the rightmost point that is not an endpoint of the support interval; we denote this point by $u^{\alpha}_n$ (it can coincide with a point in the sets $\{u^\mathrm{t}_i\}$ or $\{u^\mathrm{b}_i\}$); We denote the leftmost point of the support interval by $u_0$ and the rightmost point by $v_0$. Consider the first return map $T$ on the interval $J=[u_0, u^{\alpha}_n)$. It is easy to see that this map is also an interval exchange transformation, and the number of intervals of continuity and irreducibility of the permutation is preserved. We denote this IET by $T'$. Rauzy induction is a dynamical system defined on the space of IETs. In the case of an IET on two intervals this renormalization algorithm coincides with the Euclidean algorithm in the following sense: if the lengths of two continuity intervals of the original IET $T$ are equal to $(\lambda_A,\lambda_B)$, then after one step of Rauzy induction we obtain an IET with lengths of intervals
$$
\begin{equation*}
(\lambda'_A,\lambda'_B)=(\lambda_A-\lambda_B,\lambda_B)\quad (\text{if } \lambda_A>\lambda_B)
\end{equation*}
\notag
$$
or
$$
\begin{equation*}
(\lambda'_A,\lambda'_B)=(\lambda_A,\lambda_B-\lambda_A)\quad (\text{if } \lambda_B>\lambda_A).
\end{equation*}
\notag
$$
The matrix of the linear map that describes the transformation of lengths is either
$$
\begin{equation}
\begin{pmatrix} 1 &-1 \\ 0 & \hphantom{-}1 \end{pmatrix}
\end{equation}
\tag{3}
$$
or
$$
\begin{equation}
\begin{pmatrix} \hphantom{-}1 & 0 \\ -1 & 1 \end{pmatrix}.
\end{equation}
\tag{4}
$$
We can add the renormalization condition $\lambda_A+\lambda_B=1$. Set
$$
\begin{equation*}
x=\frac{\lambda_B}{\lambda_A} \quad\text{if}\ \ \lambda_B<\lambda_A,\quad\text{and}\quad x=\frac{\lambda_A}{\lambda_B} \quad\text{if}\ \ \lambda_A<\lambda_B.
\end{equation*}
\notag
$$
Then the projectivization of the map described above (the renormalization) is described by the formulae
$$
\begin{equation}
g(x)=\begin{cases} \dfrac{x}{1-x} =: g_1(x) & \text{if}\ 0<x<\dfrac{1}{2}\,, \\ \dfrac{1-x}{x}=g_1(1-x) & \text{if}\ \dfrac{1}{2}<x<1. \end{cases}
\end{equation}
\tag{5}
$$
In order to prove this it is sufficient to see that the new lengths (which we obtain after renormalization) are $\dfrac{\lambda'_A}{\lambda'_A+\lambda'_B}$ and $\dfrac{\lambda'_B}{\lambda'_A+\lambda'_B}$ and to compute their ratio. The Gaussian map $G(x)=\{\frac{1}{n}\}$ is an acceleration of this renormalization in the following sense: where $n$ is the smallest positive integer such that $g^{n-1}(x)\in[1/2,1)$.Consider one step of Rauzy induction for an IET on an arbitrary number of intervals: Since every IET is specified by a permutation and a vector of lengths, we can suppose that Rauzy induction acts on the space of pairs $(\pi,\lambda)$:Thus, Rauzy induction $\mathcal{R}$ has two components: the combinatorial one (an action on the permutation) and the metric one (an action on the vector). Namely, one can check that
$$
\begin{equation}
T'=(\lambda',\pi')=\begin{cases} ((I_a(\pi))^{-1}\lambda, a(\pi)) & \text{if}\ \lambda_{\alpha_0}<\lambda_{\alpha_1}, \\ ((I_b(\pi))^{-1}\lambda, b(\pi)) & \text{if}\ \lambda_{\alpha_0}>\lambda_{\alpha_1}, \end{cases}
\end{equation}
\tag{6}
$$
where $\alpha_0$ and $\alpha_1$ are the symbols of the rightmost intervals of the preimage and image of the interval exchange transformation $T$, and the matrices $I_a(\pi),I_b(\pi)\in \operatorname{SL}(n,\mathbb Z)$ relating the lengths of intervals of $T'$ and $T$ are defined in the following way:
$$
\begin{equation*}
I_a(\pi)=E+E_{\alpha_0,\alpha_1}\quad\text{and}\quad I_b(\pi)=E+E_{\alpha_1,\alpha_0}
\end{equation*}
\notag
$$
($E$ is the identity matrix, and $E_{\alpha,\beta}$ is the elementary matrix that contains $1$ in the position $(\alpha,\beta)$). The combinatorial part depends on whether $u^\alpha_n$ belongs to the set $\{u^\mathrm{t}_i\}$ or $\{u_i^\mathrm{b}\}$ (this depends on which of the two rightmost intervals is longer, the interval in the original partition or its image). The longer interval, which preserves its position, is called the winner, while the shorter one, which preserves its length but changes its position after the permutation is called the loser. The new permutations look as follows:
$$
\begin{equation*}
\begin{aligned} \, a(\pi)&:=\begin{cases} (\pi_0(\alpha), \pi_1(\alpha)) & \text{ if}\ |\pi_1|(\alpha) \leqslant \pi_1(\alpha_0), \\ (\pi_0(\alpha),\alpha_1) & \text{ if}\ |\pi_1|(\alpha)=\pi_1 (\alpha_0)+1, \\ (\pi_0(\alpha),\pi_1(\alpha)+1) & \text{otherwise}; \end{cases} \\ b(\pi)&:=\begin{cases} (\pi_0(\alpha),\pi_1(\alpha)) & \text{ if}\ \pi_0(\alpha) \leqslant \pi_0( \alpha_1), \\ (\alpha_0,\pi_1(\alpha)) & \text{ if}\ \pi_0(\alpha)=\pi_0 (\alpha_1)+1, \\ (\pi_0(\alpha)+1,\pi_1(\alpha)) & \text{otherwise}. \end{cases} \end{aligned}
\end{equation*}
\notag
$$
A detailed description of Rauzy induction (also known as Rauzy–Veech induction) is provided, for example, in [70]. Rauzy induction can be viewed as an example of the multidimensional fraction algorithm (see § 3). It is easy to see that the combinatorial part of Rauzy induction can be represented by the Rauzy diagram: this is a finite graph whose vertices are the irreducible permutations on $n$ elements and two vertices $\pi$ and $\pi'$ are connected by an edge if and only if one of the permutations can be obtained from the other using the algorithm described above: The metric part (the change of components of the vector of lengths) is described by a linear transformation, and the lengths of intervals of the original IET $T$ can be expressed in terms of the lengths of intervals of the resulting IET $T'$ as a linear combination, where all coefficients are equal to $0$ or to $1$:
$$
\begin{equation*}
\lambda=I_a(\pi)\lambda'\quad\text{or}\quad \lambda=I_b(\pi)\lambda'.
\end{equation*}
\notag
$$
If we add the renormalization condition ($\sum_i \lambda_i=1$), then the cone of lengths $\mathbb{R}_{+}^n$ is replaced by a finite-dimensional simplex and linear transformations are replaced by their projectivizations. At the same time Rauzy induction cannot be defined for IETs in which the lengths satisfy a relation with rational coefficients, and thus the IET does not satisfy Keane’s condition: Rauzy induction requires the comparison of the lengths of intervals with different names (and a choice between two options, $a(\pi)$ and $b(\pi)$, can be made only if the intervals have different lengths). However, in many important topological and dynamical problems one needs to deal with IETs that do not meet the requirement of rational independence of all lengths. For example, one of the most well-known families with this feature is the family of Arnoux–Rauzy IETs (they are described in details in § 8 and [3]). In full generality the question of the ergodic properties of typical representatives of these exceptional families remains open. One possible approach to this problem is based on topological arguments and is described in [30]. It is easy to see that a non-trivial integer linear equality that holds for the lengths can be interpreted as an element of the relative homology group for the suspension surface for the IET under consideration. The linear spaces of restrictions determined by the rational relations holding between the lengths of intervals can be divided into two groups, of rich and poor spaces, in accordance with whether or not they contain a so-called asymptotic cycle. The definition of an asymptotic cycle was introduced in [62]; the role of this notion for IETs was described in [73] and [74]. In [30] it was shown that if a poor restriction space contains a minimal uniquely ergodic IET, then minimality and unique ergodicity are locally stable properties; in [30] we also stated the conjecture that in the case of a rich restriction space minimality is not a stable property. In addition to the applications listed above, Rauzy induction is an efficient instrument to study the geometry of moduli spaces of abelian differentials. For example, in [47] some combinatorial analysis of the connected components of the Rauzy diagram resulted in the classification of connected components of the moduli spaces of abelian differentials. In [31] a full classification of the closures of orbits of $\operatorname{SL}(2,\mathbb{R})$ in the moduli space was obtained. 3. Markovian multidimensional fraction algorithmsMultidimensional continued fractions (MCF) are algorithms designed to approximate a vector of real (irrational) numbers by a rational vector whose denominators are uniformly bounded. All algorithms described here belong to the class of so-called Markovian multidimensional continued fractions. These algorithms were introduced by Lagarias [48] and are determined by two piecewise continuous maps: and
$$
\begin{equation*}
A\colon [0,1]^d \to \operatorname{GL}(d+1,\mathbb Z).
