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This article is cited in 3 scientific papers (total in 3 papers) Geometry of quasiperiodic functions on the plane I. A. Dynnikova, A. Ya. Mal'tsevb, S. P. Novikovab a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences
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     To Iskander Taimanov on his 60th birthday 1. IntroductionThe theory of quasiperiodic functions originated in works of Bohr and Besikovich [1], [2]; it plays an important role in the description of a great variety of phenomena in various fields of theoretical and applied science. As a rule, a quasiperiodic function $f(x^{1},\dots, x^{n})$ with $N$ quasiperiods in the space $\mathbb{R}^{n}$ is the restriction of a ‘good enough’ (for example, smooth) $N$-periodic function $F(z^{1}, \dots, z^{N})$ to the image of the space $\mathbb R^n$ under some affine embedding $\iota\colon\mathbb{R}^{n} \hookrightarrow \mathbb{R}^{N}$, that is, $f=F\circ\iota$. In the case of a generic embedding $\iota$ the function $f(x^{1}, \dots, x^{n})$ has no exact periods in $\mathbb{R}^{n}$. Such periods appear when the intersection of the subspace $\iota(\mathbb R^n)$ with the lattice $\mathbb Z^N$ is not empty. It can also turn out that the image $\iota(\mathbb R^n)$ lies in a non-trivial subspace having an integral direction. In this case the number of quasiperiods of $f$ is less than $N$. In this paper both the generic cases and some special cases of quasiperiodic functions corresponding to embeddings $\iota\colon \mathbb{R}^{n} \hookrightarrow\mathbb{R}^{N}$ of various types are important to us. It is well known that the theory of quasiperiodic functions is extremely important in the description of quasicrystals. In this context, the physically important cases are $n = 2$ and $n = 3$, and $N$ is most often equal to $2n$. In addition, the theory of quasiperiodic functions underlies the description of solutions to integrable dynamical systems (both finite- and infinite-dimensional). In this survey we consider qualitative questions of the geometry of quasiperiodic functions on the plane. We mean by this the global behaviour of their level curves $f(x,y) = \mathrm{const}$, which plays a very important role in the description of a large number of physical phenomena. The problem of the qualitative description of the geometry of level curves of quasiperiodic functions on the plane (Novikov problem) is quite non-trivial, and its complexity grows rapidly with the number of quasiperiods. Here we attempt to give an overview of the most recent results obtained in this area. The most fundamental, from the point of view of physical applications, is the Novikov problem for functions with three quasiperiods. This problem was first stated in [3] and can also be considered as the problem of the qualitative description of the geometry of intersections of an arbitrary two-dimensional periodic surface in $\mathbb{R}^{3}$ with a family of planes having a fixed direction (see Fig. 1). In this formulation, the Novikov problem is related most directly to the description of galvanomagnetic phenomena in metals in a constant uniform magnetic field at low temperature. The role of the periodic surface is played here by the Fermi surface in the space of quasimomenta. The intersections of the Fermi surface with planes orthogonal to the magnetic field determine the geometry of quasiclassical electron trajectories in this space, and the corresponding quasiperiodic functions $f(x,y)$ arise as the restrictions of a periodic (three-dimensional) dispersion relation $\epsilon(\mathbf p)$ to these planes. As shown previously in a number of important examples (see [4]–[7]), the behaviour of transport phenomena (magnetic conductivity) in a metal, in the limit of strong magnetic fields, depends most significantly on the geometry of the trajectories described, which allows one to conduct their experimental study. The most interesting magnetic transport phenomena in this case are associated with the presence of non-closed (open) electron trajectories on the Fermi surface; therefore, the classification of open level curves of the corresponding functions $f(x,y)$ is most important. To date, the Novikov problem has been investigated most deeply in the case of three quasiperiods (see [8]–[14]). In particular, now we have a qualitative classification of open level curves of the functions $f(x,y)$, which establishes their division into topologically regular [8], [9], [11] and chaotic ones [10], [12], [13]. A detailed study of topologically regular level curves has made it possible to introduce non-trivial topological characteristics associated with them, which were previously unknown. Such characteristics have the form of irreducible integer triples $(m^{1}, m^{2}, m^{3})$ and are defined for each stable family of topologically regular level curves (trajectories). In [15] and [16] these characteristics were defined as new topological numbers observed in the conductivity of normal metals with sufficiently complex Fermi surfaces. As regards chaotic level curves of functions $f(x,y)$ with three quasiperiods, their existence had not been known before [10], [12], and [13]. All such level curves are unstable with respect to small variations of the problem parameters, and their geometry is very complicated. Chaotic level curves can be divided into two main types, Tsarev-type trajectories and Dynnikov-type ones. The description of the global behaviour of the trajectories in the second case is particularly difficult, and the appearance of such trajectories on the Fermi surface leads to the most non-trivial, not considered previously, behaviour of the magnetic conductivity in strong magnetic fields [17], [18]. The study of chaotic level curves of functions with three quasiperiods has been continuing since their discovery (see, for instance, [19]–[36]). In this article we try, in particular, to give the most detailed description of the current state of this area. The theory of galvanomagnetic phenomena in metals, however, is not the only field of physical applications of the general Novikov problem. It is natural that numerous applications of the theory of quasiperiodic functions on the plane arise also in the physics of two-dimensional systems (see [37]–[42]). As a rule, quasiperiodic functions $f(x,y)$ play the role of potentials in which the dynamics of particles localized in two dimensions is observed. Here applications of the Novikov problem are related primarily to the description of transport phenomena in such systems (both in the presence of a magnetic field and without it). In the description of such phenomena, both the geometry of the level lines of the potential $f(x,y)$ (see, for example, [38]) and the geometry of the domains $f(x,y)\leqslant\epsilon_{0}$ (see [42]), which is related directly to the geometry of level curves, can come to the forefront. In addition to purely applied relevance, the study of the Novikov problem also has an important general theoretical value. Namely, quasiperiodic potentials on the plane that have a sufficiently large number of quasiperiods can be considered as a link between ‘ordered’ and random potentials, as they have the properties of potentials of both types. To describe random potentials authors use various models, which can differ from one another in a number of properties. Some of the features of random potentials, however, are usually considered universal and related to the behaviour of level curves of the potential. In particular, random potentials are characterized by the presence of open level curves only at a single level of energy $V(x,y)=\epsilon_{0}$, while at the other energies all level curves are closed (see [43]–[46], for example). Open level curves of random potentials have a rather complex geometry as a rule, wandering around the plane in a chaotic manner. Considering the Novikov problem from the point of view of random potential models, even in the case of three quasiperiods one can observe both rich families of ‘regular’ potentials (having topologically regular open level lines in a finite energy interval) and non-trivial examples of ‘random’ potentials (with chaotic level curves existing only at a single energy level). Experimental techniques make it possible as a rule to create families of quasiperiodic potentials with a prescribed number of quasiperiods, depending on a finite number of parameters $\mathbf U = (U^{1}, \dots, U^{N})$. In most of these cases the results of the study of the Novikov problem for three quasiperiods lead to a universal (albeit rather non-trivial) description of the sets of ‘regular’ and ‘random’ potentials within the full family $V(x,y,\mathbf U)$. Namely, the parameter space contains an everywhere dense subset consisting of domains with piecewise smooth boundaries, each of which is a ‘stability zone’ and corresponds to potentials with topologically regular level curves. Each stability zone is determined by values of the topological invariants $(m^{1}, m^{2}, m^{3})$ specific to it. The complement of the union of all stability zones in the parameter space $\mathbf U$ is a set of fractal type which parametrizes potentials with chaotic level curves. This set can be regarded as a realization of the model of quasiperiodic potentials with the properties of random potentials. As the investigations of the Novikov problem with four quasiperiods [47], [48] show, in this case the set of potentials is also naturally divided into the subsets of potentials with topologically regular and chaotic open level curves. As in the case of three quasiperiods, there are topological invariants associated with topologically regular open level curves; now they have the form of irreducible integer quadruples $(m^{1}, m^{2}, m^{3}, m^{4})$. As follows from [47] and [48], in the generic case smooth families of quasiperiodic potentials $V(x,y,\mathbf U)$ with four quasiperiods must also contain an everywhere dense subset that is the union of the stability zones corresponding to potentials with topologically regular level curves, and the fractal complement to this set, which parametrizes the potentials with chaotic level curves. The question of whether chaotic level curves exist in a non-degenerate energy interval or only at a single level of energy remains open for potentials with four quasiperiods. Note that potentials with four quasiperiods are most closely related to the theory of two-dimensional quasicrystals, which we have mentioned above. As for functions with a larger number of quasiperiods, there are practically no rigorous general results for them at the moment. For any number of quasiperiods it is easy to construct functions that have stable topologically regular level curves. However, the question of whether they are everywhere dense in smooth families of quasiperiodic potentials $V(x,y,\mathbf U)$ for $N>4$ is still open. Also, at the moment there are no rigorous results on the description of chaotic level curves of such potentials. In this article we describe the situation arising here using a number of examples, and we also formulate and prove a number of general statements concerning level curves of functions with an arbitrary number of quasiperiods. 2. Novikov problem in the case of three quasiperiods and angular diagrams of magnetic conductivity in metalsIn this section we dwell on the Novikov problem with three quasi periods and its main application, the description of galvanomagnetic phenomena in metals in the presence of strong magnetic fields. Many key consequences of the study of the Novikov problem for the theory of galvanomagnetic phenomena were already discovered and discussed in a number of papers some time ago (see, for example, [15]–[17] and [49]–[51]). After that, however, a number of new important aspects were revealed, which supplemented the general picture essentially, both in terms of rigorous mathematical results and in the field of applications. In the setting under consideration the role of a periodic function in the ambient space is played by the dispersion relation $\epsilon(\mathbf p)$ in the space of quasimomenta $\mathbf p = (p_{1}, p_{2}, p_{3})$. The function $\epsilon(\mathbf p)$ is periodic with respect to the reciprocal lattice, whose basis vectors $\mathbf a_{1}$, $\mathbf a_{2}$, and $\mathbf a_{3}$ are connected with the basis of the crystal lattice $(\mathbf l_{1},\mathbf l_{2}, \mathbf l_{3})$ by the relations
$$
\begin{equation*}
\mathbf a_{1}=2 \pi \hbar\,\frac{\mathbf l_{2} \times \mathbf l_{3}} {(\mathbf l_{1}, \mathbf l_{2}, \mathbf l_{3})}\,, \quad \mathbf a_{2}=2 \pi \hbar\,\frac{\mathbf l_{3} \times \mathbf l_{1}} {(\mathbf l_{1}, \mathbf l_{2}, \mathbf l_{3})}\,, \quad \mathbf a_{3}=2 \pi \hbar\,\frac{\mathbf l_{1} \times \mathbf l_{2}} {(\mathbf l_{1}, \mathbf l_{2}, \mathbf l_{3})}\,.
\end{equation*}
\notag
$$
In the presence of an external magnetic field a non-trivial semiclassical dynamics of electronic states arises in the space of quasimomenta. It is described by the system
$$
\begin{equation}
\dot{\mathbf p}=\frac{e}{c}[\mathbf v_{\mathrm{gr}} \times \mathbf B ]= \frac{e}{c}[\nabla \epsilon (\mathbf p) \times \mathbf B]
\end{equation}
\tag{2.1}
$$
