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This article is cited in 2 scientific papers (total in 2 papers) Behaviour of Birkhoff sums generated by rotations of the circle A. B. Antonevicha, A. V. Kocherginb, A. A. Shukurca a Belarusian State University, Minsk, Belarus b Faculty of Economics, Lomonosov Moscow State University, Moscow, Russia c Faculty of Computer Science and Mathematics, University of Kufa, Kufa, Iraq
Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 7, DOI: https://doi.org/10.4213/sm9356 
Bibliographic databases:
    § 1. Introduction. Birkhoff sumsLet $ \mathbb{T}^m=\mathbb R^m /\mathbb{Z}^m$ be an $m$-dimensional torus and let $h=(h_1,\dots,h_m)\in \mathbb T^{m}$. The vector $h$ defines the shift
$$
\begin{equation}
\alpha_h\colon \mathbb T^{m}\to \mathbb T^{m}, \qquad \alpha_h (x)=x+h,
\end{equation}
\tag{1.1}
$$
on the torus. This mapping defines the discrete-time dynamical system (cascade) Let $f$ be a function on $\mathbb T^{m}$, which can be interpreted as a function of $m$ variables with period 1 in each variable. The $n$th Birkhoff sum constructed from $\alpha_h$ and $f$ is defined by
$$
\begin{equation*}
f(n,x, h)=\begin{cases} f(x)+f(x+h)+\dots+f(x +(n-1)h), & n>0, \\ 0, & n=0, \\ -( f(x-h)+f(x-2h)+\dots+f(x+nh)), & n<0. \end{cases}
\end{equation*}
\notag
$$
The mapping (1.1) is ergodic with respect to the Lebesgue measure if and only if the numbers $1, h_1, h_2, \dots, h_m$ are linearly independent over the field of rational numbers, and in this case it is strictly ergodic, that is, there exists a unique $\alpha_h$-invariant probability measure, namely, the Lebesgue measure. The mapping being strictly ergodic, the sequence of Birkhoff means for $f\in C(\mathbb{T}^m)$ converges uniformly to the mean value of the function $f$ (see, for example, [1] and [2]):
$$
\begin{equation}
\frac{1}{n} f(n,x,h) \to \int_{\mathbb T^{m}} f(x)\, dx,\qquad n\to \infty.
\end{equation}
\tag{1.2}
$$
Hence, for functions with zero mean and ergodic $\alpha_h$ the means of Birkhoff sums $\frac{1}{n} f(n, x,h)$ converge uniformly to zero. Given a vector $h$, one can define a linear flow, which is the continuous-time dynamical system This flow is ergodic with respect to the Lebesgue measure if and only if the numbers $h_1, h_2, \dots, h_m$ are linearly independent over the rationals, and in this case this flow is strictly ergodic.An analogue of a Birkhoff sum for the linear flow (1.3) is the path integral The study of Birkhoff sums and the study of path integrals complement each other. Poincaré [3] proposed to reduce the investigation of a flow to the study of a mapping via a flow-transversal section on which the first-recurrence mapping is induced (there are also other names for this mapping). For example, for a linear flow on $\mathbb T^{m}$, where $m\geqslant 2$, one can consider the intersection of the hyperplane $x_1=0$ with the torus $\mathbb T^{m}$ as such a section ($x_1$ is the first coordinate of the vector $x=(x_1,\dots,x_m) \in \mathbb{T}^m$). For $x=(x_1,\dots,x_m) \in \mathbb{T}^m$ we set ${x'=(x_2, \dots, x_m) \in \mathbb{T}^{m-1}}$ and $h'=({h_2}/{h_1},\dots,{h_m}/{h_1})\in \mathbb T^{m-1}$. Let
$$
\begin{equation*}
\begin{gathered} \, x'\mapsto x'+h' \text{ be a shift on }\mathbb T^{m-1}, \\ \varphi(x')=\int_0^{1/h_1} f(0+s h_1,x_2+sh_2,\dots,x_m+sh_m)\,ds. \end{gathered}
\end{equation*}
\notag
$$
We have
$$
\begin{equation*}
\alpha_h^{1/h_1}(0,x_2,\dots,x_m) =\biggl(0+1,x_2+\frac{h_2}{h_1},\dots,x_n+\frac{h_n}{h_1}\biggr)=(0,x'+h'),
\end{equation*}
\notag
$$
and hence, for times $t_n={n}/{h_1}$, $n \in \mathbb{Z}$,
$$
\begin{equation*}
{\mathscr I}(t_n,x, h)=\sum_{k=0}^{n-1} \varphi(x'+k h'),
\end{equation*}
\notag
$$
that is, these are Birkhoff sums for the function $\varphi$ that are constructed from a shift on the torus $\mathbb{T}^{m-1}$ of smaller dimension. The main objects of investigation in the present paper are the Birkhoff sums $f(n,x,h)$ constructed from an irrational shift $h$ on a one-dimensional torus (circle) and a continuous function $f$ with zero mean on the circle: We study the growth rate as $n\to\infty$ of the values of the Birkhoff sums $f(n,x,h)$ at a fixed point, as well as the growth rate of the maxima of Birkhoff sums, which we denote by By strict ergodicity of irrational rotations of a circle, for continuous functions with zero mean we have which imposes a constraint on the growth rate: $ \Phi(n, h;f)=o(n)$.The behaviour of Birkhoff sums and path integrals in various problems was investigated by Poincaré (see [3] and [4]). In particular, he gave an example of a linear flow (1.3) on a two-dimensional torus $\mathbb T^{2} $ and a continuous but nowhere differentiable function $f$ with zero mean on the torus, such that
$$
\begin{equation*}
\lim_{t\to +\infty}\int_0^t f(sh_1, sh_2)\,ds=+\infty.
\end{equation*}
\notag
$$
For an analysis of this example, see [5]. It follows from it, in particular, that for $h=\sqrt{2} $ there exists a continuous function with zero mean for which the Birkhoff sums at 0 tend to infinity. On the other hand, the phenomenon of recurrence of trajectories was manifest in some examples. Namely, there are points $x$ on the circle such that $f(n_k,x, h) \to 0$ for some subsequence $ n_k$. Kozlov [6] discovered the stronger synchronous recurrence of Birkhoff sums for $C^{2}$-functions, to the following effect: for any irrational rotation there exists a subsequence $n_k$ such that $f(n_k,x, h)$ tends to zero uniformly, that is, ${\max_x |f(n_k,x, h) | \to 0}$ as $k \to \infty$. Subsequently, Krygin [7] established a similar fact for $C^{1}$-functions, and Sidorov [8] extended this result to absolutely continuous functions on the circle. However, for linear flows on tori of dimension $m\geqslant 3$ and for shifts on tori of dimension $m\geqslant 2$ synchronous recurrence can fail to hold even for smooth functions. In 1995, Moshchevitin [9] (see also [5]) showed that for any generic continuous function with zero mean (but with nonzero Fourier coefficients) on the torus $\mathbb T^{3} $ the quantities
$$
\begin{equation*}
J(t)=\sup_{x_1,x_2,x_3}\int_0^t f(sh_1 +x_1, s h_2+x_2, sh_3+x_3)\, ds
\end{equation*}
\notag
$$
are growing for a special choice of $h$; moreover, these quantities can grow arbitrarily fast within the above constraints:for any function $F(t)$ that is monotonically decreasing to zero arbitrarily slowly as $t\to+\infty$ there exits a set of frequencies $h_1$, $h_2$, $h_3 $ (linearly independent over $\mathbb{Z}$) such that, staring from some $t_0$, This implies, in particular, that on the torus $\mathbb T^{2} $, for any generic continuous function with zero mean (in particulary, for an arbitrarily smooth one) and any monotone null sequence $\sigma_n$ there exist $h=(h_1,h_2)$ such that $\max_x |f(n,x,h)| \geqslant n \sigma_n$ and, in particular, the Birkhoff sums feature no synchronous recurrence. However, for a smooth function $f$ on the circle $\mathbb T $ we have synchronous recurrence for any irrational rotation, so that a similar assertion on the growth of the Birkhoff sums is not true. Hence in the case of the circle the problem of the behaviour of Birkhoff sums requires a further analysis, which is presented below. In § 3 and § 4 we prove the following main results supplementing the well-known facts about the possible behaviour of Birkhoff sums on the circle. Let $\sigma_n$ be an arbitrary monotone null numerical sequence. Then:
Assertion (1) extends the result in Poincaré’s example by showing that the growth of the Birkhoff sums is possible not only for $h=\sqrt{2} $, but for any irrational $h$. Moreover, this assertion describes the possible growth rate, that is, it improves Poincaré’s result slightly. Assertion (2) specifies the possible behaviour of Birkhoff sums: by the recurrence property, for smooth $f$ and any irrational $h$ there exists a subsequence $m_k$ such that $f(m_k,x, h) \to 0$, but at the same time there exist $h$ for which a prescribed growth rate takes place for another subsequence $n_k$. The behaviour of Birkhoff sums over a rotation of the circle is closely related to the dynamics of cylindrical mappings. Thus, in § 2 we give a survey of the known results describing the dynamical properties of cylindrical mappings and provide some auxiliary results. The shift operators and weighted shift operators generated by dynamical systems appear naturally in the theory of dynamical systems. As an application, we show below that the resolvents of weighted shift operators generated by irrational rotations can grow arbitrarily fast as the spectral parameter approaches the spectrum, which is indicative of the involved structure of such operators. It should be noted that one motivation for this paper was the question of the possible growth rate of the resolvents of weighted shift operators. § 2. Cylindrical cascadesVarious variants of the behaviour of Birkhoff sums can be clearly visualized using the example of a cylindrical mapping. Let $ \alpha\colon X \to X$ be an invertible continuous mapping, let $Y=X \times \mathbb R$, and let $ f $ be a real-valued continuous function on $ X$. The mapping $\beta\colon Y \to Y$ given by defines an extension of the original dynamical system (the so-called skew product). Iterates of this mapping are described in terms of Birkhoff sums: In particular, if $ X=\mathbb{T}$ and $\alpha_h $ is a rotation of the circle, then $\mathbb{T} \times \mathbb{R}$ is a cylinder and is called a cylindrical mapping, or, following Anosov [10], a cylindrical cascade. Its iterates are as follows: So the dynamical properties of cylindrical cascades are closely related to the behaviour of the Birkhoff sums $f(n,x, h)$.A cylinder is homeomorphic to a punctured plane, and so cylindrical mappings can be looked upon as mappings of the plane. Depending on the form of $f$ and $h$, cylindrical cascades can have fairly involved dynamical properties. This was noted by Poincaré [3] himself, who considered them as nontrivial examples of mappings of the plane. Subsequently, the dynamical properties of cylindrical cascades have been studied extensively. The main case of interest is when the function $f$ has zero mean, since otherwise all orbits tend to infinity. The dynamics of a cylindrical cascade is most simple in the case when $f$ is a coboundary over $\alpha_{h}$. A function $f$ is called a coboundary over $\alpha_{h}$ if the cohomological equation has a continuous solution $u$. In this case Birkhoff sums are given by moreover, they are uniformly bounded, each trajectory of a cylindrical cascade is bounded, and the orbit of an arbitrary point $ (x_{0},t_{0}) $ is contained in the closed invariant curve