\end{equation*}
\notag
$$
In order to define a Markovian multidimensional continued fraction, first we need to recall the definition of a cocycle in dynamical systems: given an autonomous dynamical systems with phase space $X$ and a group $T$ acting on $X$ and coinciding with $\mathbb Z$ or $\mathbb R$, a cocycle is a map $C\colon X\times T\to \mathbb{R}^{n\times n}$, that satisfies the following two conditions:
The algorithms that we are going to study act as follows: given a vector with real components $\theta=(\theta_1,\dots,\theta_d)$, one defines a sequence of $(d+ 1)\times(d+1)$ matrices, such that the rows of these matrices provide Diophantine approximations to $\theta$. For any $\theta\in [0,1]^d$ we introduce the notation and define the cocycle so that $C^{n}(\theta)=A^{(n)}(\theta)C^{n-1}(\theta)$ (in this case the group $T$ coincides with $\mathbb Z$, and the phase space is simply $[0,1]^d$). We will consider cocycles for which the rows $C_{i}^{(n)}=(c^{n}_{i,1},\dots,c^{n}_{i,d+1})$ provide a simultaneous approximation of the components of $\theta$, namely, the following condition is satisfied: for any $\theta=(\theta_1,\dots,\theta_d)$, as $n\to\infty$, we have
$$
\begin{equation*}
w^{(n)}_i\to\theta,\quad\text{where}\ \ w^{(n)}_i=\biggl(\frac{c^{(n)}_{i,1}}{c^{(n)}_{i,d+1}}\,,\dots, \frac{c^{(n)}_{i,d}}{c^{(n)}_{i,d+1}}\biggr).
\end{equation*}
\notag
$$
This condition is called weak convergence of multidimensional continued fraction); if it is satisfied, then the pair $(f,A)$ is called a Markovian multidimensional continued fraction or a Markovian algorithm. This name is motivated by the fact that the action of the algorithm at some $(n+1)$st step is fully determined by the quantity $\theta^{(n)}=f^{n}(\theta)$ and does not depend on the previous steps. In this paper we study a special class of these algorithms, which is known as linear simplex splitting algorithms. Consider a homogenous cone in $\mathbb{R}^{d+1}$:
$$
\begin{equation*}
\mathbb{R}_{++}^{d+1}=\Bigl\{y\in\mathbb{R}^{d+1}\colon y_i\geqslant0,\ y_{d+1}=\max_{i=1,\dots,d}{y_i}\Bigr\}.
\end{equation*}
\notag
$$
This cone is subdivided into a finite or countable number of homogeneous subcones, and for each subcone one can construct a homogeneous linear map $\widehat f$ by the formula $\widehat f(x)=\widehat A(x)$ for some $\widehat A\in \operatorname{GL}(d+1,\mathbb{Z})$. We set Note that for each ray $[y]= \{\lambda(y_1,\dots,y_{d+1})\colon\lambda>0\}$ the function $\widehat f$ determines a point in the homogeneous space that does not depend on $\lambda$. Therefore, for each such ray it is sufficient to choose one representative (for example, the one with the property $y_{d+1}=1$), while the set of rays can be identified with the space $[0,1]^{d}$. Then we construct the map $f(x)$ from $\widehat f$ : and the couple $(f,A)$ determines a Markovian algorithm (in the sense of the definition above).Thus, linear simplex splitting can be defined as follows: one takes a $d$-dimensional simplex (parameter space) $\Delta=\{x\in\mathbb{R}^{d+1}\colon \|x\|=1\}$ and its partition into a finite or countable number of subsimplices $\Delta_\alpha$. For every $\alpha$ we fix the matrix $A_{\alpha}$ and define $f\colon \Delta_{\alpha}\subset\Delta$ by the following equality: If we want the algorithm to be Markovian, then the partition must be Markovian too, namely, the image of each $\Delta_{\alpha}$ must be the union of a finite or countable number of partition subsimplices.The list of simplex splitting algorithms contains not only the Euclid algorithm or Rauzy induction mentioned above, but also, for example, the Brun, Selmer, and Jacobi–Perron algorithms (see [63]). Their ergodic and spectral properties are actively studied in the theory of Diophantine approximation and in symbolic dynamics, since, on the one hand, as we saw above, these properties are an important source of information about the ergodic properties of the system that has this algorithm as the renormalization and, on other hand, the quality of approximation by the algorithm is determined by its spectral properties. A non-homogeneous (renormalized) simplex splitting algorithm $f(x)$ determines a Markov shift on a finite or countable alphabet. We recall the definition. Definition 5. A topological Markov shift (TMS) with the set of states $\mathcal{S}$ and transition matrix $\mathbb{A}=(t_{a,b})_{\mathcal{S}\times\mathcal{S}}$ is the set With any topological Markov shift one can associate a graph which we call Rauzy induction (as in the case of Teichmüller dynamics). The vertices of this graph are the letters of the alphabet $\mathcal{S}$ given by the set of states of the shift; two vertices $a$ and $b$ are connected by a directed edge if there exist $x\in [a]$ and $y \in[b]$ such that $\sigma(x)=y$. In the case of the simplex splitting algorithm the set of states is exactly the set of indices of the subsimplices $\Delta_\alpha$ that form the partition of the original simplex $\Delta$. The orbit of each point $x$ under the action of $f$ can be coded by an infinite sequence of letters of the alphabet formed by the state set; the action of $f$ corresponds to the application of $\sigma$ to such a sequence. One can construct a Rauzy diagram for this shift $\sigma$, Thus, with each edge $\gamma_1$ of the Rauzy diagram one can associate a matrix $A_{\gamma_1}$ using the definition of the Markov multidimensional continued fraction $(f,A)$; with a path $\gamma$ on the Rauzy diagram we associate the matrix $A_{\gamma}$ that is the product of the matrices associated with each edge in $\gamma$: if $\gamma=\gamma_1\gamma_2\cdots\gamma_n$, where $\gamma_i$ is an edge and $A_{\gamma_i}$ is the associated matrix, then $A_{\gamma}=A_{\gamma_1}A_{\gamma_2}\cdots A_{\gamma_n}$. Definition 6. A path on the Rauzy diagram is called positive if the matrix $A_{\gamma}$ associated with this path contains only strictly positive elements. Definition 7. A path on the Rauzy diagram is called a loop if it starts and ends at the same vertex of the diagram. Fix the matrix $A_{\gamma}$ associated with a path $\gamma$. We consider a subset $\Delta_\gamma$ of $ \Delta$ such that $f(\Delta_\gamma)=\Delta$, where
$$
\begin{equation*}
f(x)=\frac{A_{\gamma}^{-1}x}{\|A_{\gamma}^{-1}x\|}\,.