(see, for example, [7] and [52]–[54]). Geometrically, the trajectories of (2.1) are the intersections of surfaces of constant energy $\epsilon(\mathbf p) = \mathrm{const}$ with planes orthogonal to the magnetic field, or, in other words, by the level curves of the restrictions of $\epsilon(\mathbf p)$ to these planes. Given a dispersion relation $\epsilon(\mathbf p)$, we thus obtain a family of quasiperiodic functions on the plane, with the direction of the magnetic field $\mathbf B$ and the shift of the plane relative to the origin as parameters. (Moreover, for the questions of interest to us the shift does not play any role in the case of a generic direction $\mathbf B$.) Many of our results are formulated for generic functions. This means that the function belongs to some fixed open everywhere dense subset of the space of all smooth functions. The trajectories of system (2.1) in the $\mathbf p$-space can, of course, be closed and open alike. The following property of closed trajectories is specific for level curves of quasiperiodic functions with three quasiperiods. Lemma 2.1 (see [11]). For any fixed direction $\mathbf B$ and value of energy $\epsilon(\mathbf p)=\epsilon_0$ the diameters of all closed trajectories of system (2.1) in the $\mathbf p$-space are bounded by the same constant (depending on $\mathbf B$ and $\epsilon_0$). The value of the corresponding constant, however, can depend on the direction $\mathbf B$ and on $\epsilon_{0}$, and it can become arbitrarily large as these parameters vary. As we have already said, we are interested in phenomena associated with the presence of non-closed trajectories of (2.1). Contributions to the magnetic conductivity come from trajectories in all planes orthogonal to the magnetic field. However, if the direction of $\mathbf B$ is not proportional to an integral direction (that is, to a direct lattice vector), then the image of any plane orthogonal to $\mathbf B$ is everywhere dense in the torus $\mathbb T^3=\mathbb R^3/\mathbb Z^3$. Therefore, in different planes orthogonal to $\mathbf B$ the level curves of the dispersion relation $\epsilon$ behave similarly. We divide open trajectories into two types, topologically regular and chaotic. Topologically regular ones are open trajectories whose projections onto $\mathbb T^3$ are contained in an embedded two-dimensional torus. The topological characteristics of such a trajectory include the possible homology classes this torus can have. (If the trajectory is not periodic, then such a homology class is defined uniquely up to a coefficient.) Any topologically regular trajectory lies in a straight strip of finite width in some plane orthogonal to $\mathbf B$ and traverses it (see Fig. 4 below). The direction of the strip is determined by the topological characteristics of the trajectory. An open trajectory that is not topologically regular is called chaotic. Lemma 2.2 (see [13]). For any fixed direction $\mathbf B$ not proportional to an integral direction, all open trajectories of system (2.1) in all planes orthogonal to $\mathbf B$ have the same type and topological characteristics, which moreover do not depend on the value of energy $\epsilon_{0}$ (provided that not all trajectories at this level of energy are closed). If the direction $\mathbf B$ is integral, then open trajectories of system (2.1) have a relatively simple description, namely, they can only be periodic. We divide the directions $\mathbf B$ into rational (proportional to integral ones), partially irrational (such that the plane orthogonal to $\mathbf B$ contains only one reciprocal lattice vector up to a factor), and generic directions (the plane orthogonal to $\mathbf B$ does not contain non-trivial reciprocal lattice vectors). It is natural in our situation to introduce an angular diagram showing the dependence of the type of open trajectories of system (2.1) on the direction $\mathbf{B}$ of the magnetic field. For simplicity, here by open trajectories we mean not only non-closed non-singular trajectories of (2.1), but also connected complexes in the $\mathbf p$-space that consist of stationary points and separatrices connecting them, provided that they are unbounded (Fig. 2). Similarly, in addition to closed non-singular trajectories of (2.1), by closed trajectories we also mean bounded connected complexes in the $\mathbf p$-space that consist of stationary points and separatrices connecting them (Fig. 3). For this definition Lemma 2.1 remains valid, and the following assertion also holds.  Figure 2.Unbounded connected complexes of stationary points and separatrices of system (2.1) in the $\mathbf p$-space.  