$$
\begin{equation}
\bigl\{ (x,t)\colon t=t_0+u(x)-u(x_0),\ x\in \mathbb{T}\bigr\}
\end{equation}
\tag{2.3}
$$
as a dense subset. Furthermore, the cylinder $ \mathbb T\times\mathbb R $ is foliated by such curves, and the cylindrical mapping is equivalent to a rotation of a cylinder about its axis. The conjugacy
$$
\begin{equation}
\psi\colon (\mathbb T\times\mathbb R)\to(\mathbb T\times\mathbb R), \qquad \psi(x,t)=(x,t-u(x)+\mathrm{const})
\end{equation}
\tag{2.4}
$$
is defined so that on each generator of the cylinder the reference point is moved to the point of intersection with some fixed invariant curve of the form (2.3). In this case one says that the cylindrical mapping is integrable. It was shown in [5] that the study of certain Hamiltonian systems can be reduced to an analysis of cohomological equation (2.2). Moreover, the solvability of (2.2) leads to the appearance of additional integrals of the original system. Note that the property of synchronous recurrence holds if the cohomological equation is solvable: if the fractional parts of the numbers $n_kh$ tend to zero, then $\max_x|f(n_k,x, h)| \to 0$. In § 5 we describe some special properties of weighted shift operators in the case when the cohomological equation is solvable. The cohomological equation was widely used by Poincaré (see [3]) and many other scientists in dealing, for example, with problems in celestial mechanics. Among most recent studies we mention, for example, [10]–[14]. Note that equality (1.4) holds automatically for coboundaries; however, it is not sufficient for the solvability of (2.2). The set of coboundaries is a non-closed vector subspace, which depends substantially on the number $h$ and has no explicit description. In general, the problem of the solvability of (2.2) is quite involved. On the one hand, a trigonometric polynomial with zero mean is an ‘ideal’ coboundary: for any irrational $h$ equation (2.2) is solvable in the class of trigonometric polynomials. On the other hand, in the general case the solution of the cohomological equation can turn out to be a nonmeasurable unbounded function or there may be no measurable solutions whatsoever (see [10] and [12]). Moreover, for example, for the function
$$
\begin{equation*}
f(x)=\sum_{|k|\geqslant 2} \frac{1}{2ik\sqrt{\ln|k|}}\, e^{i2\pi kx}
\end{equation*}
\notag
$$
the cohomological equation has no solution for any irrational $h$ (see [15], [5]). In stating his fifth problem, Hilbert gave this equation already as an example showing that the solution can be nondifferentiable for smooth (even analytic) functions $f$. The difficulties with the solution of (2.2) are related to the appearance of so-called small denominators (see [11]). Consider the Fourier series of a function satisfying (1.4): For irrational $h$ a formal solution of the cohomological equation is given by the series
$$
\begin{equation}
u(x) \sim -\sum_{k \ne 0}\frac{C_k}{1-e^{i2\pi k h}}\, e^{i2\pi k x}.
\end{equation}
\tag{2.5}
$$
The numbers $1-e^{i2\pi k h}$ in the above denominators are distinct from zero, but some of them can be quite small. As a result, the corresponding coefficients of the series (2.5) can exceed the $C_k$ substantially. In particular, it can happen that the coefficients $C_k/(1-e^{i2\pi k h})$ do not tend to zero. In this case (2.5) cannot be a Fourier series of an integrable function. For a sufficiently smooth function $f$ its Fourier coefficients have estimates. As a result, equation (2.2) can be shown to be solvable for almost all $h$. A typical result in this direction (see [5]) can be formulated as follows. Let $\mathbf K_2 $ be the set of numbers $h$ that are slowly approximable by rational numbers in the following sense: for any rational fraction $m/n$, for sufficiently large $n$,
$$
\begin{equation*}
\biggl| h-\frac{m}{n}\biggr| \geqslant \frac{C}{n^{5/2}}.
\end{equation*}
\notag
$$
If $ f \in C^2(\mathbb T)$, then for any $h \in \mathbf K_2 $ the cohomological equation has a continuous solution and the set $\mathbf K_2$ has measure 1. Similar conditions and arguments on small denominators were used by Kolmogorov [16] in his proof of linearizability of a smooth flow with integral invariant on a two-dimensional torus in the general case. A more general class of functions was described in [14] in terms of Fourier coefficients: for each of these functions the set of $h$ such that the cohomological equation has a continuous solution also has measure $1$. Here the relaxed smoothness requirements on $f$ are compensated for by a reduction of the set of $h$ for which the equation is shown to have a solution, but this set still has full Lebesgue measure. So for sufficiently smooth $f $ and almost all $h$ (in the sense of the Lebesgue measure) the cohomological equation has a continuous solution, and for such pairs $f $ and $h$ the sequence of Birkhoff sums is bounded. In addition, we note that the smoothness of the function $f $ and slow approximation of the number $h$ by rational numbers are not necessary for $f $ to be a coboundary. This is because it is not necessary for the convergence of the series (2.5) that all the Fourier coefficients of $f$ converge rapidly to zero, but only those with indices $k$ for which the denominators $1-e^{i2\pi k h}$ are small. In the case when the cohomological equation is not solvable the behaviour of the Birkhoff sums (and of the trajectories of the cylindrical cascades) can be more sophisticated and versatile. Schnirelmann [17] (1930) and Besicovitch [18] (1937) constructed topologically transitive cylindrical cascades, that is, cascades with dense (transitive) orbits. A transitive orbit is not bounded, and a necessary condition for the existence of transitive orbits is the unboundedness of the sequence $ \Phi(n, h;f)$ of maxima of Birkhoff sums. On the other hand, a transitive orbit has the recurrence property, that is, if a point $x$ has a transitive orbit, then there exits a sequence of numbers $n_k$ such that $x+n_k h\to x$ and $f(n_k,x,h)\to 0$. Note that the structure of the set of transitive orbits was not known at that time, in particular, it was not known whether all orbits can be transitive. Subsequently, it was shown (see, for example, [10]) that there are both cases when the set of points on the circle $\mathbb T\times \{0\}$ with transitive orbits is a nullset and when, on the contrary, it has full measure. In 1955 Gottschalk and Hedlund [19] proved the following alternative for a continuous function $f$ with zero mean: if $h$ is irrational, then either the cylindrical cascade (2.1) is topologically transitive or $ f $ is a coboundary over $\alpha_h$ in the class of continuous functions. In 1951 Besicovitch [20] found out that a cylindrical cascade must have some nontransitive orbits. Moreover, in the same paper, he extended an example when an orbit goes off to infinity, which was in the actual fact constructed by Poincaré in [4], and proved the existence of a topologically transitive cascade that has discrete orbits (which are minimal sets) for any irrational rotation of the circle. Note that a point $(x,t)$ has a discrete orbit if and only if the Birkhoff sums for this point satisfy The set of points on the circle $\mathbb{T}\times \{0\}$ that have discrete orbits is a Lebesgue nullset; however, for cylindrical cascades this set can be quite thick (in a certain sense), and, in particular, its Hausdorff dimension can be positive or even equal to 1 (see, for example, [21] and [22]). The behaviour of Birkhoff sums depends on that of the approximations of $h$ by rational numbers. We recall some facts from the theory of continued fractions (see [23]). Each irrational number $h\in (0,1)$ can be expanded in an infinite continued fraction where $k_1, k_2,\dots$ are natural numbers known as partial fractions. This can briefly be written in the form $h=[k_1,\dots,k_s,\dots]$. An irreducible fraction $p_s/q_s$ written as a finite continued fraction is known as the $s$th convergent to $h$. The convergents are defined recursively:
$$
\begin{equation*}
\begin{gathered} \, p_{s+1}=k_{s+1}p_s+p_{s-1}, \qquad s\geqslant1, \qquad p_{-1}=1, \qquad p_0=1, \\ q_{s+1}=k_{s+1}q_s+q_{s-1}, \qquad s\geqslant1, \qquad q_{-1}=0, \qquad q_0=1. \end{gathered}
\end{equation*}
\notag
$$
Setting $\delta_s=|h-p_s/q_s|$ we have
$$
\begin{equation*}
h=\frac{p_s}{q_s}+(-1)^{s}\delta_s, \qquad \frac{1}{2q_{s+1}q_s}<\delta_s<\frac{1}{q_{s+1}q_s}.
\end{equation*}
\notag
$$
Convergents provide best approximations to the number $h$: any other fraction with denominator not exceeding $q_s$ differs from $h$ by more than $\delta_s$. The rate of approximation depends on the growth rate of the sequence of denominators of the convergents. If a function $f$ has bounded variation on $\mathbb{T}$ (note that $f$ can even be discontinuous) and if $p_s/q_s$ is a convergent to $h$, then for any $x,y\in \mathbb{T}$,
$$
\begin{equation}
|f(q_s,x,h)-f(q_s,y,h)|\leqslant \operatorname{Var} f.