\end{equation*}
\notag
$$
Then $\Delta_\gamma$ coincides exactly with the set of points whose symbolic description starts with $\gamma$. Moreover, if $A_{\gamma}$ contains only strictly positive elements, then the corresponding subsimplex $\Delta_\gamma$ is compactly embedded in $\Delta$. This enables us to introduce so-called special acceleration of the algorithm. The idea is as follows: for an ergodic algorithm one can consider the first return map to a subsimplex contained compactly in the parameter space, and according to Poincaré’s return theorem almost all orbits will indeed return. For a $\mathcal C^1$-map $T\colon X \to Y$ we denote by $d_x T$ the differential of this map at the point $x \in X$. The following definition can be found in [13]. Definition 8. projective expanding map is a map where $\Delta_{*}$ is a simplex contained compactly in the standard simplex and corresponding to a path $\gamma_{*}$ on the Rauzy diagram, the $\{\Delta^{(l)}\}$ form a finite or countable family of pairwise disjoint simplexes in $\Delta_{*}$ which covers almost all points in $\Delta_{*}$, and $T^{l}=T\colon \Delta^{(l)}\to\Delta_{*}$ is a bijection such that its inverse is a restriction of a projective contraction, so that it is represented by a positive matrix. Remark 2. It is clear from the definition that special acceleration of any ergodic simplex-splitting multidimensional continued fraction is a projective expanding map. A projective expanding map $T$ is strongly expanding (the term is borrowed from [9]) if the following condition holds: there exists $k >1$ such that for the Jacobian of $T$ the inequality $\|d_x T\|_\infty \geqslant k$ is satisfied for all $x \in \Delta_*$. Strongly expanding Markov maps were extensively studied in ergodic theory (see, for example, [1]). Jointly with Fougeron, we proved the following statement, which has appeared to be very useful in many examples (see [36]). Proposition 9 (Fougeron and Skripchenko). Any special acceleration of a simplex splitting algorithm is strongly expanding. It is known that any uniformly expanding map admits a unique absolutely continuous invariant measure that is ergodic and, moreover, mixing. In [35] Fougeron proposed a very efficient criterion that allows one to establish unique ergodicity for a large class of multidimensional continued fraction using their combinatorial description. The key notions in this context are a simplicial system and a quickly escaping simplicial system. The first means that there exists a Rauzy diagram and a Markovian multidimensional continued fraction associated with this diagram, while the second can be seen as a combinatorial description of the bounded distortion property. The bounded distortion property was introduced by Kerckhoff in [45] as a tool used for an alternative proof of the Masur–Veech theorem about the unique ergodicity of almost all IETs. Informally, we mean the following property. In using Rauzy induction we change (increase) the norms of the columns corresponding to losers of the matrix of induction, while the norms of other columns remain unchanged. Thus, if the winners and losers alternate regularly and all letters appear regularly in both roles, then the matrix of Rauzy induction is balanced after a sufficient number o iterations, meaning that the ratio of the norms of any two columns is bounded by a uniform constant. But if the winner or the loser remains the same for a long sequence of iterations of induction, then balancedness can be violated. However, this can only occur for a small subset of the parameter space, and the measure of this subset can be estimated effectively. The bounded distortion property for IETs was deeply studied in [10] by Avila, Gouëzel, and Yoccoz. Their estimates are much sharper than the ones in [45] and are used to show the existence of an exponential tail for the roof function that is used to define the suspension flow (in their case they deal with the Veech flow) on IETs (the definitions of the suspension flow and roof function can be found in § 7). In [12] the same approach was applied to systems of isometries and their renormalization. Fougeron showed that the same ideas can be used in a much broader context of a special class of multidimensional continued fractions. Namely, [35] contains combinatorial conditions that are sufficient for the bounded distortion property for a Markovian multidimensional continued fraction. Classical instruments of ergodic theory enable one in this case to prove ergodicity, while the application of thermodynamical formalism for Markov shifts leads to the construction of invariant ergodic measure for the algorithm under investigation. The spectral properties of Markovian multidimensional continued fractions are much poorer studied in comparison with the ergodic ones. An application of the multiplicative ergodic theorem allows one to define Lyapunov exponents. Then the spectrum of the algorithm is formed by the Lyapunov exponents of the corresponding cocycle. This spectrum is simple if all Lyapunov exponents are different. The spectrum satisfies the Pisot property if one of the Lyapunov exponents is strictly positive while the others are strictly negative. In [36] we proposed a simple strategy to check whether or not a Lyapunov spectrum is simple. The proof is based on ideas described previously in [13] (the Avila–Viana theorem on the simplicity of the spectrum of the Kontsevich–Zorich cocycle) and in [56] (the Galois version of the Avila–Viana theorem, proved by Matheus, Möller, and Yoccoz). The tools that we developed for abstract multidimensional continued fractions are used for particular examples in §§ 4 and 8. 4. Interval exchange transformations with flipsInterval exchange transformations with flips (fIET) are a natural generalization of the notion of IETs. Informally, a fIET is determined by a finite partition $I_{\alpha}$ of a half-open interval $I\subset\mathbb{R}$ and a piecewise continuous map $f$ that takes $\bigcup_{\alpha \in \mathcal{A}} \operatorname{int} (I_{\alpha})$ to $I$ and is organized as follows: on each $I_{\alpha}$ the map $f$ is an isometry, so that the interior parts of the images of the partition $\{I_{\alpha}\}$ do not intersect, and $f$ reverses the orientation on at least one $\{I_{\alpha}\}$. Note that due to the orientation reversal one cannot extend $f$ to a bijection $I \to I$. Remark 3. IETs have a stronger property than minimality: they are forward and backward minimal, that is, every forward orbit and every backward orbit is dense. For a fIET not all orbits are bi-infinite, thus this cannot hold. Nonetheless our proof shows that this stronger property is almost true for fIETs: every orbit infinite forwards is dense, and the same for orbits infinite backwards. A topological motivation to study fIETs comes from the fact that each fIET is the first return map on a transversal to a vector field on a non-orientable surface. More precisely, one can associate each fIET with an oriented, but not co-orientable measured foliation on a non-orientable surface. We proceed with the precise definition. Consider the interval $I=[0,1)$ and a partition of $I$ into subintervals $I_\alpha$, which are labelled by an alphabet $\mathcal{A}$. Denote the lengths of the intervals $I_\alpha$ by $\lambda_\alpha$ and consider the vector $\lambda=(\lambda_{\alpha})_{\alpha \in \mathcal{A}}$ (each component $\lambda_\alpha$ is positive). Then we define a pair of maps called a generalized permutation, where $\pi_0\colon \mathcal{A}\to \{1,\dots,n\}$ is a bijection and $\pi_1\colon \mathcal{A}\to \{-n,\dots,-1,1,\dots,n\}$ is a map such that its absolute value $|\pi_1|$ is a bijection. The pair of maps $\pi_0$, $|\pi_1|$ describes the ordering of the subintervals $I_\alpha$ before and after the map is iterated. The map $\pi_1$ can be viewed as a signed permutation $\theta|\pi_1|$, where $\theta\in\{-1,1\}^{n})$. The signed permutation $\pi_1$ describes additionally the set of flipped intervals, which are marked by $\theta_\alpha=-1$. The set is called the flip set. Definition 10. The fIET given by $(\widehat \pi,\lambda)$, is the following map: The dynamics of fIETs is diametrically opposite to the dynamics of classical IETs: a typical fIET has periodic points (see [57]). Theorem 10 (Nogueira). For a fixed permutation almost all fIETs (with respect to the Lebesgue measure on the parameter space) have finite orbits. Therefore, the set of parameters that give rise to a minimal fIET on $n$ intervals (we denote this set by $\operatorname{MF}(n)$) has zero Lebesgue measure. In [66] we improved this statement in the following way. Theorem 11 (Skripchenko–Troubetzkoy). The Hausdorff dimension of the set $\operatorname{MF}(n)$ satisfies As a corollary of this observation one obtains a lower bound for the Hausdorff dimension of the set of non-uniquely ergodic minimal fIETs (we denote this set by $\operatorname{NUE}(n)$). Theorem 12 (Skripchenko–Troubetzkoy). For each $n > 4$ the Hausdorff dimension of the set of non-uniquely ergodic minimal $n$-fIETs $\operatorname{NUE}(n)$ satisfies The key instrument used in the proof of these statements (as also in the proof of Theorem 10) is the renormalization algorithm called Rauzy induction (just as for IETs). A detailed description of Rauzy induction for fIEts can be found in [40]. As in the case of IETs, one step of induction is the first return map on a subinterval of the support interval; the matrices expressing the lengths of the intervals of the resulting map in terms of the length of the original ones are exactly the same as for IETs; however, the combinatorial part is different since the important property that the resulting permutation is necessarily irreducible if the original one is does not always hold for fIETs. Namely, one can start with an irreducible permutation and obtain a reducible one. In this case induction stops. Thus, for the proof of Theorem 11 we have to estimate the Hausdorff dimension of the sets of parameters such that Rauzy induction never stops for the fIETs determined by these parameters. The strategy of the proof of Theorem 11 has a lot in common with the approach described in § 3 in connection with Fougeron’s criterion. A similar statement is also mentioned in § 7 (see Theorem 24). The proof contains the following steps: 1) first one needs to describe explicitly the Markov shift associated with the renormalization and the corresponding suspension flow (the latter notion is defined in § 7); 2) then one needs to verify the bounded distortion property; 3) at the next step we recall the definition of the roof function from the first step (it is the first return map to a subsimplex compactly contained in the parameter space) and check that the roof function has an exponential tail: a function $f$ defined on a parameter space $\Delta$ has an exponential tail is there exists $\sigma>0$ such that $\displaystyle\int_{\Delta}e^{\sigma f}\,d\operatorname{Leb}<\infty$; 4) one has to check that the Markov shift associated with the renormalization algorithm is fast decaying (this property follows from the existence of an exponential tail of the roof function): a Markov map $T\colon \Delta\to\Delta$, with the associated Markov partition $\Delta^{(l)}$, is called fast decaying if there exist constants $C_1>0$ and $\alpha_1>0$, such that $\sum_{\mu(\Delta^{(l)})\leqslant \varepsilon}\mu(\Delta^{(l)})\leqslant C_1\varepsilon^{\alpha_1}$ for any $1>\varepsilon>0$; 5) an upper bound for the Hausdorff dimension is established using the Avila–Delecroix theorem (see [8]) and follows from the fast decay statement from the previous block (see 4)); 6) in order to obtain a lower bound one needs to construct explicitly a family of minimal fIETs and estimate the Hausdorff dimension of this set. The question about the construction of a measure that is invariant with respect to the renormalization described above and whose support set is $\operatorname{MF}(n)$ has so far been answered only for fIETs on four intervals. The key obstacle for an application of the thermodynamical formalism strategy from [12] (which was subsequently generalized in [35] and is described in § 7) is related to the complicated combinatorial structure of the Rauzy diagram associated with the renormalization procedure. Namely, in the case of fIET it is not necessarily true that for any vertex the numbers of incoming and outcoming edges are equal. Thus, it is much harder to check that the Markov shift is topologically transitive (moreover, numerical experiments reveal that, starting from $n=5$, this property holds only asymptotically). Still, the following theorem holds. Theorem 13. Let $\mu$ be an ergodic measure with support set $\operatorname{MF}(n)$ that is invariant with respect to Rauzy induction. Then almost all (with respect to $\mu$) minimal fIETs are uniquely ergodic. Since we have a renormalization algorithm which can be viewed as a Markov multidimensional continued fraction, this theorem follows basically from Proposition 9. Another way to prove it exploits symbolic dynamics: it is easy to see that fIETs (just as IETs) have a linear complexity and meet the assumptions of Boshernitzan’s criterion (see [16]). Note that the statement of Theorem 13 assumes the existence of a measure $\mu$, which has not been constructed yet for $n\geqslant 5$. There are some examples of non-uniquely ergodic fIETs (see [51] and [66]). It is also known that the number of ergodic invariant measures for a fIET does not exceed the number of intervals of continuity (this follows from [19]). The symbolic approach toward fIETs enables us to refine this estimate: using a theorem in [20], which is based on an estimate for the number of bispecial words of a fixed dynamical system with linear complexity, one can show that the number of ergodic invariant measures does not exceed $[n/2]$, where $n$ is the number of continuity intervals. However, the question of whether this bound is sharp remains open. 5. Linear involutions and quadratic differentials5.1. Linear involutions: the definition and propertiesLinear involutions appear to be the first return maps on transversals to the foliations on Riemann surfaces defined by quadratic differentials (rather than by Abelian differentials as in the case of IETs). Definition 11. Let $\widehat X=X\times \{0,1\}$ be two disjoint copies of an open interval $X$. A linear involution $T$ on $\widehat X$ is a map such that (a) $T'$ is a smooth involution without fixed points defined on $\widehat{X}\setminus \Sigma$, where $\Sigma$ is a finite subset of $\widehat{X}$; (b) if $(x,\varepsilon)$ and $T(x,\varepsilon)$ belong to the same connected component of $\widehat X$, then the derivative $T'$ at $(x,\varepsilon)$ is equal to $-1$, and otherwise it is equal to $1$; (c) $f$ is the involution $(x,\varepsilon)\mapsto(x,1-\varepsilon)$. It is easy to see that if $G$ is an IET on $X$, then $G$ (or, more precisely, two copies of it) can be viewed as a linear involution $T$ on $\widehat X$:
$$
\begin{equation}
\begin{cases} T(x,0)=(G(x),0), \\ T(x,1)=(G^{-1}(x),1), \end{cases} \qquad x\in X.