Figure 3.Examples of bounded connected complexes of stationary points and sepatarices of system (2.1) in the $\mathbf p$-space. Lemma 2.3 (see [13] and [14]). For any fixed direction $\mathbf B$ the set of values of energy $\epsilon$ for which system (2.1) has open trajectories has the form of a closed interval $[\epsilon_{1}(\mathbf B), \epsilon_{2}(\mathbf B)]$, which can degenerate to a point $\epsilon_{0}(\mathbf B) = \epsilon_{1}(\mathbf B) = \epsilon_{2}(\mathbf B)$. The main feature of the angular diagram describing the open trajectories of system (2.1) is the presence of an everywhere dense set which has the form of a union of ‘stability zones’ each of which is a domain with piecewise smooth boundary on the unit sphere and consists of directions $\mathbf B$ corresponding to topologically regular open trajectories (see [8], [9], [11], [13], and [14]). Namely, the following assertion is true. Theorem 2.4. For every generic dispersion law $\epsilon$ there is an everywhere dense set on the sphere $\mathbb S^2$ which is the union of at most countably many closed domains $\Omega_\alpha$ with the following properties: Thus, given a generic direction $\mathbf B\in\Omega_\alpha$, all topologically regular trajectories of system (2.1) (see Fig. 4) have the same mean direction given by the intersection of the plane orthogonal to $\mathbf B$ with some integral plane $\Gamma_{\alpha}$, which is always the same throughout the fixed stability zone $\Omega_{\alpha}$. This plane is orthogonal to the vector
$$
\begin{equation*}
\mathbf l_{\alpha}=m^{1}_{\alpha}\mathbf l_{1}+m^{2}_{\alpha}\mathbf l_{2}+ m^{3}_{\alpha}\mathbf l_{3}
\end{equation*}
\notag
$$
of the crystal lattice.  Figure 4.A topologically regular open trajectory of system (2.1) in the plane orthogonal to $\mathbf B$ (schematically). In the general case topologically regular open trajectories of (2.1) are quasiperiodic in a certain sense. However, each stability zone $\Omega_\alpha$ contains an infinite set of directions $\mathbf B$ for which the open trajectories of system (2.1) are periodic. This occurs whenever the intersection of the plane orthogonal to $\mathbf B$ with $\Gamma_{\alpha}$ has a rational direction in the $\mathbf p$-space. Note that topologically regular trajectories are naturally divided into stable, unstable, and semi-stable ones. Namely, a topologically regular trajectory is called stable if for any point on it and any open neighbourhood $V$ of this point, for any sufficiently small perturbations of the dispersion law $\epsilon$, the energy level $\epsilon_0$, and thedirection of magnetic field $\mathbf B$, some topologically regular trajectory of the perturbed system intersects $V$. If making an arbitrarily small perturbation of the system we can achieve that a stable topologically regular trajectory traverses $V$, then the original trajectory is said to be semi-stable. In the other cases topologically regular trajectories are called unstable. Using the remarkable properties of topologically regular open trajectories of (2.1) new topological characteristics were introduced in [15] and [16], which can be observed in the conductivity of normal metals whenever stable trajectories of this type are present on the Fermi surface. The introduction of these characteristics is based on certain features of the contributions of such trajectories to the conductivity tensor in the limit of strong magnetic fields, among which the most important is the strong anisotropy of conductivity in the plane orthogonal to $\mathbf B$. Measuring the conductivity in this plane enables one to measure the mean direction of open trajectories in the $\mathbf p$-space directly, as the direction of greatest suppression of the conductivity in the limit as $B\to \infty$. Thus, in the presence of open topologically regular trajectories on the Fermi surface, which are stable under small rotations of $\mathbf B$, measuring conductivity in strong magnetic fields enables us to determine the integer invariants $(m^{1}_{\alpha}, m^{2}_{\alpha}, m^{3}_{\alpha})$ corresponding to the family of open trajectories under consideration. As follows from [14], the full angular diagrams for periodic dispersion relations $\epsilon(\mathbf p)$ can belong to only one of the following two types. Figure 5 shows an example of an angular diagram of the second type. The next statement, from which the above theorem follows, describes a key property of such diagrams, namely, their structure near the boundary of each stability zone.  