\end{equation}
\tag{2.6}
$$
The function $f(q_s,x,h)$ has zero mean. Hence the sequence of functions $|f(q_s,x,h)|$ is uniformly bounded by the quantity $ \operatorname{Var} f$. So the $\beta$-images of the circle $\mathbb{T}\times \{0\}$ have the property of recurrence to the neighborhood of size ${\operatorname{Var} f}$ of the same circle. As pointed out in § 1, for smooth functions there is a stronger synchronous recurrence of orbits into any neighbourhood of the initial point (see [6]–[8]). More precisely, the subsequence $f(q_s,x,h)$, where $q_s$ is the sequence of denominators of convergents to $h$, converges uniformly to zero. For irrational $h$ an estimate similar to (2.6) holds not only for the denominators of convergents. This was shown in [24], in the proof of the absence of mixing in a smooth flow on a two-dimensional torus. Namely, given some $q$, if there exists an irreducible fraction $p/q$ satisfying $|h-p/q|<1/q^{2}$, then As pointed out by the referee, this inequality can also be obtained from Koksma’s well-known inequality, but the proof in [24] is more straightforward and transparent.Thus, the sequence $\max_x f(n,x,h)$ can go off to infinity only for functions $f$ of unbounded variation, while for smooth $f$ growth is in principle possible only for some subsequences $n_k$ separated (in a certain sense) from the sequence $q_s$ of denominators of convergents. Convergence to infinity of some subsequence $\Phi(n_k, h;f)$ means that the graphs of Birkhoff sums $ f(n_k,x, h)$ are stretched in the vertical direction. But discrete orbits (when a given point goes off to infinity) can appear only under a stronger condition that the values of all Birkhoff sums grow at a fixed point. The constructions underlying the results mentioned in the introduction are based on the following considerations. If $f$ is a $1/N$-periodic function, then for rational translations $ m/N$ the Birkhoff sums grow linearly: $f(n,x, m/N)=nf(x)$. Moreover, if $ m/N$ is a good approximant to an irrational $h$ (for instance, if $ m/N$ is a convergent to $h$), then in some interval of values of $n$ the Birkhoff sums $f(n, x, h)$ grow nearly linearly. In the proof of assertion (1) this enables us, given an irrational $h$, to construct a function $f$ with rapidly stretching Birkhoff sums as the sum of a series with smooth or Lipschitz terms $f_k$, each of which is invariant under the shift by the corresponding convergent to $h$. Each term provides for the growth of the Birkhoff sums on a certain interval of values of $n$, and it is possible to construct these terms so that, by the time the Birkhoff sums of some term cease to grow, the Birkhoff sums of the next term have already grown sufficiently to replace them. As a result, we have a growing sequence of amplitudes of the Birkhoff sums, and this growth can be arbitrarily fast. The functions $f$ thus constructed cannot be smooth, but nevertheless they can satisfy the Hölder condition and can even be almost Lipschitz in a certain sense (see [25]). Constructions of functions which ‘resonate’ with the mapping, thereby providing for the growth of the amplitudes of Birkhoff sums, are also used in different circumstances. For example, in a similar way one can construct special mixing flows over fairly general ergodic automorphisms (see [26]). In the proof of assertion (2), by expanding a given function $f$ in a Fourier series, we succeed in constructing an increasing sequence $ N_k$ such that the Birkhoff sums grow linearly for rational shifts of the form $ m_k/N_k$. Hence, for irrational $h$ that are well approximable by rational numbers $ m_k/N_k$ one can guarantee the growth of the amplitudes of the Birkhoff sums for $f$ on some interval of values of $n$, which expands with increasing $k$. Unlike in the first case, for a smooth function $f$, because of synchronous recurrence, among the values of $n$ for which the sums $f(n,x,h)$ have ceased to be close to $f(n,x, m_k/N_k)$ there are $n$ such that the sums $f(n,x,h)$ are small and the sums $f(n,x, m_{k+1}/N_{k+1})$ are also small. However, for even larger values of $n$ the growth of the sums $f(n,x, m_{k+1}/N_{k+1})$ turns out to be substantial, and this leads to larger values of $\max_x f(n,x,h)$ for some $n$. As $n$ increases further, the sums $f(n,x,h)$ cease to be close to $f(n,x, m_{k+1}/N_{k+1})$ and there appear $n$ such that the sums $f(n,x,h)$ are small. With further increase of $n$ the growth of the sums $f(n,x, m_{k+2}/N_{k+2})$ makes its contribution, and so on. As a result, for irrational numbers $h$ that are sufficiently well approximable by rational numbers $ m_k/N_k$ some subsequence $\max_x f(n_k,x,h)$ shows growth. § 3. On the growth of Birkhoff sums for a given rotationTheorem 1. For any irrational number $h$ and any strictly decreasing null sequence $\{ \sigma_n\colon\sigma_1< 1\}$, there exist a continuous function $f$ on ${\mathbb{T}}$ with norm 1 and zero mean such that the following lower estimate for Birkhoff sums holds at $x=0$ for any ${n\in \mathbb{N}}$: Proof. Let $h$ be an irrational number and ${p_k}/{q_k}$ be the sequence of convergents to $h$. We set $\delta_k=|h-{p_k}/{q_k}|$. Recall that (see [23]) First, for each convergent $p_k/q_k$, we construct a function $f_k$ with the following properties:
Let us show that the functions where $ F_k(x)$ is the function with period $1/q_k$ such that
$$
\begin{equation*}
F_k(x)=\frac{1}{\delta_k}|x| \quad\text{for }\ x\in \biggl[-\frac{1}{2q_k},\frac{1}{2q_k}\biggr],
\end{equation*}
\notag
$$
have these properties.
The function $ F_k(x)$ can equivalently be defined by
$$
\begin{equation*}
F_k(x)=\begin{cases} \dfrac{1}{\delta_k}x, & x\in\biggl[0,\dfrac{1}{2q_k}\biggr], \\ \dfrac{1}{\delta_k}\biggl(\dfrac{1}{q_k}-x\biggr), & x\in\biggl[\dfrac{1}{2q_k},\dfrac{1}{q_k}\biggr], \end{cases}
\end{equation*}
\notag
$$
or, what is the same,
$$
\begin{equation*}
F_k(x)=\frac{1}{\delta_k}\biggl(\frac{1}{2q_k}-\biggl|x-\frac{1}{2q_k}\biggr|\biggr), \qquad x\in \biggl[0,\frac{1}{q_k}\biggr].
\end{equation*}
\notag
$$
We have and therefore, since the function $F_k$ is $1/q_k$-periodic, we have
$$
\begin{equation*}
f_k(x)=F_k(x+h)-F_k(x)=F_k(x+(-1)^{k}\delta_k)-F_k(x).
\end{equation*}
\notag
$$
The function $f_k$ is continuous, $1/q_k$-periodic, and has zero mean. For even $k$ this function is given by
$$
\begin{equation*}
f_k(x)= \begin{cases} 1, & x\in \biggl[0, \dfrac{1}{2q_k}-\delta_k\biggr], \\ -\dfrac{2}{\delta_k}\biggl(x-\dfrac{1}{2q_k} \biggr)-1, & x\in \biggl[\dfrac{1}{2q_k}-\delta_k, \dfrac{1}{2q_k}\biggr], \\ -f_k\biggl(x-\dfrac{1}{2q_k}\biggr), & x\in \biggl[ \dfrac{1}{2q_k}, \dfrac{1}{q_k}\biggr]. \end{cases}
\end{equation*}
\notag
$$
(To verify this it suffices, for example, to evaluate the function at cusp points.) For odd $k$ the graph is symmetric to the above one with respect to the line $x=0$. In both cases $\|f_k\|_C=1$.
It is easily seen that for these functions the Birkhoff sums corresponding to a given $h$ can be evaluated by
$$
\begin{equation*}
f_k(n,x,h)=F_k(x+nh)-F_k(x)=F_k(x+(-1)^{k}n\delta_k)-F_k(x).
\end{equation*}
\notag
$$
Since $F_k(0)=0$ is the smallest value of $F_k$, the last equality implies that $f_k(n,0,h)\geqslant 0$ for all $n$.
If $0\leqslant n\leqslant q_{k+1}/2$, then
$$
\begin{equation*}
0\leqslant n\delta_k< \frac{q_{k+1}}{2}\,\frac{1}{q_kq_{k+1}}=\frac{1}{2q_k},
\end{equation*}
\notag
$$
which implies the equality
$$
\begin{equation*}
f_k(n,0,h)=F_k(0+(-1)^{k}n\delta_k)=\frac{1}{\delta_k}n\delta_k=n.
\end{equation*}
\notag
$$
Consider a number series with decreasing positive terms and unit sum: Let $\{ \sigma_n\} $ be a decreasing null sequence. Without loss of generality we can assume that $\sigma_1<a_1$.We have $a_j>0$ and $ \sigma_n \to 0$, hence for any $j$ there exists $n_j$ such that $\sigma_n<a_j$ for all $n\geqslant n_j$. We can assume without loss of generality that $n_1=1$ and that the sequence of numbers $\{n_j\}$ is strictly increasing. Note that if the sequence $\sigma_n $ is slowly decreasing, then the sequence $\{n_j\}$ is rapidly increasing. Consider now a strictly increasing sequence of integers $\{k_j\}$ such that and setThis series is uniformly convergent, so the function $f$ is continuous and $\|f\|=1$. First, we have $f_k(0)=1$ for any $k$, hence $f(0)=1$ by (3.1). As a result, $\|f\|\geqslant 1$. On the other hand, from (3.1) and since $\|f_{k_j}\|=1$ we obtain $\|f\|\leqslant 1$. Now we take an arbitrary $n$ and consider $m$ such that $n_m\leqslant n<n_{m+1}$. We have $n<n_{m+1}<{q_{k_{m}+1}}/{2}$, and so $f_{k_m}(n,0,h)=n$ by construction. Taking into account that $f_k(n, 0,h)\geqslant 0$ for all $k$ and $n$, we get that
$$
\begin{equation*}
f(n, 0,h)=\sum_{j=1}^{\infty}a_jf_{k_j}(n, x,h) \geqslant \sum_{j=1}^{\infty}a_jf_{k_j}(n, 0,h) >n a_{m}.
\end{equation*}
\notag
$$
On the other hand, since $ n \geqslant n_m $, we have $a_{m} > \sigma_{n}$, from which the required result follows, namely, Theorem 1 is proved. For a cylindrical cascade constructed from $h$ and a function $f$, the last inequality implies that the trajectory of the point $(0;0)$ goes off to infinity faster than $n\sigma_n$. Remark 1. The function thus constructed is closely related to the denominators of convergents to $h$, whose numerators satisfy the same recurrence relations as the denominators. This leads to fairly stringent constrains when, given a function, we attempt to construct other angles of rotation with rapidly increasing values of the Birkhoff sums (for example, by changing the numerators of fractions). Terms which are coboundaries for a fixed $h$ need not be coboundaries for other angles, so it is difficult to control their Birkhoff sums with large indices. Remark 2. If from the conclusion of the theorem one removes the condition that the function $f$ has unit norm, then the condition $\sigma_1<1$ can also be removed. Indeed, it suffices to multiply the function constructed above by an appropriate number. § 4. On the growth of maxima of the Birkhoff sums for a given functionThe next theorem refines one result in [27] to a certain extent. Theorem 2. Assume that a continuous function $f$ on ${\mathbb{T}}$ has zero mean and is not a trigonometric polynomial. Then for any decreasing null sequence $\sigma_n$ there exist continuum many irrational $h$ such that for some subsequence $n_k$ common for all $h$’s. Before proceeding with the construction of the required $h$ we need to establish some properties of Birkhoff sums for a rational rotation. Let $f$ be a continuous function on $\mathbb T$ with zero mean, and let $h={m}/{N}$, where ${m}/{N}$ is an irreducible fraction. We set
$$
\begin{equation}
L(f, h)=\frac{1}{N} \max_x f(N,x, h)=\frac{1}{N}\Phi(N, h;f).