\end{equation}
\tag{7}
$$
Linear involutions were introduced by Danthony and Nogueira in [21]; in the same paper the renormalization algorithm for linear involutions was described; it was given the name of Rauzy induction or Rauzy–Veech induction as in the case of IETs. It was also shown that if there is a flip (the orientation is reversed on at least one of the continuity intervals), then almost all linear involutions have periodic orbit, while almost all linear involutions without flips are uniquely ergodic. Linear involutions are determined by the set of combinatorial data (that are represented by generalized permutations) and continuous data, namely, the lengths of continuity subintervals. A generalized permutation of type $(l,m)$ is in this case a ‘2-to-1’ map $\pi\colon\{1,\dots, 2d\}\to\mathcal{A}$ (here $d$ is the number of continuity intervals and $\mathcal{A}$ is an alphabet). It is important to mention that the class of linear involutions defined in this way is wider than the set of maps that can be viewed as the first return maps for the foliations defined by some quadratic differentials. A generalized permutation is called irreducible if there exists a linear involutions associated with this generalized permutation which represents an appropriate cross-section of the vertical foliation of some quadratic differential. A generalized permutation is dynamically irreducible if there exists a minimal linear involution associated with this permutation. These definitions were introduced by Boissy and Lanneau in [15]. In the same paper they provided a combinatorial interpretation of irreducibility (it does not coincide with the one for classical IETs) and described the connection between the number of connected components of the Rauzy diagram (for irreducible permutations) and the number of connected components of the moduli space of quadratic differentials. Linear involutions defined by dynamically irreducible permutations were shown to be almost always minimal, and it was proved that the sufficient condition for minimality is a precise analogue of Keane’s condition for IETs. 5.2. Cohomological equations and linear involutionsWe generalize the results by Marmi, Moussa and Yoccoz mentioned above on cohomological equations for IETs to the case of irreducible linear involutions. In order to do this, first we define Roth-type linear involutions and prove a result on the solvability of the cohomological equation for this class and then show that almost all irreducible linear involutions belong to this class. The definition of Roth-type linear involutions is presented below (see § 5.3). Let us focus on functions defined on $\widehat X$. We denote by $\widehat{\operatorname{BV}}{}_{*}^1\bigl(\,\bigsqcup A_\alpha\bigr)$ the set of functions $\widehat\Phi$ in the class $\mathcal C^1$ defined on each continuity interval, with a derivative of bounded variation, and whose integral over the domain vanishes. In [49] the following statements were proved. Theorem 14 (Lanneau, Marmi, and Skripchenko). Let $T$ be a minimal linear involution of Roth type. Then for any $\widehat\Phi\in \widehat{\operatorname{BV}}{}_{*}^1\bigl(\,\bigsqcup A_\alpha\bigr)$ there exists a function $\chi$ that is constant on each interval $A_\alpha$, and a bounded function $\Psi$ on $\widehat X$ such that Theorem 15 (Lanneau, Marmi, and Skripchenko). Roth-type linear involutions form a full-measure set in the space of all irreducible linear involutions. 5.3. Linear involutions of Roth typeFor irreducible linear involutions the definition of the Roth-type property basically coincides with the definition for IETs. In both cases there are three key conditions that must be satisfied, and these conditions are expressed in terms of Rauzy–Veech induction: the growth rate condition, the spectral gap condition, and coherence. The formal definition exploits acceleration of Rauzy induction similar to the acceleration proposed by Zorich in the case of IETs. The idea is the same as in the definition of the fast Euclid algorithm; namely, all the consecutive steps of the algorithm are joined in a single iteration if the winner for these iterations remains the same (in the case of an IET on two intervals this is exactly the Gauss map). This acceleration of Rauzy induction is called Zorich induction. As in the case of the classical Rauzy induction for IETs, an application of one step of Zorich induction to an IET results in a new IET, and the lengths of intervals of the new and old IETs are related by a linear transformation with integer coefficients. The matrix of this transformation is the product of the elementary matrices associated with the steps of the classical Rauzy induction that are included in the step of accelerated induction. At the same time the special acceleration described in § 3 can be seen as acceleration of Zorich induction. Let us consider acceleration such that one step of it comprises all the steps of Rauzy induction at which all intervals but one can win (therefore, the set of winners has cardinality $d-1$). The matrix of the linear transformation associated with this acceleration (the product of the elementary matrices associated with edges of a path on the Rauzy diagram) will be denoted by $Z(k)$. We denote the original linear involution by $T$ and the resulting linear involution by $T^{(k)}$. We also add the following notation: for $k<l$ we set
$$
\begin{equation*}
Q(k,l)=Z(k)\cdots Z(l) \quad \text{and}\quad Q(k)=Q(0,k).
\end{equation*}
\notag
$$
Now we are ready to describe three conditions mentioned above formally: (a) A growth rate condition: for every $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that for every $k\geqslant0$ one has
$$
\begin{equation}
\|Z(k+1)\|\leqslant C_{\varepsilon}\|Q(k)\|^{\varepsilon}.
\end{equation}
\tag{8}
$$
This condition is basically the bounded distortion property for the special acceleration of Rauzy induction. (b) A spectral gap. The second condition is a spectral way to express unique ergodicity for a given linear involution. The formal description of this condition is as follows: we take an irreducible linear involution; for each $k\geqslant 0$ we denote by $\Gamma^{(k)}$ the space of functions that are constant on each continuity interval of the linear involution $T^{(k)}$ that is the result of the application to $T$ of the acceleration of induction described before (a). For $0\leqslant k\leqslant l$ we denote by $S(k,l)$ the linear map from $\Gamma^{(k)}$ to $\Gamma^{(l)}$ whose matrix in the canonical basis is $(Q(k,l))^\top$; we write $S(k)$ instead of $S(0,k)$. As mentioned above, the combinatorial part of a linear involution is given by a generalized permutation: each generalized permutation contains the letters of some alphabet and every letter appears twice in the permutation. We denote the alphabet by $\mathcal{A}$, and its double copy is denoted by $\widehat{\mathcal{A}}$. For every $\phi=(\phi_\alpha)_{\alpha\in\widehat{\mathcal{A}}}\in\Gamma^{(k)}$ we set
$$
\begin{equation*}
I_k(\phi)= \sum_{\alpha\in\widehat{\mathcal{A}}}\lambda_\alpha^{(k)}\phi_\alpha.
\end{equation*}
\notag
$$
Then We denote by $\Gamma^{(k)}_*$ the kernel of the linear form $I_k$. Then the spectral gap condition is as follows: there exist $\theta >0$ and $C>0$ such that for every $k\geqslant 0$ we have
$$
\begin{equation*}
\bigl\|S(k)|_{\Gamma^{(0)}_*}\bigr\|\leqslant C\|S(k)\|^{1-\theta}=C\|Q(k)\|^{1-\theta}.
\end{equation*}
\notag
$$
(c) Coherence. The third condition is also expressed in terms of the operators $S(k,l)\colon \Gamma^{(k)}\to\Gamma^{(l)}$. Let $\Gamma^{(k)}_{\rm s}$ be the linear subspace of the space $\Gamma^{(k)}$ whose elements $v$ satisfy the following condition: there exist $\sigma=\sigma(v)>0$ and $C=C(v)>0$ such that for all $l\geqslant k$ we have We say that $\Gamma^{(k)}_{\rm s}$ is the stable subspace of $\Gamma^{(k)}$. The operator $S(k,l)$ maps $\Gamma^{(k)}_{\rm s}$ to $\Gamma^{(l)}_{\rm s}$. Then one can define the map
$$
\begin{equation*}
S_{(k,l)}\colon\Gamma^{(k)}/\Gamma^{(k)}_{\rm s}\to \Gamma^{(l)}/\Gamma^{(l)}_{\rm s}.