Figure 5.The stability zones on a sphere for the dispersion relation $\epsilon(\mathbf p)=\cos p_{x}\cos p_{y}+\cos p_{y}\cos p_{z}+ \cos p_{z}\cos p_{x}$ (in the stereographic projection) [55]. Theorem 2.6. Let $\mathbf B$ be a point of smoothness of the boundary of some stability zone $\Omega_{\alpha}$ such that all open trajectories of system (2.1) are periodic. Then $\mathbf B$ also belongs to the boundary of another zone, $\Omega_{\beta}$, such that $\partial\Omega_\beta$ has a corner point at $\mathbf B$, and $\mathbf B$ is an isolated point of intersection of $\Omega_\alpha$ and $\Omega_\beta$. The set of such points is dense on the boundary of each stability zone (see Fig. 6). One can see that the presence of periodic trajectories of (2.1) means that the direction $\mathbf B$ is orthogonal to some vector of the reciprocal lattice, namely, the one such that the trajectories are invariant under shifts by this vector. This means that $\mathbf B$ is contained in a plane spanned by two vectors of the direct lattice. If $\mathbf B\in\Omega_\alpha$, then the vector $\mathbf l_\alpha$ corresponding to the stability zone in question can be taken as one of these vectors, since the mean direction of the open trajectories is orthogonal to it.  Figure 7.The everywhere dense families of directions $\mathbf B$ in stability zones for which system (2.1) has periodic open trajectories: (a) the zone does not contain the vector $\mathbf l_\alpha$; (b) the zone contains the vector $\mathbf l_\alpha$. Thus, on the angular diagram the directions $\mathbf B\in\Omega_\alpha$ for which system (2.1) has periodic trajectories form an everywhere dense union of arcs (geodesics) passing through the point $\mathbf l_\alpha$ (see Fig. 7). The points at which these arcs intersect the boundary of the zone, except the corner points of the boundary, are the ones at which other stability zones adjoin $\Omega_\alpha$. Moreover, if $\mathbf B\in\Omega_\alpha\cap\Omega_\beta$, then the vector $\mathbf B$ is a linear combination of the vectors $\mathbf l_\alpha$ and $\mathbf l_\beta$, and the corresponding geodesic arc through $\mathbf l_\alpha$ extends from $\Omega_\alpha$ to $\Omega_\beta$ (Fig. 8).  Figure 8.Curves of partially irrational directions $\mathbf B$ which correspond to the presence of periodic trajectories of (2.1) in adjacent stability zones. We are not aware of any other general results on the geometry of an individual stability zone. It can be seen from Fig. 5 that stability zones are not necessarily convex. Moreover, stability zones can be non-simply connected: a relevant example can be constructed as follows. Example 2.7. Let the level surface of the dispersion law have the shape shown in Fig. 9. Namely, the surface consists of a family of parallel planes connected by thin curved tubes, contained each in a small neighbourhood of a plane parallel to a fixed plane $\Pi_0$ and symmetric with respect to it. Then any family of parallel planes forming a not too small angle with $\Pi_0$ cuts all these tubes along closed curves. One can see from the constructions in [11] and [13] that in this case all non-closed components of sections of this surface by planes in this family are topologically regular. Thus, in this case all directions $\mathbf B$ that are not too close to the normal to the plane $\Pi_0$ belong to the same stability zone. At the same time, it can be shown that this stability zone does cover the whole sphere. Generally speaking, the functions $\epsilon_1$ and $\epsilon_2$ introduced in Lemma 2.3 need not be continuous on the whole sphere. However, as noted in [14], they become continuous when restricted to the set of completely irrational directions $\mathbf B$; on this set they coincide with the restrictions of some functions $\widetilde{\epsilon}_{1}(\mathbf B)$ and $\widetilde{\epsilon}_{2}(\mathbf B)$ which are defined and continuous on the whole of $\mathbb{S}^{2}$. Inside stability zones these functions are also uniquely characterized by the fact that $(\widetilde\epsilon_1(\mathbf B),\widetilde\epsilon_2(\mathbf B))$ is the largest range of energies $\epsilon_0$ for which system (2.1) has stable topologically regular trajectories. For rational and partially irrational directions $\mathbf B$ the values of $\epsilon_{s} (\mathbf B)$ and $\widetilde{\epsilon}_{s} (\mathbf B)$ may not coincide due to the presence of unstable topologically regular trajectories at energy levels outside the interval $(\widetilde\epsilon_1(\mathbf B),\widetilde\epsilon_2(\mathbf B))$. In this case we always have the inequalities
$$
\begin{equation*}
\epsilon_{1}(\mathbf B)\leqslant\widetilde{\epsilon}_{1}(\mathbf B)\leqslant \widetilde{\epsilon}_{2}(\mathbf B)\leqslant\epsilon_{2}(\mathbf B).