\end{equation}
\tag{4.1}
$$
Lemma 1. Let $h=m/N$ be an irreducible fraction. Then the following assertions hold. 1. The Birkhoff sum $f(N,x, h)$ is $1/N$-periodic in $x$. 2. $L(f, h) $ depends only on the denominator of the irreducible fraction $h={m}/{N}$:
$$
\begin{equation*}
L\biggl( f, \frac{m}{N}\biggr)=L\biggl( f, \frac{1}{N}\biggr).
\end{equation*}
\notag
$$
3. $L( f,{1}/{N}) \geqslant 0$ for any natural number $N$. Moreover, 4. If $s\in \mathbb N$, then
$$
\begin{equation*}
\begin{gathered} \, f(sN,x,h)=sf(N,x,h), \\ \Phi(sN,h;f)=sN L(f,h). \end{gathered}
\end{equation*}
\notag
$$
5. If $n=sN+j$, where $s,j\in \mathbb N$, $0\leqslant j<N$, then Proof. 1–2. Since $m$ and $N$ are coprime, we have
$$
\begin{equation*}
\begin{aligned} \, &f\biggl( N,x,\frac{m}{N}\biggr) =f(x)+f\biggl( x +\frac{m}{N}\biggr)+ f\biggl( x +2\frac{m}{N}\biggr) +\dots+ f\biggl( x +(n-1)\frac{m}{N}\biggr) \\ &\quad=f(x)+f\biggl( x +\frac{1}{N}\biggr)+f\biggl( x +\frac{2}{N}\biggr) +\dots+ f\biggl( x +\frac{n-1}{N}\biggr)=f\biggl( N,x, \frac{1}{N}\biggr). \end{aligned}
\end{equation*}
\notag
$$
As a result, $f(N,x, h)$ is periodic, and moreover, which implies that
3. The Birkhoff sum $f(N,x, h)$ has zero mean, hence its maximum is nonnegative. This maximum is attained at some point $x_0$, and
$$
\begin{equation*}
L\biggl( f, \frac{1}{N}\biggr)=\frac{1}{N}\biggl( f(x_0)+f\biggl( x_0 +\frac{1}{N}\biggr) +f\biggl( x_0 +\frac{2}{N}\biggr) +\dots+ f\biggl( x_0 +\frac{n-1}{N}\biggr)\biggr)
\end{equation*}
\notag
$$
is an integral sum for the function $ f(x)$. Hence
$$
\begin{equation*}
\lim_{N \to \infty}L\biggl( f, \frac{1}{N}\biggr)=\int_{\mathbb{T}} f(x) \,dx=0.
\end{equation*}
\notag
$$
4. The rotation of the circle by $2\pi h$ is $N$-periodic, hence
$$
\begin{equation*}
f(sN,x,h)=f(N,x,h)+f((s-1)N,x+Nh,h)=f(N,x,h)+f((s-1)N,x,h),
\end{equation*}
\notag
$$
when the first equality follows by induction. The second equality is a consequence of the first and the definition of $L(f,h)$.
5. First, for an arbitrary natural number $n$ we establish the lower estimate Indeed, the Birkhoff sum $f(N,x,h)$ is $1/N$-periodic in $x$, so it attains the maximum at points of the form $x'+k/N$, $k=0,\dots,N-1$. As a result,
$$
\begin{equation*}
L(f, h)=\frac{1}{N} \max_x f(N,x, h)=f\biggl(x'+\frac kN\biggr).
\end{equation*}
\notag
$$
Let us find the mean value of the Birkhoff sum $f(n,x,h)$ over the set of all points of the form $x'+k/N$, $k=0,\dots,N-1$. We have
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{N}\sum_{k=0}^{N-1}f\biggl( n,x'+\frac{k}{N},h\biggr) = \frac{1}{N}\sum_{k=0}^{N-1}\sum_{l=0}^{n-1}f\biggl( x'+\frac{k}{N}+\frac{lm}{N}\biggr) \\ &\qquad =\frac{1}{N}\sum_{l=0}^{n-1}\sum_{k=0}^{N-1}f\biggl( x'+\frac{lm}{N} +\frac{k}{N}\biggr)=\frac{1}{N}\sum_{l=0}^{n-1}f\biggl( N,x'+\frac{lm}{N},\frac{1}{N}\biggr) \\ &\qquad =\sum_{k=0}^{n-1}\frac{1}{N}f(N,x',h)=nL(f, h). \end{aligned}
\end{equation*}
\notag
$$
Hence, among the points $x'+k/N$ there exists $x''$ such that which yields the required inequality.
To find an upper estimate, we write $n$ in the form $ n=sN+j $. Then we have
$$
\begin{equation*}
f(n,x, h)=s f(N,x, h)+f(j,x, h)\leqslant \frac{n}{N}f(N,x, h)+j\|f\|.
\end{equation*}
\notag
$$
Setting $ n=(s+1)N-(N-j) $, we arrive at another estimate:
$$
\begin{equation*}
\begin{aligned} \, f(n,x, h) &=(s+1)f(N,x, h)-\sum_{k=j+1}^{N-1}f(x +kh) \\ &=\frac{n}{N}f(N,x, h)+ \frac{N-j}{N}f(N,x, h)- \sum_{k=j+1}^{N-1} f(x +kh) \\ &=\frac{n}{N}f(N,x, h)+ \sum_{k=j+1}^{N-1}\biggl( \frac{f(N,x, h)}{N}-f(x +kh)\biggr) \\ &\leqslant n L(f, h)+2(N-j)\|f\|. \end{aligned}
\end{equation*}
\notag
$$
Now (4.3) follows by combining these two estimates.
Lemma 1 is proved. A principal difference between irrational and rational rotations is that for irrational $ h$ the Birkhoff sums grow slower that $ n$, while for rational $ h=m/N$ the Birkhoff sums with numbers dividing $N$ stretch proportionally, But the growth is linear in $ n $ only in the case when $f(N,x,m/N)$ is a nonzero function (which is equivalent to saying that $ L(f, {1}/{N})\neq 0$).Lemma 2. The set consists of the indices of nonzero Fourier coefficients of the function $f$ and their divisors. Proof. The equality $ L(f, {1}/{N})=0 $ implies that $ f(N,x, {1}/{N})=0$ for all $x$, and this is possible if and only if all the Fourier coefficients of $f$ with indices dividing $N$ vanish. Indeed
$$
\begin{equation*}
\begin{aligned} \, 0 &=\int_{0}^{1}f\biggl(N,x, \frac{1}{N}\biggr)e^{-i2\pi sNx}\,dx =\sum_{k=0}^{N-1}\int_{0}^{1}f\biggl(x+ \frac{k}{N}\biggr)e^{-i2\pi sNx}\,dx \\ &=\sum_{k=0}^{N-1}\int_{0}^{1}f(x)e^{-i2\pi sN(x-k/N)}\,dx= \sum_{k=0}^{N-1}\int_{0}^{1}f(x)e^{-i2\pi sNx}e^{i2\pi (sNk)/N}\,dx \\ &=N \int_{0}^{1}f(x)e^{-i2\pi sNx}\,dx. \end{aligned}
\end{equation*}
\notag
$$
So, if the Fourier coefficient $\widehat f_{sN}$ is nonzero, then $ L(f, {1}/{N})\ne 0 $ and $ L(f, {1}/{s})\ne 0 $, which proves Lemma 2. We see that if a function $f$ with zero mean is not a trigonometric polynomial, then the set $\mathcal{N}_f$ is infinite. Using this fact, given a function, we can construct a required irrational number $h$ as the limit of a sequence of rational numbers with denominators in the set $\mathcal{N}_f$. Proof of Theorem 2. For brevity, for $ \Phi({n, h;f})$ we write
According to Lemma 2, the set $\mathcal{N}_f$ of natural numbers $N$ such that $ L(f,{1}/{N}) > 0$ consists of the indices of nonzero Fourier coefficients of the function $f$ and their divisors. Hence this set is infinite. Fix a decreasing null sequence $\sigma_n$. We construct the set of required $h$ as a Cantor-type set. Each of these $h$ is the limit of a sequence of irreducible rational fractions. Moreover, the sequence of denominators $N_1, N_2,\dots$ of these fractions is the same for all $h$; it is a subsequence of indices from $\mathcal{N}_f$. Fix $N_1 \in \mathcal{N}_f$. By (4.3) we have
$$
\begin{equation*}
\Phi\biggl( n, \frac{1}{N_1}\biggr)\geqslant n L\biggl( f, \frac{1}{N_1}\biggr).
\end{equation*}
\notag
$$
Consider the following inequality for $n$:
$$
\begin{equation*}
\frac{1}{n} \Phi\biggl( n, \frac{1}{N_1}\biggr) \geqslant 2 \sigma_n.
\end{equation*}
\notag
$$
The right-hand side tends to zero and the left-hand side is bounded below by a positive constant $ L(f, {1}/{N_1})$, hence there exists $n_1$ such that
$$
\begin{equation*}
\Phi \biggl( n_1, \frac{1}{N_1}\biggr) \geqslant 2 n_1 \sigma_{n_1}.
\end{equation*}
\notag
$$
A similar inequality holds for all $m_1$ coprime to $N_1$:
$$
\begin{equation}
\Phi \biggl( n_1, \frac{m_1}{N_1}\biggr) \geqslant 2 n_1\sigma_{n_1}.
\end{equation}
\tag{4.5}
$$
Let
$$
\begin{equation*}
W( \delta)=\sup_{|x_1-x_2|\leqslant \delta}|f(x_1)-f(x_2)|
\end{equation*}
\notag
$$
be the modulus of continuity of the function $f$. For any $h$ and $\rho$, for $0\leqslant j<n$ we have $|jh -j\rho |\leqslant n|h-\rho|$, which gives the following estimate for the difference of two Birkhoff sums for arbitrary $h$ and $\rho$:
$$
\begin{equation*}
\biggl| \sum_{j=0}^{n-1} f(x+jh)-\sum_{j=0}^{n-1} f(x+j \rho)\biggr| \leqslant n W(n|h-\rho|).
\end{equation*}
\notag
$$
Let $\delta_1>0$ be such that If we have
$$
\begin{equation*}
\biggl| h-\frac{m_1}{N_1}\biggr| \leqslant\frac{\delta_{1}}{n_1}
\end{equation*}
\notag
$$
for some $m_1$, then we obtain an estimate for the difference of Birkhoff sums:
$$
\begin{equation}
\biggl| \sum_{j=0}^{n_1-1} f(x+j h)-\sum_{j=0}^{n_1-1} f\biggl( x+j\frac{m_1}{N_1}\biggr) \biggr| \leqslant n_1\sigma_{n_{1}}.