\end{equation*}
\notag
$$
Our condition (c) means that the norm of the map inverse to $S_{(k,l)}$ is not too large: for every $\varepsilon >0$ there exists $C_\varepsilon >0$ such that for all $l\geqslant k$ we have
$$
\begin{equation*}
\| [S_{(k,l)}]^{-1}\| \leqslant C_\varepsilon\| Q(l)\|^\varepsilon
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\bigl\|S(k,l)|_{\Gamma^{(k)}_{\rm s}}\bigr\| \leqslant C_\varepsilon\|Q(l)\|^\varepsilon.
\end{equation*}
\notag
$$
Definition 12. A linear involution is of Roth type if both $T$ and $T^{-1}$ satisfy condtions (a)–(c). The statement of Theorem 15 follows from the detailed study of the spectral properties of linear involutions (the multiplicative ergodic theorem, the simplicity of the Lyapunov spectrum, the bounded distortion property). 6. Interval translation mappings6.1. The definition and known resultsDefinition 13. An interval translation map (ITM) is a piecewise translation map $R$ defined on an open interval $I \subset \mathbb{R}$ with values in $I$. We call $R$ an $n$-interval translation map, if $I$ has $n$ maximal open subintervals such that the restriction of $R$ to each of them is a translation. Endpoints of these intervals are called singularities of the map, and endpoints of the images of intervals are the images of singularities.. Interval translation maps are determined by a finite number of parameters: for each interval it is enough to fix the length and the translation. Interval translation maps were introduced in [17] as generalizations of IETs. The key difference between IETs and interval translation maps is the fact that, unlike IETs, interval translation maps are not necessarily surjective, the images of continuity intervals are not supposed to form a partition and are allowed to overlap. In the same paper a classification of interval translation maps was introduced: an interval translation map can be either of finite type or of infinite type, depending on the properties of the attractor of this map. Definition 14. Given an interval translation map $R$, one can define the sequence of sets If there exists $n\in\mathbb{N}$ such that the sequence $\Omega_n$ stabilizes: then we say that the map $R$ is of finite type. If such $n$ does not exist, then we say that $R$ is of infinite type. Remark 4. If $R$ is of infinite type, then $\overline\Omega$ — the closure of $\Omega=\lim_{n\to\infty}\Omega_n$ — is a Cantor set. Remark 5. The first example of an interval translation map of infinite type was constructed in [17]. The dynamics of interval translation maps of infinite type is remarkably different from the one for IETs. Conjecture 1 (Boshernitzan and Kornfeld). For $n\geqslant3$ the set of parameters that give rise to $n$-interval translation maps of infinite type has zero Lebesgue measure. A few answers on the questions posed by Boshernitzan and Kornfeld (including the conjecture above) were obtained only for a few special class of interval translation maps. We outline the known results in the next subsections. 6.2. Bruin–Troubetzkoy interval translation mapsThe first family of interval translation maps for which Conjecture 1 is proved was introduced by Bruin and Troubetzkoy in [18]. The family is described as follows: let
$$
\begin{equation*}
U:=\{(\alpha,\beta)\colon 0\leqslant\beta\leqslant\alpha\leqslant1\}\quad\text{and}\quad L:=\{(\alpha,\beta)\colon 0\leqslant\alpha\leqslant\beta+1\leqslant1\};
\end{equation*}
\notag
$$
then for every interior point $(\alpha,\beta)\in U$ we define
$$
\begin{equation}
R_{\alpha,\beta}(x)=\begin{cases} x+\alpha & \text{for }\ x\in[0,1-\alpha], \\ x+\beta & \text{for }\ x\in[1-\alpha,1-\beta], \\ x+\beta-1 & \text{for }\ x\in[1-\beta,1). \end{cases}
\end{equation}
\tag{9}
$$
In the case of $L$ everything is defined symmetrically. One can check that $R(x)=R_{\alpha,\beta}(x)\colon [0,1)\to[0,1)$ is an interval translation map. If we glue two points 0 and 1, then we obtain a so-called double rotation map, which is described in the next subsection, § 6.3. The Boshernitzan–Kornfeld example of an infinite-type interval translation map also belongs to the Bruin–Troubetzkoy family. We say that the couple $(\alpha,\beta)$ determines the map $R_{\alpha,\beta}$. In [18] Bruin and Troubetzkoy proved Conjecture 1 and also showed that the set $B$ of parameters that give rise to uniquely ergodic interval translation maps is a $G_\delta$-subset of the set $A$ that is the parameter set for Bruin–Troubetzkoy interval translation maps of infinite type. We generalize their result in the following way. Theorem 16. Let $A$ be the set of parameters $(\alpha,\beta)$ such that , the couple $(\alpha,\beta)$ determines an interval translation map $R_{\alpha,\beta}$ of infinite type, and let $B\subset A$ be the set of pairs $(\alpha,\beta)$ such that the corresponding interval translation map $R_{\alpha,\beta}$ is a uniquely ergodic interval translation map of infinite type. Then there exists a probability measure $\mu$ on $(\alpha,\beta)$ whose support set is $A$, and $B$ is a subset of full measure with respect to $\mu$. Bruin and Troubetzkoy proved their theorem using a renormalization procedure, which they called the Gauss map; this renormalization resulted in a symbolic description of interval translation maps in terms of substitutions. The strategy of the proof of Theorem 16 also exploits the renormalization scheme, but our algorithm is remarkably different: it can be viewed as renormalization for the system of isometries associated with an interval translation map (see § 7) and belongs to the class of Markovian multidimensional continued fraction. As mentioned above, Fougeron proved the unique ergodicity criterium for a wide class of these algorithms. So the proof of Theorem 16 is based on an application of this criterium. Proof of Theorem 16. We start with the description of the renormalization procedure (we denote it by $\mathcal R$). First, we change the notation in order to make the description of our family more homogeneous. Namely, we introduce new parameters: if $\alpha>\beta$, then we have
Thus $a+b+c=1$, and
$$
\begin{equation*}
\begin{aligned} \, R([0,a))&=[1-a,1), \\ R([a,a+b))&=[1-b,1) \end{aligned}
\end{equation*}
\notag
$$
and
By $a$-intervals we mean pairs whose length is equal to $a$; in a similar way we define $b$-intervals and $c$-intervals. We assume that $a$, $b$, and $c$ are rationally independent and distinguish three cases. Case 1: $a>b+c$. We consider the first return map on the subinterval $[b+c,1)$. It is an interval translation map of the same family with the following lengths of intervals: One can easily see that in this case the matrix of induction coincides with the one defined by Arnoux and Rauzy (it is also mentioned as the renormalization algorithm for a special system of isometries in § 7.2).The interval translation we have used can be interpreted in the following way: the pair of intervals of length $b$ does not face any change (in its position or in its length), while one of the $c$-intervals moves (but preserves its length), and both $a$-intervals are cut (but preserve their position). Case 2: $c<a<b+c$. One can check that in this case the interval translation map can be reduced to an interval translation map on two intervals and thus is of finite type. Case 3: $a<c$. We consider the first return map to the subinterval $[a,1)$. It results in the following interval translation map:
$$
\begin{equation*}
\begin{aligned} \, R([0,b))&=[1-b,1); \\ R([b,a+b))&=[1-a,1); \\ R([a+b,1-a))&=[0,c-a). \end{aligned}
\end{equation*}
\notag
$$
In this case the geometric interpretation is as follows: once again nothing changes for $b$-intervals, but both $c$- intervals are shortened, and one $a$-interval is moved. Thus, we obtain a partition of the original parameter space $\Delta$ into three subsimplices (we denote them by $\Delta_1$, $\Delta_2$, and $\Delta_3$, respectively). The projectivization of the map $R$ induces on each $\Delta_i$ a transformation that maps $\Delta_i$ onto the whole of $\Delta$ so that vertices are mapped to vertices and edges are the images of edges. The Rauzy diagram of the corresponding algorithm contains three vertices: two of them correspond to Case 1 and Case 3, respectively, while the last vertex corresponds to the second case, when induction stops (we call this simplex the hole). The following statement is obvious. Lemma 17. The set $A$ of parameters such that for $(\alpha,\beta)\in A$ $R_{\alpha,\beta}$ is an infinite- type Bruin–Troubetzoy interval translation map coincides with the set of parameters that do not enter the hole during the induction procedure The next step is to establish an analogue of the Marmi–Moussa–Yoccoz statement on the positivity of the matrix of induction after a sufficient number of iterations (see [53], § 1.2). We use the terminology introduced for Rauzy induction for IETs: the longest interval, which is cut, is called the winner, and the shorter interval, which is moved, is called the loser. Therefore, in Case 1 an $a$-interval is a winner and a $c$-interval is a loser, while in Case 3 it is the opposite. Sometimes we simplify our wording and say that in Case 1 letter $a$ wins and letter $c$ loses (and similarly in other cases). The previous lemma implies the following statement. Lemma 18. A Bruin–Troubetzkoy interval translation map is of infinite type if and only if each letter wins and loses an infinite number of times. Now keeping in mind all these observations we introduce the following notation: let $\lambda=(a,b,c)$ be the vector of original lengths and $\lambda'=(a',b',c')$ be the vector produced after one step of induction; then where
$$
\begin{equation*}
\mathcal{R}=R_1^{k_1}R_2^{k_2}R_1^{k_3}\cdots\quad\text{for some}\quad k_1,k_2,\ldots \in\mathbb{N}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
R_1=\begin{pmatrix} 1 & 1 & 1\\ 0 & 1& 0 \\ 0 & 0& 1 \end{pmatrix},\qquad R_2=\begin{pmatrix} 0 & 1 & 0\\ 1 & 0& 0 \\ 0 & 1& 1 \end{pmatrix}.