\end{equation}
\tag{4.6}
$$
Let $H_1$ be the union of closed intervals of the form $[m_1\mkern-1mu/\mkern-1muN_1-\delta_{1}\mkern-1mu/\mkern-1mu n_1,m_1\mkern-1mu/\mkern-1mu N_1+ \delta_{1}\mkern-1mu/\mkern-1mu n_1]$, where $ 1\leqslant m_1< N_1$ and the integers $m_1$ are coprime to $N_1$. Hence, for all $h \in H_1$, using (4.5) and (4.6) we obtain the lower estimate
$$
\begin{equation*}
\Phi(n_1, h)\geqslant \Phi \biggl(n_1,\frac{m_1}{N_1}\biggr)- n_1\sigma_{n_1} \geqslant n_1 \sigma_{n_1}.
\end{equation*}
\notag
$$
We impose the additional constraint ${\delta_{1}}/{n_1}<{1}/{N_1^{2}}$ on $\delta_{1}$. It guarantees that the set $H_1$ has no irreducible fractions with denominators smaller than $N_1$. Moreover, this constraint also guarantees that the intervals composing $H_1$ are disjoint.
Next, we choose a number $N_2>N_1$ for which $L(f,{1}/{N_2})>0$. Here $N_2$ can be taken sufficiently large so that each of the closed intervals forming $H_1$ contains at least four irreducible fractions of the form ${m_2}/{N_2} $ (two of which may later be thrown away due to their proximity to endpoints of the interval). These points will be the midpoints of intervals of the next rank and, incidentally, the next in order rational approximants to the required $h$. In the choice of $N_2$ the following two cases are possible. If the set $\mathcal{N}_f$ contains an infinite number of primes, then we can take a prime number as $N_2$, and then the required result is clear. If the set $\mathcal{N}_f$ contains only a finite number of primes $ p_1, \dots, p_r$, then each $N \in \mathcal{N}_f$ can be represented in the form $N=p_1^{\nu_1}p_2^{\nu_2}\dotsb p_r^{\nu_r}$, and we can take the largest term of the form $p_k^{\nu_k}$ in the factorization of $N$ as $N_2$; this factor is greater than $N^{1/r}$. Then only multiples of $p_k$ will not be coprime to $N_2$, so the required $m_2$ exists: of any two consecutive integers at least one is coprime to $N_2$. Similarly to the first step, there exists $ n_2$ such that, for the above $m_2$,
$$
\begin{equation}
\frac{1}{n_2} \Phi\biggl( n_2, \frac{m_2}{N_2}\biggr)=\frac{1}{n_2} \Phi\biggl( n_2, \frac{1}{N_2}\biggr) \geqslant 2 \sigma_{n_2}.
\end{equation}
\tag{4.7}
$$
Let $\delta_2>0$ be such that
$$
\begin{equation*}
W( \delta_2)\leqslant\frac{\sigma_{n_{2}}}{n_2}, \qquad \delta_2<\frac{1}{N_2^{2}}.
\end{equation*}
\notag
$$
If the inequality
$$
\begin{equation*}
\biggl| h-\frac{m_2}{N_2}\biggr| \leqslant\frac{\delta_{2}}{n_2}
\end{equation*}
\notag
$$
holds for some $m_2$, then by (4.7)
$$
\begin{equation*}
\Phi(n_2, h)\geqslant \Phi\biggl( n_2, \frac{m_2}{N_2}\biggr)- n_2 \sigma_{n_2} \geqslant n_2 \sigma_{n_2}.
\end{equation*}
\notag
$$
In each of the closed intervals of the form $[m_1/N_1 -\delta_{1}/n_1,\, m_1/N_1 +\delta_{1}/n_1]$ that comprise the set $H_1$ there are at least two subintervals of the form $[m_2/N_2 - \delta_{2}/n_2,\, m_2/N_2 + \delta_{2}/n_2]$, where $ m_2$ is coprime to $N_2$. We denote the set of all such intervals that lie wholly in $H_1$ by $H_2$. Then $H_2\subset H_1$ and for all $ h \in H_2$ we have
$$
\begin{equation*}
\Phi(n_1, h)\geqslant n_1\sigma_{n_1}, \qquad \Phi(n_2, h)\geqslant n_2 \sigma_{n_2}.
\end{equation*}
\notag
$$
Continuing the process, we obtain sequences of natural numbers $N_k$ and $n_k$ and a decreasing sequence of sets $H_k\subset [0,1]$ with the following properties: 1) each set $H_k$ consists of a finite number of disjoint closed intervals; 2) each of the closed intervals comprising the set $H_{k-1}$ contains at least two closed intervals in $H_k$; 3) for all $h \in H_k$ and $i \leqslant k$ the lower estimate holds. Let Then for $h \in H$ inequality (4.8) holds for all $i$.The construction of $H$ is similar to that of the Cantor set. The set $H$ is nonempty as the intersection of a decreasing sequence of closed subsets of a closed interval. By construction, each $h \in H$ is the limit of some sequence ${m_k}/{N_k}$, where different limits correspond to different sequences $\{m_k\}$. For fixed $ m_1, m_2, \dots, m_{k-1}$, there are at least two admissible values of $m_k$. Hence $H$ has the cardinality of the continuum, all elements of $H$ being irrational because no set $H_k$ contains irreducible fractions with denominators $<N_k$. Theorem 2 is proved. It is worth pointing out that, for the function constructed in the proof of Theorem 1, the entire sequence of values of Birkhoff sums at a fixed point tends to infinity, while Theorem 2 asserts only that some subsequence $\Phi(n_i,h)$ of maxima of Birkhoff sums is increasing. This difference is principal because, for a function $f$ of bounded variation, there exists a bounded subsequence of Birkhoff sums, while for smooth functions some subsequence $\Phi(m_k,h)$ tends to zero due to synchronous recurrence. § 5. Weighted shift operators5.1. Definitions and the relation to Birkhoff sumsA mapping $\alpha\colon X \to X$ generates weighted shift operators or composition operators with weight acting in function (or vector function) spaces on $X$ by where $a(x)$ is a given complex-valued (matrix-valued) function on $X$. Operators of the form are called composition operators or shift operators. It is quite natural that the properties of such operators are closely related to the dynamics of $\alpha$ (for applications to dynamical systems, see [1], [2] and [28]).Weighted shift operators (also known under different names) appear also in other fields. Such operators, the corresponding operator algebras and related function equations have extensively been studied in various function spaces in their own right and in relation to various applications (see, for example, [29], [30] and the references there). In particular, they constitute a subclass of nonlocal function-differential operators which display a qualitative difference from differential ones, because their properties depend on the dynamics of the mapping $\alpha$. Each rotation of the circle generates the shift operator and the family of weighted shift operators where $a \in C(\mathbb{T})$. This is one of the best studied classes of weighted shift operators, for which the problems discussed below have never been examined. Formula (5.1) defines a bounded linear operator in the spaces $L_p(\mathbb{T})$ and in $ C(\mathbb{T})$. The results that follow hold in all these spaces. To be more specific, we consider operators in the Hilbert space $L_2(\mathbb{T})$.The norms of positive powers of the operator (5.1) are given by If $|a(x)| \ne 0 $ for all $x$, then the operator $B$ is invertible and
$$
\begin{equation*}
\|B^{-n}\|=\max_x \dfrac{1}{\prod_{k=0}^{n-1} |a(x+kh)|}.
\end{equation*}
\notag
$$
Hence if we set $f(x)=\ln |a(x)|$, then for all $n\in \mathbb N$,
$$
\begin{equation}
\|B^n\|=\exp\Bigl( \max_x f(n, x, h)\Bigr)=\exp \Phi(n, h;f).
\end{equation}
\tag{5.2}
$$
So, the behaviour of the norms of positive powers of the operator is controlled by the behaviour of the sequence of maxima of Birkhoff sums for the function $f$, and the information about their behaviour is given in Theorems 1 and 2. For negative $n$ we have and therefore the behaviour of the norms of negative powers of the operator is controlled by that of the sequence of minima of Birkhoff sums.Here it is worth pointing out that in the case of synchronous recurrence of Birkhoff sums there exists a subsequence $n_k$ such that $\| B^{n_k}\| \to 1$. But even in this case there can exist subsequences $m_k$ for which the norms $\| B^{m_k}\| $ increase unboundedly. The behaviour of norms of powers of weighted shift operators was considered in [27], [31] and [32]. By Gelfand’s formula, the spectral radius $R(B)$ of an operator $B$ is given by Given (5.2) and (1.2), we can find this limit, thereby obtaining a formula for the spectral radius of operator (5.1) for all irrational $h$:
$$
\begin{equation}
R(aT_h)=\exp\biggl( \int_{\mathbb{T}}\ln |a(x)|\, dx\biggr).
\end{equation}
\tag{5.3}
$$
In particular, we have $R((aT_h)^{-1})=R((aT_h))^{-1}$, which implies that the spectrum of the operator lies on the circle of radius $R(aT_h)$. Moreover, for irrational $h$ the spectrum is invariant under rotations and coincides with this circle. In particular, if $f(x)=\ln |a(x)|$ is a continuous function with zero mean, then the spectrum of the operator is the circle of unit radius. Formula (5.3) also holds in the case when $a(x)=0 $ at some points and the function $\ln |a(x)|$ is not bounded below. Moreover, if $\displaystyle\int_{\mathbb{T}}\ln |a(x)|\,dx=-\infty$, then $ R(aT_h)=0$, that is, (5.3) also holds if we set $\exp(-\infty)=0$. In this case the spectrum is the disc of radius $ R(aT_h)$. 5.2. The properties of the weighted shift operator in the case when the cohomological equation is solvableThe solvability of the corresponding cohomological equation makes it possible to single out the most simple operators among the weighted shift operators $aT_h$. Let $a(x) $ be a real-valued function such that $a(x)\ne 0 $ for all $x$, let $f(x)=\ln |a(x)|$ and $\displaystyle S=\int_{\mathbb T} f(x)\,dx$. We set $\omega=\mathrm{sign}(a(x))$ and consider the cohomological equation If this equation is solvable, we say that $f$ is cohomological to the constant $S$. If a function is cohomological to a constant, then this constant is the mean value of the function.Setting $d(x)=e^{u(x)}$ we obtain the cohomological equation in the multiplicative form: As a result, in the case of solvability, we have the representation of the coefficient which is called factorization with respect to translation in [33].Just as the change of variables (2.4) reduces a cylindrical cascade constructed from an irrational rotation and a coboundary to a rotation of the cylinder about its axis, so the mapping
$$
\begin{equation}
\Psi\colon L_{2}(\mathbb T)\to L_{2}(\mathbb T), \qquad\Psi\varphi(x)=\frac{1}{d(x)} \varphi(x)
\end{equation}
\tag{5.5}
$$
reduces a weighted shift operator to a composition of a translation and multiplication by the constant $e^{S}$: Note that the operator $T_h$ is unitary and has simple discrete spectrum, because the eigenfunctions $e_k(x)=\exp(i2\pi k x)$, $k \in \mathbb{Z}$, correspond to distinct eigenvalues $\lambda_k=\exp(i2\pi kh)$ and form a complete system. The following result is a consequence of the above properties of the cohomological equation. Proposition 1. Let $h$ be an irrational number, let $a(x)>0$, and let the function $f(x)=\ln a(x)$ have mean value zero. Then the following conditions are equivalent. Under these conditions the resolvent has the estimate Let us explain, for example, how property 4 implies the others. It is known that if the norms of all powers of an operator $A$ in a Hilbert space are uniformly bounded, then $A$ is similar to a unitary operator. For weighted shift operators this result can be refined in view of an alternative in [19]: the above similarity is defined via the operator (5.5), which acts as multiplication by a continuous function, and the corresponding unitary operator is $ T_h$. Proposition 1 can be extended to the case when the coefficient $ a(x)$ is complex- valued. Here we assume that $a(x)$ is a periodic function with period 1 on $\mathbb{R}$. Let $F(x) $ be a continuous branch of $\ln a(x)$. This function may not be periodic, its increment on the interval $[0,1]$ is a number of the form $i 2\pi \varkappa$, where the integer $ \varkappa$ is called the Cauchy index of the function $a$. In the case of complex-valued functions, in addition to the condition that the mean value of the right-hand side must be zero, there is another necessary condition for the existence of a continuous solution of equation (5.4), namely, the Cauchy index of the right-hand side must be zero. The integral of the function $F(x) $ over $[0,1]$ is a complex number $ S=S_1+i \nu$. Consider the cohomological equation with the ‘corrected’ right-hand side for which both solvability conditions are met.Proposition 2. If equation (5.7) has a continuous solution, then $aT_h$ is similar to the operator $ e^{S_1} e^{i \nu}e^{i 2\pi \varkappa x} T_h$. It follows, in particular, that for $ \varkappa=0$ the operator $aT_h$ has continuous eigenfunctions, while for $\varkappa \ne 0$ the spectrum of the operator is purely continuous. The converse theorem is also valid: if an operator $aT_h$ has at least one continuous eigenfunction, then the cohomological equation is solvable and the Cauchy index of the coefficient $a$ is zero. In the case of complex coefficients we can associate with the operator $aT_h$ the cylindrical mapping on the cylinder $\mathbb{T}\times \mathbb{C}$ with complex generator acting by $\beta(x, \xi)=(x+ h, a(x)\xi)$. In [13] a difference of the dynamical properties of this mapping for $ \varkappa=0$ from those for $ \varkappa \ne 0$ was pointed out. According to the above, the properties of the operators $aT_h$ are also different in these two cases. 5.3. A rational rotationAs already mentioned, the constructions in § 3 and § 4 are based on the fact that, for rational $h$ the Birkhoff sums behave substantially differently from those for irrational $h$. Similarly, for rational $h$ the norms of powers of the operators $aT_h$ display qualitative differences in their behaviour. Recall its description in [31] and [32]. We write a rational number $h={m}/{N}$ as an irreducible fraction. For such $h$ there is a description of the spectrum of the operator $ aT_h$ (see [29]):
$$
\begin{equation*}
\sigma(aT_h)=\biggl\{\lambda\colon\exists\,x\colon \lambda^N=\prod_{k=0}^{N-1} a(x+kh) \biggr\}.
\end{equation*}
\notag
$$
As a result,
$$
\begin{equation}
R(aT_h)=\biggl( \max_x \prod_{k=0}^{N-1} |a(x+kh)|\biggr)^{1/N}=\exp(L(f,h)),
\end{equation}
\tag{5.8}
$$
where $f(x)=\ln |a(x)|$ and $ L(f, h)=\frac{1}{N} \max_x f(N,x, h)=\frac{1}{N}\Phi(N, h;f)$ was defined in (4.1). For any operator $A$ we have
$$
\begin{equation}
R(A)^n \leqslant \|A^n\|=R(A)^n \frac{\|A^n\|}{R(A)^n}.
\end{equation}
\tag{5.9}
$$
If $h={m}/{N}$, then $(aT_h)^{N}$ acts as multiplication by the function and its norm coincides with the spectral radius. Hence, for $ n=sN$, $s \in \mathbb{N}$, we have For other powers $n$ this equality may fail to hold. If $n$ is written as $ n=sN+j$, $0\leqslant j<N$, then from (5.9) and (5.8) we obtain the estimate
$$
\begin{equation}
\begin{aligned} \, \exp(nL(f, h))&\leqslant\|(aT_h)^n\| \nonumber \\ & \leqslant \exp(nL(f, h)) \frac{\| (aT_h)^j\|}{R(aT_h)^j} \leqslant M(h, f) \exp(nL(f, h)), \end{aligned}
\end{equation}
\tag{5.10}
$$
where
$$
\begin{equation*}
M(h, f)=\max_{0\leqslant j<N} \frac{\| (aT_h)^j\|}{R(aT_h)^j}.
\end{equation*}
\notag
$$
These estimates, which correspond to estimates for Birkhoff sums in Lemma 1, give a definite operator meaning to such sums. For example, inequality (4.3) reflects the fact that the spectral radius of an operator is majorized by its norm. Let $h$ be an irrational number and $h_k=p_k/q_k$ be a sequence of irreducible fractions tending to $h$. Then it follows from (4.2), (5.3) and (5.8) that the spectral radii $R(aT_{h_k}) $ tend to $R(aT_{h})$. This result looks quite natural. But here it is worth pointing out that this fact reflects certain specific properties of weighted shifts and does not follow from general properties of operators. The thing is that the spectral radius of an operator is discontinuous on the space of all bounded linear operators: for an arbitrary sequence of operators $A_k$ the norm convergence of $A_k$ to $A$ does not imply that $ R(A_k) \to R(A) $ (see [34]). In the case under consideration the sequence of operators $aT_{h_k}$ does not even converge in norm to $aT_{h} $ (here $aT_{h_k}u $ converges to $aT_{h} u$ for any function $u,$ that is, convergence is pointwise); however, there is a convergence of spectral radii. We note in passing that the most illustrative examples demonstrating the discontinuity of the spectral radius are constructed from the weighted shift operators generated by irrational rotations (see [29] and [35]). We give an example. Let $a(x)=|x -1/2|$ for $0\leqslant x \leqslant 1$ and let
$$
\begin{equation*}
a_k(x)=\begin{cases} \biggl|x -\dfrac12\biggr|-\dfrac{1}{k} &\text{for }\ \biggl|x -\dfrac12\biggr|\geqslant \dfrac{1}{k}, \\ 0& \text{for }\ \biggl|x -\dfrac12\biggr|\leqslant\dfrac{1}{k}. \end{cases}
\end{equation*}
\notag
$$
Then $\| aT_h-a_k T_h \|\leqslant 1/k$ and, moreover, $R(aT_h)=1/(2e) >0$ and $R(a_k T_h)=0$. This shows that the spectral radius is discontinuous as a function of the operator.
§ 6. On the growth of resolvents of weighted shift operators6.1. Norms of powers of operators and the behaviour of resolventsOur study was motivated, in particular, by the question of the possible behaviour of the resolvents of weighted shift operators. First recall some general results. Let $ B$ be a bounded linear operator in a Banach space. The norm of the resolvent $\mathscr R(\lambda;B):=(B-\lambda I)^{-1}$ increases as the spectral parameter approaches the spectrum. Note that we always have
$$
\begin{equation*}
\|\mathscr R(\lambda;B)\|\geqslant \frac{1}{d(\lambda,\sigma(B))},
\end{equation*}
\notag
$$
where $d(\lambda,\sigma(B))$ is the distance from $\lambda$ to the spectrum $\sigma(B)$. In Hilbert spaces normal operators are the most simple ones, and for them the growth of the resolvent is smallest possible, namely, $\|\mathscr R(\lambda;B)\|=d(\lambda,\sigma(B))^{-1}$. If an operator $B$ is similar to a normal operator, then the norm of the resolvent has a similar growth rate:
$$
\begin{equation*}
\|\mathscr R(\lambda;B)\|\leqslant \frac{\mathrm{const}}{d(\lambda,\sigma(B))}.
\end{equation*}
\notag
$$
In particular, the estimate (5.6) for the resolvent has the same form. In the general case the norm of a resolvent can grow much faster than $1\mkern-1mu/\mkern-1mu d(\mkern-1mu\lambda,\mkern-1mu \sigma(\mkern-1mu B\mkern-1mu)\mkern-1mu)$. The growth rate of the resolvent is one of the characteristics of the complexity of an operator. This can already be seen in the case when $\sigma(B) = \{1\}$. In finite-dimensional spaces the size of the largest Jordan block, which we denote by $m$, can be looked upon as a measure of the complexity of $B$. In this case there is a clear dependence between the number $m$, the behaviour of the resolvent, and the behaviour of the norms of powers of the operator. Namely, the norm of the resolvent increases as $ {\mathrm{const}}/{|\lambda-1|^m}$, and the norms of positive powers of the operator grow as $n^{m-1}$. In particular, the boundedness of the norms of positive powers implies that $m=1$ and $ B=I$. However, no analogue of a Jordan cell exists in infinite-dimensional Banach spaces. So, in this case one can use the growth rate of the norms of powers of an operator and the growth rate of the resolvent as indicators of complexity. The links between these characteristics are much more involved than in the finite-dimensional setting. They have extensively been studied in parallel with the relations between the properties of an operator, the behaviour of its resolvent, and that of the norms of its powers. One of the first results in this direction (Gelfand [36]) asserts that if $\sigma(B)=\{1\}$ and the norms of positive and negative powers of $B$ are bounded, then $ B=I$. Shilov [37] showed that this result is nontrivial, namely, in the infinite-dimensional setting the conditions that $\sigma(B)=\{1\}$ and the norms of the positive powers alone of the operator are bounded do not imply that $ B=I$. Much attention has been devoted to the case when the norm of the resolvent is estimated in terms of some power of the function ${1}/{d(\lambda,\sigma(B))}$ and, in particular, to the case when the so-called Kreiss condition
$$
\begin{equation*}
\|\mathscr R(\lambda;B)\|\leqslant \frac{\mathrm{const}}{|\lambda|-1}
\end{equation*}
\notag
$$
holds for $ |\lambda|>1 $ (as in (5.6); see, for example, [38]–[41]). In infinite-dimensional spaces a resolvent can grow faster than some power of ${1}/{d(\lambda,\sigma(B))}$. For example, Carleman showed that if $A$ is a Hilbert-Schmidt operator and the spectrum of the operator $ B=I -A$ is the singleton $\{1\}$, then only the exponential estimate of the form
$$
\begin{equation}
\|\mathscr R( \lambda; B) \| \leqslant C\exp\biggl(\frac{\rho}{| \lambda-1|^2}\biggr)
\end{equation}
\tag{6.1}
$$
holds for the resolvent. Moreover, it is believed to be known that for an arbitrary operator the resolvent can increase arbitrarily fast as the spectral parameter approaches the spectrum. A fast growth of the resolvent of an operator in a particular class shows this class contains fairly involved operators. Information about the growth rate of the resolvent is relevant to many problems. For example, one of the classical problems in operator theory calls for a construction of functional calculus, that is, for defining functions $f(B)$ of a given operator $B$ for $f$ in some class of functions. This class of suitable functions depends substantially on the properties of the operators under consideration. The most advanced functional calculus was constructed for unitary and self-adjoint operators in Hilbert spaces, in which a function of an operator is defined in the class of all bounded Borel functions on the spectrum of the operator. In Riesz’s functional calculus, which is constructed for arbitrary bounded operators, a function of an operator is defined only for functions which are analytic in a neighbourhood of the spectrum. Dyn’kin [42] constructed a functional calculus in which a function $f(B)$ of an operator is defined for infinitely differentiable functions $f$ whose derivatives satisfy certain inequalities depending on the growth rate of the resolvent of the operator. 6.2. Lower estimates the function majorizing the resolventIf $ R(B)= 1$, then for $|\lambda|> 1$ the resolvent is defined as a series in powers of ${1}/{\lambda}$:
$$
\begin{equation}
\mathscr R(\lambda;B)=(B-\lambda I)^{-1}=-\sum_{n=0}^{\infty}\frac{1}{\lambda^{n+1}}B^n.