\end{equation*}
\notag
$$
One can check that
$$
\begin{equation*}
R_1^{k_1}=\begin{pmatrix} 1 & k_1 & k_1\\ 0 & 1& 0 \\ 0 & 0& 1 \end{pmatrix}
\end{equation*}
\notag
$$
and if $k_2=2l_2+1$, then
$$
\begin{equation*}
R_2^{k_2}= \begin{pmatrix} 0 & 1 & 0\\ 1 & 0& 0 \\ l_2 & l_2+1& 1 \end{pmatrix},
\end{equation*}
\notag
$$
thus $R_{0}=R_1^{k_1}R_2^{k_2}R_1$ is a matrix with strictly positive entries:
$$
\begin{equation*}
R_0=\begin{pmatrix} k_1+k_{1}l_2 & 2k_1(l_2+1) & 2k_1+k_{1}l_2\\ 1 & 1& 1 \\ l_2 & 2l_2+1&l_2+1 \end{pmatrix}.
\end{equation*}
\notag
$$
Now it remains to mention that Lemma 18 implies that if we wait sufficiently long, then we necessarily obtain $k_{2i-1}>0$ and an even $k_{2i}$ (otherwise $b$-intervals or $a$-intervals would not win infinitely often). Therefore, after a sufficient number of steps the induction matrix looks like $R_0$ multiplied from the left and/or from the right by matrices with nonnegative coefficients. Therefore, the following statement holds The definitions of special acceleration and strongly expanding maps were presented in § 3. The construction of the invariant measure is based on thermodynamical formalism for countable Markov shifts; it was described in [35]: one can easily see that our simplicial system is quickly escaping (in the sense of the definition in § 3.1 in [35]) and therefore meets the assumptions of Corollaries 4.4 and 4.13 in [35]. The same corollaries imply the unique ergodicity statement. This completes the proof of Theorem 16. 6.3. Double rotationsDouble rotations were introduced by Suzuki, Ito, and Aihara in [68]; in the same paper the renormalization for this class was described; this algorithm is called SIA induction. The proof of Conjecture 1 is due to Bruin and Clack; it is based on SIA induction. Moreover, they proved a theorem that is somehow similar to our Theorem 13: without presenting the invariant measure (with respect to SIA induction) explicitly (and even without checking that such a measure exist) they showed that almost all (with respect to this measure if it exists) double rotations of infinite type are uniquely ergodic; the proof is based on symbolic dynamics. In [6] we obtained the following improvement of this result. Theorem 20 (Artigiani, Hubert, Fougeron, and Skripchenko). The set $A_{D}$ of parameters $(\alpha,\beta,c)$ such that $P_{\alpha,\beta,c}$ is a double rotation of infinite type has Hausdorff dimension strictly less than 3. As a byproduct of the proof of this statement, in [6] we constructed a probability measure $\mu'$ whose support set is $A_{D}$. This measure is invariant with respect to the renormalization that we introduced there (it is different from SIA induction) and induces a maximum entropy measure for the suspension flow associated with an interval translation map. Theorem 21 (Artigiani, Hubert, Fougeron, and Skripchenko). Almost all double rotations of infinite type are uniquely ergodic. By ‘almost all’ in Theorem 21 we mean almost all with respect to the measure $\mu'$. Our key instrument is the renormalization that we mentioned above (as we already said, it does not coincide with the one in § 3). Just as Rauzy induction, this algorithm is a Markovian multidimensional continued fraction. We describe the simplicial system associated with this algorithm and check that it satisfies the conditions in Corollaries 4.4 and 4.13 in [35]. The Buzzi–Hubert theorem (see [19]) implies that double rotations cannot admit more than two ergodic invariant measures. Some examples of non-uniquely ergodic examples were constructed in [18], but all these examples are not generic since they belong to the Bruin–Troubetzkoy family of interval translation maps. Remark 6. Volk showed in [71] that interval translation maps on three intervals can be reduced to double rotations; thus, now all challenging open questions address interval translation maps on at least four intervals. Remark 7. It is easy to extend the proof of the statement on the absence of strong mixing from [42] to interval translation maps. However, the question about weak mixing for interval translation maps is much more subtle and thus remains open. It is known that interval translation maps appear as the first return maps on transversals to the billiard flow in so-called billiards with spy mirrors. Сonsider a billiard table consisting of a (not necessarily rational) polygon $Q\subset\mathbb{R}^2$ with several one-sided mirrors inside; that is, with straight line segments connection pairs of points in $Q$, each of which has two sides, a transparent side and a reflecting side; if the particle arrives at the transparent side of a mirror, then it passes through it unperturbed, while if it arrives at the reflecting side of a mirror or at the boundary of $Q$, then it is reflected in accordance with the usual law of geometric optics. In particular, a double rotation can be defined as the first return map on a transversal in the square table $[0,1]\times[0,1]$ with a vertical mirror (see [65]). Theorem 21 means that for almost all $x$, $y$, and $\theta$ billiard trajectories in the direction $\theta$ are uniquely ergodic. 7. Systems of isometries7.1. Definitions and conjecturesThe notion of systems of isometries is a kind of generalization of IETs and interval translation maps. It was introduced by Gaboriau, Paulin, and Levitt in [38] as a tool to prove Rips’s theorem about the action of finitely generated groups on $\mathbb R$-trees and rediscovered by Dynnikov in [26] under the name of ‘interval identification systems’ in connection with the Novikov problem of plane sections of triply periodic surfaces (details are presented in § 8). Definition 16. A system of isometries $S$ consists of the following data: We focus mainly on the case when $D$ is just one interval, and in this case we call $D$ the support interval. One can define the orbit of a system of isometries as follows. Each system of isometries $S$ can be associated with a graph $\Gamma(S)$ whose vertices are all the points of the interval $D$, and an edge connects two points if and only if these points are associated through one of the isometries $\phi_i$ ($i=1,\dots,n$) (in [37] this graph was called the Cayley graph of the system of isometries). We denote the connected component of this graph that contains the point $x$ by $\Gamma_x(S)$. The system of isometries $S$ determines an equivalence relation on $D$: points that belong to the same connected components of $\Gamma(S)$ are equivalent. The set of points that are equivalent to a fixed point $x$ form the orbit of $x$. In [26] the following result was proved. Proposition 22. If the sum of the lengths $|A_i|$ (or $|B_i|$) of bases is less than the length of the multi-intverval $D$, then there exists a subset $U\subset D$ of positive Lebesgue measure such that for all $x\in U$ the graph $\Gamma_{x}(S)$ us finite. If the sum of the lengths $|A_i|$ of bases exceeds the length of the support multi-interval $D$, then for all $x$ in some positive-measure subset $U\subset D$ the graph $\Gamma_{x}(S)$ contains cycles. The same was proved in [38] in terms of independent generating isometries (generators). We say that generators $\phi_j$ are independent if the following condition is satisfied: assume that no non-trivial reduced word in $\phi_i^{\pm 1}$ with non-empty domain is a restriction of the identity. It was shown in [38] that if the generators $\phi_j$ are independent, then the system of isometries is minimal only if it is balanced (the definition of balancedness is presented below). It is important to mention that any system of isometries can be replaced by an equivalent one (in the sense of the behaviour of orbits) whose generators are independent (see the precise statements and proofs in [37]). Definition 17. A balanced system of isometries is a system such that the sum of lengths of the intervals in each of the collections $A_i$ and $B_i$ ($i=1,\dots,n$) is equal to the length of the support (multi-)interval. IETs can be viewed as a special subclass of systems of isometries: it comprises the systems for which every point of support interval is covered by exactly two intervals; these systems of isometries are called systems of surface type. In this case, typically, an orbit is everywhere dense and, topologically, an orbit is a straight line. However, not all orbits (even for minimal systems of isometries) are geometrically the same.The most interesting case (from the dynamical point of view) is when systems of isometries are minimal and all orbits are infinite trees. This case is said to be thin or exotic; it was discovered by Levitt in [50]. The most important open question related to systems of isometries is as follows. Conjecture 2. The set of parameters that define minimal systems of isometries whose orbits are infinite trees is a zero-measure subset of the set of parameters that give rise to balanced systems of isometries of fixed dimension, and the Hausdorff dimension of this set is strictly smaller than $2n-1$, where $n$ is the number of intervals in the systems of isometries. Right now this conjecture is only proved for one class of systems of isometries; we describe this class below. 7.2. Special systems of isometries of order 3 Definition 18. A system of isometries $S$ is called special if the following conditions are satisfied: We denote by $\Delta$ the standard 2-dimensional simplex and introduce the following barycentric coordinates $\lambda_1$, $\lambda_2$, and $\lambda_3$:
$$
\begin{equation*}
\Delta=\{(\lambda_1,\lambda_2,\lambda_3)\in \mathbb{R}^3_{\geqslant 0}\colon \lambda_1+\lambda_2+\lambda_3=1\}.
\end{equation*}
\notag
$$
Consider the linear maps $\mathbb R^3_{\geqslant0}\to\mathbb R^3_{\geqslant0}$ given by the matrices
$$
\begin{equation}
M_1=\begin{pmatrix}1&1&1\\0&1&0\\0&0&1\end{pmatrix},\quad M_2=\begin{pmatrix}1&0&0\\1&1&1\\0&0&1\end{pmatrix},\quad\text{and}\quad M_3=\begin{pmatrix}1&0&0\\0&1&0\\1&1&1\end{pmatrix},
\end{equation}
\tag{11}
$$
and let $f_1$, $f_2$, and $f_3$ be the corresponding projective maps of $\Delta$ to itself:
$$
\begin{equation*}
\begin{aligned} \, f_1(\lambda_1,\lambda_2,\lambda_3)&=\biggl(\frac{1}{1+\lambda_2+\lambda_3}\,, \frac{\lambda_2}{1+\lambda_2+\lambda_3}\,, \frac{\lambda_3}{1+\lambda_2+\lambda_3}\biggr), \\ f_2(\lambda_1,\lambda_2,\lambda_3)&= \biggl(\frac{\lambda_1}{1+\lambda_1+\lambda_3}\,, \frac{1}{1+\lambda_1+\lambda_3}\,, \frac{\lambda_3}{1+\lambda_1+\lambda_3}\biggr), \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
f_3(\lambda_1,\lambda_2,\lambda_3)= \biggl(\frac{\lambda_1}{1+\lambda_1+\lambda_2}\,, \frac{\lambda_2}{1+\lambda_1+\lambda_2}\,, \frac{1}{1+\lambda_1+\lambda_2}\biggr).
\end{equation*}
\notag
$$
The images $f_1(\Delta)$, $f_2(\Delta)$, and $f_3(\Delta)$ are the three smaller triangles, cut off from $\Delta$ by the midlines. Given a word $i_1i_2\ldots i_k$ in the alphabet $\{1,2,3\}$, we denote by $f_{i_1i_2\ldots i_k}$ the composition $f_{i_1}\circ f_{i_2}\circ\cdots\circ f_{i_k}$. In particular, $f_\varnothing=\mathrm{id}$. Definition 19. The Rauzy gasket is the maximal subset $\mathcal{R}_\mathrm{g}$ of the parameter space $\Delta$ satisfying the relation
This parameter space was defined several times in different contexts: in symbolic dynamics (as the set of parameters that give rise to Arnoux–Rauzy IETs [3]), in geometric group theory (in connection with pseudogroups of rotations [50]), and in low-dimensional topology (as the set of parameters that give rise to chaotic regimes in the case of centrally symmetric surface of genus three carrying a foliation with two double saddles [22]). Each of the papers mentioned ([4], [50], and [22]) contains the proof of the following statement. Theorem 23 (Arnoux–Starosta, Levitt, and Dynnikov–De Leo). The Rauzy gasket has zero Lebesgue measure. In [11] we obtained a stronger version of this result. Theorem 24 (Avila, Hubert, and Skripchenko). The Hausdorff dimension of the Rauzy gasket is strictly less than 2. Remark 8. Subsequently, our result has been updated several times: first, by Fougeron [35] and then by Policott and Sewell [59], whose upper bound of 1.74 is the sharpest so far. This estimate matches well with the results of numerical experiment in [22]. Remark 9. The best lower bound of 1.19 is due to Matheus and Gutierréz- Romo [39]. The proof of Theorem 24 is based on the same ideas that where used in the proof of Theorem 11. The key step is the description of renormalization: starting from the original system of isometries, we construct an equivalent one but with a smaller support interval, and then provide a linear map that allows us to express the lengths of intervals in the original system in terms of the lengths of the new ones. In the case of a special system of isometries this linear map coincides with a famous multidimensional fraction algorithm, the Arnoux–Rauzy multidimensional continued fraction; this algorithm was originally defined as the first return map to a subinterval for Arnoux–Rauzy IETs; this family of IETs is defined in the next section (see Definition 22). One step of renormalization for systems of isometries (an analogue of Rauzy induction for IETs) consists of two moves, called translation (on the right or on the left) and reduction (on the right or on the left). This map was introduced in [26] (see Fig. 1). This renormalization results in a sequence of systems of isometries, and each system in this sequence is equivalent to the original one (from the point of view of the behaviour of orbits), and the number of intervals remains the same. It is important to mention that translation affects only the combinatorial part (the order of critical points), while reduction changes also the lengths of some intervals. In the case of a special system of isometries the matrix of the linear map that expresses the lengths of the original system in terms of the lengths of the renormalized one contains only non-negative entries, and this linear map is conjugate to the famous Markovian multidimensional continued fraction named the Fully subtractive algorithm. Renormalization is defined only if the length of the longest pair of interval exceeds one half of the length of the support interval; if this condition does not hold, then the support interval contains some points that are not covered by any interval from the system of isometries; thus the system is obviously non-minimal. The set of points such that renormalization never stops coincides with the Rauzy gasket. We also apply renormalization to construct an ergodic measure whose support set is the Rauzy gasket. This measure is invariant with respect to renormalization. More precisely, in [12] we proved the following statement. Theorem 25 (Avila, Hubert, and Skripchenko). There exists a measure of maximum entropy $\nu$ for the suspension flow associated with the renormalization algorithm, and this measure is unique. Before we proceed with a sketch of the proof, we recall the key notions. We start with the roof function: it is the first return map to the subsimplex $\Delta$ defined by a positive path $\gamma$ on the Rauzy diagram:
$$
\begin{equation*}
r(\lambda,\pi)=-\log\|(B^\top_{\gamma})^{-1}\lambda\|.
\end{equation*}
\notag
$$
Here $B_{\gamma}$ is the matrix of the cocycle related to the matrix of induction $A_{\gamma}$ corresponding to the path $\gamma$ on the Rauzy diagram ($B_{\gamma}=A^{\top}_{\gamma}$), $\pi$ is the starting vertex of $\gamma$, and $\lambda$ is the vector of lengths of the original system of isometries. Now we define the suspension flow determined by the shift $\sigma$ with roof function $r$. This flow extends the interval obtained after the application of the renormalization to 1. The flow $\Phi(\sigma,r)$ is defined on the space
$$
\begin{equation*}
Y=\bigl\{(\lambda,\pi,t)\in \sigma \times \mathbb R\colon 0 \leqslant t \leqslant r(\lambda,\pi)\bigr\}
\end{equation*}
\notag
$$
and identifies points $(\lambda,\pi,r(\lambda,\pi))$ with the points $(\sigma(\lambda,\pi),0)$. It acts as follows: for all $t+s\in [0,r(\lambda,\pi)]$. In the case of IETs a suspension flow with the same construction is known as the Veech flow, and it is a discrete version of the Teichmüller flow (the geodesic flow in the moduli space). In our case the geometric sense of the suspension flow is not known yet. The construction of the measure $\nu$ is based on thermodynamical formalism for countable Markov shifts developed by Sarig in [60]. The measure we construct is called the Gibbs measure (in the sense of Bowen). Definition 20. A Gibbs measure (in the sense of Bowen) for a weakly continuous function $\phi\colon \Sigma\to\mathbb R$ is a measure $\mu$ invariant with respect to the topological Markov shift $\sigma$ such that there exist constants $M>1$ and $P\in\mathbb{R}$ with the following property: for every cylinder $[a_0,\dots,a_{n-1}]$ and every $n\in\mathbb{R}$ the inequality holds for $x\in[a_0,\dots,a_{n-1}]$. The function $\phi$ in the above definition is called the potential. In [60] it was shown that a Gibbs measure exists under certain sufficient conditions. The proof of Theorem 25 contains several steps: 1) first, one needs to check that the renormalization algorithm generates a countable Markov partition and to define the Markov shift associated with the suspension flow; 2) the next step is to check that in our case the Markov shift satisfies several combinatorial requirements; namely, in order to apply Sarig’s theorem one needs to verify that the shift is topologically transitive and has the BIP (big images and preimages) property; 3) the next part is dedicated to the roof function: it should be separated from zero and be weakly Hölder continuous (and therefore it has summable variations), and we prove that it has exponential tails (in the sense of the definition in § 4); 4) we consider the family of potentials $\phi_{\kappa}=-\kappa r$ obtained by multiplication of the roof function $r$ by some constant $\kappa>0$; using the properties of roof functions checked before one sees that the transfer operator is locally compact when $\kappa>3$; 5) it follows from Theorem 4.9 in [60] that a Gibbs measure exists for every $\phi_{\kappa}$ for any $\kappa>3$; 6) among the Gibbs measures with potential $\phi_{\kappa}$ there exists one measure of maximum entropy (uniqueness and existence of the measure of maximum entropy was shown in [60]; all definitions were presented in the same paper). Then we apply the Abramov–Rokhlin formula [2] in order to obtain the following result. Theorem 26 (Avila–Hubert–Skripchenko). There exists a probability measure $\mu_{\mathcal{R}}$, whose support set is the Rauzy gasket, that is invariant with respect to the renormalization algorithm that determines the Rauzy gasket. It is important that due to the general properties of Gibbs measures $\mu_{\mathcal{R}}$ is necessarily ergodic and, moreover, the cocycle associated with the renormalization algorithm is log-integrable. Therefore, one can apply the multiplicative ergodic theorem and find the Lyapunov exponents. The ideas of the proof of Theorem 26 inspired Fougeron, who generalized it and extended the approach we had proposed to the large class of Markovian multidimensional continued fractions. There exists also an alternative renormalization algorithm for systems of isometries; it is called the Rips machine and was defined in geometric group theory. The Rips machine comprises five different moves, which can be applied to a system of isometries directly or to the band complex, associated with it. Given a system of isometries, one can construct the band complex as follows: we glue rectangular bands to the support intervals; the width of each band is equal to the length of the corresponding pair of intervals; each band carries a vertical foliation (see Fig. 2). An application of the Rips machine to the system of isometries results in a new system of isometries with an equivalent behaviour of orbits but the number of intervals may not be preserved. Among the five