\end{equation}
\tag{6.2}
$$
Usually, the behaviour of an analytic function can be described in terms of majorants. For the resolvent, this is the function
$$
\begin{equation}
{M_B}(r)=\max_{|\lambda|=r}\|\mathscr R(\lambda;B)\|.
\end{equation}
\tag{6.3}
$$
Lemma 3. Let $B$ be a bounded linear operator and let $ R(B)=1$.Then for $r>1$ the function (6.3) is estimated in terms of functions constructed using only the norms of positive powers of the operator: here Proof. The upper estimate is clear. Since the resolvent is analytic, we have the Cauchy inequality From this inequality, for fixed $n$, we have an upper estimate for the norms of powers in terms of the function $ M_B(r)$: On the other hand, for fixed $r$ inequality (6.6) gives the required lower estimate in (6.4). The maximum in (6.5) exists because $ \|B^{n-1}\| /r^{n}\to 0$ as $n \to \infty$. Lemma 3 is proved. Unlike in the finite-dimensional case, the behaviour of the resolvent is not uniquely determined by the sequence $\|B^n\|$ because of the gap between the lower and upper estimates in (6.4): in general, $\psi_B(r)$ grows slower than $\psi^B(r)$ as ${r \to 1+0}$. 6.3. On the growth of the resolvent of a weighted shift operator at a prescribed rateSetting $\xi=\ln r$, consider the function
$$
\begin{equation}
\Psi_B(\xi)=\ln \psi_B(e^{\xi})=\sup_{n\geqslant 1} (-n\xi+b_n), \qquad b_n=\ln \|B^{n-1}\|.
\end{equation}
\tag{6.7}
$$
We assume that $r>1$, and so $\xi>0$. The function $\Psi_B(\xi)$ is defined for all $\xi>0$ if and only if $b_n/n\to 0$. In this case $\Psi_B(\xi)$ is a convex piecewise linear function with negative integer slopes and its graph is a convex polygonal line. We consider this function on the half-open interval $(0,\xi_{1}]$. Leaving aside the meaning of the $b_n$ for a while, we assume that $ b_1=0 $, so that for $n\geqslant 2$ this is a sequence of positive numbers (denoted by $B$). Let $F(\xi) $ be a decreasing positive convex function defined on some half-open interval $(0,\xi_{1}]$ and tending to $+ \infty$ as $\xi \to +0$. On $(0,\xi_{1}]$ we construct a piecewise linear function, which will be referred to as a polygonal line inscribed in the graph of $F$. We proceed as follows. The vertices of the polygonal line are the points $(\xi_n; F(\xi_n))$, $n=2, 3,\dots$, $\xi_{n}\leqslant \xi_{n-1}$, that is, they lie on the graph of $y=F(\xi)$ from right to left (some of these points can coincide). The slope of the link with vertices $(\xi_{n-1}; F(\xi_{n-1}))$ and $(\xi_n; F(\xi_n))$ is $-n$ (provided that these vertices are distinct). The vertices are constructed recursively, starting from the point $({\xi_{1}; F(\xi_1)}) $: $\bullet$ if the left-hand derivative $F'_{-}(\xi_{n-1})\leqslant -n$, then $\xi_n=\xi_{n-1} $; $\bullet$ if $F'_{-}(\xi_{n-1})> -n$, then $(\xi_n; F(\xi_n))$ is the point of intersection of the graph of $y=F(\xi)$ with the straight line $y-F(\xi_{n-1})=-n(\xi-\xi_{n-1})$ that lies to the left of $\xi_{n-1}$; the point of intersection exists (and is in fact unique) as $F'_{-}(\xi_{n-1}) > {-}n$, so that in some left-hand punctured neighbourhood of $\xi_{n-1}$ the graph of $y=F(\xi)$ lies strictly below the straight line, and furthermore, $F(\xi)\to +\infty$ as $\xi\to 0$. Note that in both cases Indeed, in the first case $F'_{-}(\xi_{n})=F'_{-}(\xi_{n-1})\leqslant -n$. In the second case $F'_{-}(\xi_{n})\leqslant F'_{+}(\xi_{n})\leqslant -n$ (in the actual fact, $<-n$), because in some right-hand punctured neighbourhood of $\xi_{n}$ the graph of $y=F(\xi)$ lies below the chord with slope $-n$.We set $\widehat b_1=0$, and for $n\geqslant 2$ we define $y=\widehat {b}_n$ is the point of intersection of the support line with slope $-n$ for the polygonal line thus constructed with the ordinate axis. By construction $\widehat {b}_n>0$.Lemma 4. The polygonal line inscribed in the graph of $F$ is well defined on the interval $(0,\xi_{1}] $ as the graph of the function It has the following properties. 1. The abscissas of vertices of the inscribed polygonal line tend monotonically to zero: 2. The sequence $\widehat B=\{\widehat b_n\}$ increases unboundedly, and the sequence $\{\widehat b_n/(n-1)\}$ tends monotonically to zero. 3. If $B=\{b_n\}$ is a sequence such that $b_n\,{\geqslant}\, \widehat{b}_n$ for all $n$, then on the interval $(0,\xi_1)$
$$
\begin{equation*}
\Psi_B(\xi)\geqslant \Psi_{\widehat B}(\xi)\geqslant F(\xi).
\end{equation*}
\notag
$$
4. If $B=\{b_n\}$ is a sequence such that there exists $k\geqslant 2$ such that $b_k\geqslant \widehat{b}_k$, then for $\xi\in [\xi_{k},\xi_{k-1}]$ in particular, $ \Psi_B(\xi_k) \geqslant F(\xi_k) $.Remark 3. The algorithm for constructing the polygonal line and the proof of the lemma are based on the following facts: the continuity of the function $F(\xi)$, which is convex on the interval, the existence of left-hand and right-hand derivatives of this function at each point, which are related by $F'_{-}(\xi)\leqslant F'_{+}(\xi)$, and the nondecreasing monotonicity of both derivatives. At the right-hand endpoint $\xi_{1}$ the derivative $ F'_{-}(\xi_1) $ exists and is continuous because $F$ is decreasing and convex. Proof of Lemma 4. By the definition of $\widehat b_n$, the link with endpoints $(\xi_{n},F(\xi_{n}))$ and $(\xi_{n-1},F(\xi_{n-1}))$ (even when it degenerates into a point) lies on the straight line $y=-n\xi+\widehat b_n$. Hence, as the polygonal line is convex, we have a formula for the corresponding piecewise linear function.
That the piecewise linear function is defined on the whole of $(0,\xi_1] $ follows from property 1 of the polygonal line. 1. That $\xi_{n}$ is monotone follows from the construction. From (6.8) we see that $F'_{-}(\xi_{n})\to -\infty$, and therefore, since $F'_{-}(\xi)$ is nondecreasing on the interval $(0,\xi_1)$, we get that $\xi_{n}$ tends to zero. 2. From (6.9) we obtain $\widehat b_n> F(\xi_{n})$, so that $\widehat b_n\to +\infty$. Let us show that By construction
$$
\begin{equation*}
F(\xi_{n})=F(\xi_1)-2(\xi_2-\xi_1)-\dots-n(\xi_n-\xi_{n-1}).
\end{equation*}
\notag
$$
Simple transformations yield
$$
\begin{equation*}
\begin{gathered} \, \widehat{b}_n=F(\xi_{n})+n\xi_{n}=F(\xi_1)+2\xi_1+\xi_2+\dots+\xi_{n-1}, \\ \frac{\widehat{b}_n}{n-1}=\frac{F(\xi_1)+\xi_1}{n-1}+\frac{\xi_1+\dots+\xi_{n-1}}{n-1}. \end{gathered}
\end{equation*}
\notag
$$
It is clear that the first term tends monotonically to zero. The second term is nonincreasing since so is the sequence $\xi_{n}$:
$$
\begin{equation*}
\begin{aligned} \, &\frac{\xi_1+\dots+\xi_{n}}{{n}}=\frac{\xi_1+\dots+\xi_{n-1}}{{n-1}}\,\frac{n-1}{n} +\xi_{n}\,\frac{1}{{n}} \\ &\qquad \leqslant\frac{\xi_1+\dots+\xi_{n-1}}{{n-1}} \biggl({\frac{n-1}{n}+\frac{1}{n}} \biggr) =\frac{\xi_1+\dots+\xi_{n-1}}{{n-1}}. \end{aligned}
\end{equation*}
\notag
$$
By Stolz’s theorem, the limit of the last expression is $\lim_{n \to \infty}\xi_{n-1}=0$.
3. For any $n$ the condition $b_n \geqslant \widehat b_n$ implies that $-n\xi+b_n\geqslant -n\xi+\widehat b_n$ for any $\xi$, hence
$$
\begin{equation*}
\Psi_B(\xi)=\sup_{n\geqslant 1} (-n\xi+b_n)\geqslant \sup_{n\geqslant 1} (-n\xi+\widehat b_n)=\Psi_{\widehat B}(\xi).
\end{equation*}
\notag
$$
4. For $\xi\in [\xi_{k},\xi_{k-1}]$ (or at the point $\xi_{k}$ if the vertices coincide), the equality $ \Psi_{\widehat B}(\xi)=\sup_{n\geqslant 1} (-n\xi+\widehat b_n)=-k\xi+\widehat b_k$ holds. Hence, if $b_k\geqslant \widehat b_k$, then This proves the lemma.Remark 4. From the point of view of convex analysis (see [43]) formula (6.10) means that the piecewise linear function $ \Psi_{\widehat B}(\xi)$ can be expressed as the Legendre transform of some function defined in terms of the numbers $b_n$. So, in the actual fact, Lemma 4 establishes a link between the behaviour of the original function and that if its Legendre transform. Remark 5. In the proofs of the above estimates we used only one property of the resolvent, namely, that the resolvent is a function of $\lambda$ with values in a Banach space and can be expanded in a power series (6.2). Hence similar lower estimates in terms of the norms of coefficients for the majorant (6.3) hold for all such functions, and, in particular, for ordinary analytic functions. Now we can prove our main theorems on the growth rate of the resolvent. First of all we show that, for the resolvents of the weighted shift operators (5.1) generated by irrational rotations, the norm is invariant under rotations in the complex $\lambda$-plane. Proof. If $S_k$ is the operator of multiplication by the function $\exp(i 2\pi k x)$, then
$$
\begin{equation*}
S_k^{-1}B S_k=\omega_k B, \quad \text{where }\ \omega_k=\exp(i 2\pi k h).
\end{equation*}
\notag
$$
For a fixed point $\lambda_0$, from the equality we obtain
$$
\begin{equation*}
\begin{gathered} \, (S_k^{-1}(B-\lambda_0 I)S_k)(S_k^{-1}\mathscr R(\lambda_0;B)S_k)=I, \\ (\omega_k B-\lambda_0 I)(S_k^{-1}\mathscr R(\lambda_0;B)S_k)=I \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
(B-\overline{\omega}_k\lambda_0 I)(\omega_k S_k^{-1}\mathscr R(\lambda_0;B)S_k)=I.
\end{equation*}
\notag
$$
Here the last equality means that We have $\|S_k\|=\|S_k^{-1}\|=1$ for all $k \in \mathbb{Z}$, and hence The points $\overline{\omega}_k\lambda_0=\exp(-i 2\pi k h)\lambda_0 $ are dense on the circle $|\lambda|=|\lambda_0|$, hence for all points on this circle the norms are the same. This proves the lemma. In view of this lemma lower estimates for the function $ \Psi_{B}(\xi)$ are also lower estimates for the norm of the resolvent. Theorem 3. Let $F(\xi) $ be a positive decreasing convex function defined on some half-open interval $(0,\xi_1]$ and tending to $+ \infty$ as $\xi \to +0$. Then, for any irrational $ h$, there exists a function $a \in C(\mathbb{T})$ such that the spectral radius of the weighted shift operator $aT_h$ is $R(aT_h)=1$, and, as $|\lambda| \to 1+0 $, its resolvent increases not slower than $e^{F(\ln |\lambda|)}$, namely, there exists $\delta>0$ such that Proof. On the interval $(0,\xi_1]$ we construct a polygonal line inscribed in the graph of $y=F(\xi)$ and the corresponding piecewise linear function $\Psi_{\widehat B}(\xi) $ (more precisely, the set $\widehat B=\{\widehat b_n\}$).
We set $\sigma_{n-1}=\widehat b_n/{(n-1)}$. According to assertion 2 of Lemma 4, this sequence tends to zero monotonically. In this case, by Theorem 1 and Remark 2, given an irrational $ h$, there exists a function $ f \in C(\mathbb{T})$ with zero mean such that, for the Birkhoff sums, for all $n\in \mathbb N$,
$$
\begin{equation*}
b_{n+1}=\max_x f(n,x,h) \geqslant n \sigma_n=\widehat b_{n+1}.
\end{equation*}
\notag
$$
Hence by (5.2), for the function $a(x)=\exp(f(x))$ the weighted shift operator satisfies for all $n\in \mathbb N$. Taking into account that $b_{n+1}=\ln\|B_{h,a}^{n}\|$ we obtain the set $B=\{b_n\}$ of logarithms of the norms of powers of the operator such that $b_n\geqslant \widehat b_n$ for all $n$. Hence, by assertion 3 of Lemma 4, for all $\xi\in (0,\xi_1]$ we have $\Psi_{\widehat B}(\xi)\geqslant F(\xi)$ and
$$
\begin{equation*}
\Psi_{B}(\xi)\geqslant \Psi_{\widehat B}(\xi)=F(\xi).
\end{equation*}
\notag
$$
Setting $1+\delta=e^{\xi_1}$, from (6.4), (6.7) and (6.11) we obtain the required lower estimate for the norm of the resolvent. This proves the theorem. Theorem 4. Let $F(\xi) $ be an arbitrary decreasing positive convex function defined on some interval $(0,\xi_1]$ and tending to $+ \infty$ as $\xi \to +0$. Let $a \in C(\mathbb{T})$, $a(x) \ne 0$ for all $x$, and assume that $f(x)=\ln |a(x)|$ is not a trigonometric polynomial and has zero mean. Then there exist irrational $h$ such that the norm of the resolvent of the weighted shift operator $B_{h,a}=aT_{h}$ increases not slower than $e^{F(\ln |\lambda|)}$ as $|\lambda|\to 1+0$ in the following sense: there exists a sequence of radii $r_k \to 1+0$ such that, for any index $k$, Proof. As in the previous proof, from the function $F(\xi)$ we construct an inscribed polygonal line, the corresponding set of numbers $\widehat B=\{\widehat b_n\}$ and the function $\Psi_{\widehat B}(\xi)$. As before, we set $\sigma_n=\widehat b_{n+1}/n$.
The function $f(x)=\ln |a(x)|$ has zero mean and is not a trigonometric polynomial. Hence by Theorem 2 there exist continuum many irrational $h$ and a subsequence $\{n_k \}$ such that, for any such $h$ and any $k$,
$$
\begin{equation*}
\max_x f({n_k},x,h) \geqslant {n_k} \sigma_{n_k}=\widehat b_{n_k+1}.
\end{equation*}
\notag
$$
We fix one such $h$ and consider the operator $B_{h,a}=aT_h$. According to (5.2), for any $n_k$ Setting $b_{n+1}\!=\!\ln\|B_{h,a}^{n}\|$ we obtain a set of numbers $B\!=\!\{b_n\}$ such that ${b_{n_k+1}\!\mkern-1mu\geqslant\! \widehat b_{n_k+1}}$ for all $n_k$. Next, by assertion 4 of Lemma 4, for all $\xi_{n_k+1}$ we have Setting $r_k=\exp\{\xi_{n_k+1}\}$ we obtain the required estimate. This proves the theorem. 6.4. Exponential estimates for the resolventsEstimates for the resolvent in terms of functions of a form similar to (6.1) (namely, $\exp({\rho}/(|\lambda|-1)^{\gamma})$) are of great value in some problems. In [44] such exponential lower and upper estimates were analyzed, and a sufficiently explicit description of the dependence between the behaviour of the norms of powers of the operator and the growth of the resolvent was given. Let us formulate the corresponding result. Given a function $F(r)$, $r >1$, we introduce for it the concepts of order and type as $ r \to 1$, similarly to the classical concepts of the order and type of an entire function (see [45]). The order $\gamma $ of a function $F(r)$ is defined to be the infimum of the numbers $\alpha$ for which
$$
\begin{equation*}
F(r) \leqslant \exp\biggl(\frac{1}{(r-1)^{\alpha}}\biggr).
\end{equation*}
\notag
$$
The type $\rho=\rho(F)$ of a function $F(r)$ of finite order $\gamma $ is defined to be the infimum of the numbers $\eta$ for which
$$
\begin{equation*}
F(r) \leqslant \exp\biggl(\frac{\eta}{(r-1)^{\gamma}}\biggr).
\end{equation*}
\notag
$$
Similarly, the order (or order of growth) of a sequence $\varphi_n$ is the infimum $\beta$ of the numbers $\zeta$ for which The type of a sequence $\varphi_n$ of finite order $\beta$ is the infimum $\omega$ of the numbers $\eta$ for which In particular, if $ R(B)=1$ and the sequence $\|B^n\|$ has order of growth $\beta$, then $\beta \leqslant 1$, and the class of such operators is singled out by the condition $\beta< 1 $. It turns out that for operators with $\beta< 1$ the difference between the lower and upper estimates in (6.4) is not too large; namely, the functions $\psi_B(r)$ and $\psi^B(r)$ given by (6.5) have the same order and type, even though $\psi_B(r)$ increases slower than $\psi^B(r)$. Theorem 5 (see [44]). Let $B$ be an arbitrary operator such that $ R(B)=1$. The function $M_B(r)$, as given by (6.3), has finite nonzero order $\gamma$ and type$\rho>0$ as ${r \to 1}$ if and only if the sequence $\|B^{n}\|$ has finite order $\beta$, $0<\beta< 1$, and type ${\omega>0}$, where the corresponding orders and types are related by In conclusion, a few words about the role of the above results from the point of view of the general theory of operators are appropriate here. There are certain effects for operators in infinite-dimensional Banach spaces that are not shared by operators in finite-dimensional spaces or by ‘simple’ (for example, self-adjoint) operators. Accordingly, fairly involved operators are required for the demonstration of such effects. Moreover, it turns out that a considerable number of such examples are constructed via weighted shift operators, including those generated by irrational rotations. In particular, in § 5 we give an example demonstrating that the spectral radius can be discontinuous. The result that the resolvent can increase arbitrarily fast has various interpretations. For example, in [46] this result is understood in the following sense: for Toeplitz operators whose spectrum is the unit circle it is shown that, given a sequence of regular values $\lambda_k$ converging to a spectral value and an arbitrary sequence $M_k$ growing arbitrarily fast, there exists a Toeplitz operators $T$ such that $ \|(T-\lambda_k I)^{-1}\|\geqslant M_k$. Theorems 3 and 4 show that the resolvent can grow fast in a stronger sense than in [46], namely, the norm of the resolvent of the weighted shift operator can grow arbitrarily fast not only on a sequence of regular values (as in [46]), but also uniformly in all directions in approaching the circle coinciding with the spectrum of the operator. These facts give another evidence that the weighted shift operators generated by irrational rotations (and admitting a simple representation) can have quite involved structure determined by the nature of the irrationality of the number $h$ and the properties of the coefficient $a$. AcknowledgementThe authors are sincerely grateful to the referee, who pointed out to us a number of references closely related to the topic of this paper and whose comments helped us to improve significantly the structure of the paper and avoid some inaccuracies.
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\Bibitem{AntKocShu22} |
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