moves of the Rips machine we focus on one, named a collapse of a free subarc. A subinterval of the multi-interval is called a free subarc if it is covered by the only base of the system of isometries. If such a free subarc exists, then one can collapse it, namely, construct a new system of isometries such that its support multi-interval does not contain this subarc. Therefore, the base of the original system (we denote it by $A_i$) subdivides into several subintervals; the same division is applied to the corresponding base $B_i$. (A formal definition can be found in [38] and [14].) Systems of isometries of thin type can be viewed as ones for which a free subarc always exists (see [14], Proposition 8.13). Using the Rips machine Gaboriau showed in [37] that the orbits of systems of isometries of thin type are quasi-isometric to infinite trees with a finite number of topological ends, and almost all of them have one or two topological ends (the definition of a topological end can be found in [67]). In [28] we proved several updates of Gaboriau’s result. To state them, we need the following definition. Definition 21. A system of isometries is self-similar if an application of the renormalization algorithm results in a scaled-down version of the same system. It was shown in [28] that self-similarity does not depend on the choice of the renomalization algorithm (systems that are self-similar with respect to Rauzy induction are also self-similar with respect to the Rips machine and vice versa). In [28] the following theorem was proved. Theorem 27 (Dynnikov and Skripchenko). Almost all orbits of self-similar systems of isometries of thin type have exactly one topological end. Previously, a similar result was established in [64] for some particular examples of systems of isometries. Theorem 28 (Dynnikov and Skripchenko). There exist thin systems of isometries such that almost orbits have two topological ends. The proofs of Theorems 27 and 28 are based on the study of ergodic properties of the Rips machine as the renormalization algorithm. 8. Applications to the Novikov problem of plane sections of triply periodic surfacesIn 1982 Novikov (see [58]) posed the following problem in connection with conductivity theory in monocrystals. In the 3-torus $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ one fixes a level surface $M$ of some smooth function. Let $\pi\colon \mathbb{R}^3\to \mathbb{Z}^3$ be the standard projection; thus, $\widehat M=\pi^{-1}(M)$ is a 3-triply periodic surface. We consider the plane sections of $\widehat M$ by a family of parallel planes that are orthogonal to a vector $H=(H_1,H_2,H_3)$; what can one say about the asymptotic behaviour of open connected components of these sections (we use the term $H$-sections for them)? This problem can be reformulated in the language of measured foliations: consider the foliation $\mathcal{F}$, defined on $M$ by the restriction of the 1-form with constant coefficients $H_1\,dx^1+H_2\,dx^2+H_3\,dx^3$. Leaves of this foliation are the images of sections under consideration under the projection $\pi$. Since we are dealing with a measured foliation, the first return map on a transversal is given by an IET but since this surface lies on a torus, this IET is not generic from Keane’s point of view (we have some rational relations between the lengths) and thus the standard renormalization cannot be applied. First observations about $H$-sections were made by Zorich and Dynnikov, who used topological arguments (see [23]–[25] and [72]). In particular, they showed that the following substantially distinct types of the behaviour of $H$-sections and structures of the corresponding foliation can occur (we focus on the generic case $\dim_{\mathbb{Q}}(H_1,H_2,H_3)= 3$):
The first example of a chaotic regime was constructed by Dynnikov in [25] (previously, Tsarev had discovered but had not published a kind of an intermediate example that is not integrable but not yet chaotic, in which the components of $H$ were not completely incommensurable). Currently, all challenging open questions are related to the chaotic case. It is known that chaotic regimes are rare in the following sense ([25]). Theorem 29 (Dynnikov). In the space of pairs $(M,H)$, where $M$ is a null- homologous surface in $\mathbb{T}^3$ and $H$ is a covector in $\mathbb{R}^3$, all pairs giving rise to a chaotic foliation $\mathcal{F}$ are contained in a subset $\mathcal{R}$ of codimension 1 and, moreover, form a nowhere dense subset of it. In [52] Novikov formulated two conjectures. The first conjecture claims that for a fixed generic surface homologous to zero the Hausdorff dimension of the set of covectors $H$ that give rise to chaotic sections is strictly less than 1. The second conjecture claims that for a fixed function $f$ whose level surface is a 1-parameter family of surfaces $M_c$ homologous to zero, the Hausdorff dimension of the directions $H$ such that the sections of $M_c$ by planes orthogonal to $H$ are chaotic, is strictly less than 2. In [26] Dynnikov showed that in the case of genus 3 chaotic regimes can be modelled by systems of isometries of order 3 of thin type. Namely, he presented an explicit description of how to build a surface and define a covector in terms of the parameters of a system of isometries of thin type so that the following condition is satisfied: the plane sections of the band complex associated with the thin system of isometries by planes orthogonal to the vector chosen is a deformation retract of the sections of the filled handlebody (one of the two into which the surface cuts the 3-torus) by the same family of planes. In this procedure special systems of isometries of order 3 correspond to chaotic regimes on a symmetric surface with two double saddles. It was shown that for a certain transversal the first return map of the foliation is described by the interesting family of IETs that had previously been introduced by Arnoux and Rauzy in symbolic dynamics. Definition 22. For $\lambda\in\Delta$ define the Arnoux–Rauzy interval exchange transformation $T_\lambda^{\mathrm{AR}}$ as the composition where $\Phi$ is the circle translation by $1/2$ (translation by half the circle), so that $\Phi(x)=x+1/2$, and $T_\lambda'$ is an IET on $[0,1)$ defined by the permutation The first known representative of this class of IET was the Arnoux–Yoccoz interval exchange transformation defined in [5] in terms of a foliation on $\mathbb{R}P^2$. Using Veech’s suspension construction (also known as the zippered rectangle model), one can define a surface of genus 3 such that the Arnoux–Rauzy IETs appear as the first return map on a transversal to the directional flow on this surface. A detailed study of the connection between chaotic regimes in the Novikov problem and Arnoux–Rauzy IETs can be found in [27]. It is important to mention that Theorem 24 implies the proof of Novikov’s conjecture (namely, the second statement) in the case of the Arnoux–Rauzy surface. Using the renormallization methods described above we obtain the following result (see [27]). Theorem 30 (Dynnikov, Hubert, and Skripchenko). An Arnoux–Rauzy IET admits at most two measures, and almost all (with respect to $\mu_{\mathcal{R}}$) Arnoux–Rauzy IETs are uniquely ergodic. Also the following statement holds (see [27]). Theorem 31. Almost all (with respect to $\mu_{\mathcal{R}}$) Arnoux–Rauzy IETs are not weakly mixing. Remark 10. The statements of Theorems 30 and 31 hold not for all, but for almost all Arnoux–Rauzy IETs: there are some examples of Arnoux–Rauzy IETs that admit two ergodic measures ([29]), and there are some weak mixing examples. The relationships between the two exceptional sets (the non-uniquely ergodic IETs and the weakly mixing IETs from the considered class) have not been studied yet: it is a challenging open question. The study of the spectral properties of the cocycle of the renormalization algorithm for special systems of isometries results in the following observations: the Lyapunov spectrum is almost always simple (this means that all Lyapunov exponents are different) and Pisot (there is one positive and two negative exponents). The simplicity of the spectrum can be checked using the technique described in [36], while the Pisot property follows from the same statement for the fully subtractive algorithm that is conjugate to renormalization in our case (see [7]). These observations immediately imply the following statement. Theorem 32 (Avila, Hubert, and Skripchenko). For the Arnoux–Rauzy surface the diffusion rate of the chaotic trajectory is almost always contained strictly between $1/2$ and 1: where $d(x,y)$ is the standard Eulidean distance between the points $x$ and $y$ on the plane, $x$ is an initial point on a chaotic section, and $x_t$ is the position of this initial point after time $t$. Renormalization procedures are useful not only for the baby example of a symmetric surface with two double saddles. Theorem 33 (Skripchenko). There exist a 3-periodic surface $\widehat M$ and a vector $H$ such that sections of $\widehat M$ by almost all planes orthogonal to $H$ consist of exactly one connected component. This theorem is a corollary to Theorem 27. The proof is based on the following elementary topological arguments: taking the system of isometries of thin type $S$ we construct a centrally symmetric surface $\widehat M$ that is the level surface of some function $f$; assume that there is a plane section of $\widehat M$ by a plane orthogonal to $H$, which consists of at least two connected components. Then
$$
\begin{equation*}
\chi({\mathbb R}^2)=\chi(f>0)+\chi(f<0)-\chi(\delta),
\end{equation*}
\notag
$$
where $\delta$ is the boundary corresponding to $f=0$. Since we know from Theorem 27 that $\Gamma_x(S)$ has a unique topological end, we conclude that However, we assume that $\chi(\delta)\geqslant 2$, so we obtain a contradiction. In the same way, applying Theorem 28 instead of Theorem 27 we obtain the following result. Theorem 34 (Dynnikov and Skripchenko). There exist a 3-periodic surface $\widehat M$ and a vector $H$ such that the pair $(M,H)$ determines the chaotic regime and the plane section of $\widehat M$ by almost all sections by the planes orthogonal to $H$ consist of an infinite number of connected components.
Citation in format AMSBIB
\Bibitem{Skr23} |
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