\end{equation*}
\notag
$$
The interior points of the stability zones $\Omega_{\alpha}$ are defined by the relation $\widetilde{\epsilon}_{1}(\mathbf B) < \widetilde{\epsilon}_{2}(\mathbf B)$. The equality $\widetilde{\epsilon}_{1}(\mathbf B) = \widetilde{\epsilon}_{2}(\mathbf B)$ holds on the boundaries of stability zones and at the points of accumulation of an infinite number of zones $\Omega_{\alpha}$ which decrease in size. We would like to describe the structure of complex angular diagrams in slightly greater detail and point out a number of important additional features of such diagrams. We start by considering rational directions of the magnetic field, which, in a certain sense, occupy a special place on an angular diagram. As we have already said, rational directions $\mathbf B$ differ from other directions, in particular, in that the patterns of trajectories in distinct planes orthogonal to $\mathbf B$ can differ significantly from one another when $\mathbf B$ is rational. On the other hand all non-singular open trajectories of (2.1), as well as open trajectories in our generalized sense, are always periodic on such planes. For rational directions $\mathbf B$ it is typical that at some energy levels in the interval $[\epsilon_{1}(\mathbf B), \epsilon_{2}(\mathbf B)]$ there are periodic complexes of stationary points and separatrices (Fig. 2). To such a complex we assign its rank which is the dimension of the sublattice of $H_1(\mathbb T^3;\mathbb Z)$ generated by all separatrix cycles in the image of this complex under the projection onto the torus $\mathbb T^3=\mathbb R^3/\mathbb Z^3$. For example, for the complexes in Fig. 2, (a)–(c), this rank is one, and for the complex in Fig. 2, (d), it is two. For a rational direction $\mathbf B$ the presence of such complexes of rank zero and one is typical, while the presence of complexes of rank two requires some additional conditions (certain symmetry or conditions of codimension 1) to hold. As follows from results in [11], [13], and [14], if, given a rational direction $\mathbf B$, at least for one value of energy
$$
\begin{equation*}
\epsilon \in [\epsilon_{1}(\mathbf B),\epsilon_{2}(\mathbf B)]
\end{equation*}
\notag
$$
there are only regular trajectories and/or complexes of stationary points and separatrices of rank $\leqslant1$ in all planes orthogonal to $\mathbf B$, then this direction $\mathbf B$ lies in the interior of some stability zone $\Omega_{\alpha}$. It is natural to call rational directions $\mathbf B$ of this type ordinary rational directions. We call rational directions $\mathbf B$ such that, for any value of $\epsilon\in [\epsilon_{1}(\mathbf B), \epsilon_{2}(\mathbf B)]$, a complex of stationary points and separatrices of rank 2 exists in at least one plane orthogonal to $\mathbf B$, special rational directions. Special rational directions $\mathbf B$ can appear in various parts of an angular diagram. For example, assume that for some rational direction $\mathbf B$ there is a periodic complex of separatrices shown in Fig. 2, (d), in some plane $\Pi$ orthogonal to $\mathbf B$. This complex in the $\mathbf p$-space contains, up to a shift by a reciprocal lattice vector, two saddle points connected by separatrices. First assume that the gradients of $\epsilon(\mathbf p)$ at these points are co-directed. Then for small parallel shifts of the plane the complex shown splits into closed non-singular trajectories of system (2.1). However, for shifts in opposite directions such trajectories have different (electron or hole) types. It can be shown that in this case the direction $\mathbf B$ lies in the interior of some stability zone $\Omega_{\alpha}$, and the corresponding plane $\Gamma_{\alpha}$ is parallel to $\Pi$. The situation described above provides an example when a stability zone $\Omega_{\alpha}$ contains the direction orthogonal to the corresponding plane $\Gamma_{\alpha}$. In this case the planes orthogonal to $\mathbf B$ can contain periodic trajectories of different directions, periodic trajectories of the same prescribed direction, or no regular periodic trajectories at all. Different cases arising in this situation bring about, in particular, different regimes of the behaviour of the conductivity tensor in strong magnetic fields (see [16], for example). Consider now the situation when the gradients $\epsilon(\mathbf p)$ at inequivalent saddle points in Fig. 2, (d), are opposite to each other. Then, for small parallel shifts of the plane $\Pi$ the complex shown splits into periodic regular trajectories of system (2.1), which are stable under small changes of the energy $\epsilon$, as well as under small rotations of the direction $\mathbf B$ around the mean direction. For close partially irrational directions $\mathbf B$ obtained by such rotations the non-degeneracy of the interval $[\epsilon_{1}(\mathbf B), \epsilon_{2}(\mathbf B)]$ means that such a direction belongs to some stability zone $\Omega_{\alpha}$ or its boundary. It can be seen, therefore, that when the original direction $\mathbf B$ does not belong to any stability zone or its boundary, it always represents a point of accumulation of stability zones on the angular diagram. The above argument is, in fact, of general nature. Thus we can conclude that any special rational direction $\mathbf B$ either belongs to some stability zone (or its boundary) or is an accumulation point of stability zones $\Omega_{\alpha}$ on the angular diagram (see Fig. 10). In particular, the above arguments imply the assertion of Theorem 2.4 that the union of all stability zones $\Omega_{\alpha}$ is everywhere dense on the unit sphere $\mathbb{S}^{2}$. Apart from special rational directions $\mathbf B$, directions $\mathbf B$ for which the open trajectories of system (2.1) are chaotic are points of accumulation of stability zones too. As noted above, for such a direction $\mathbf B$ of the magnetic field chaotic trajectories exist only at one energy level $\epsilon_{0}(\mathbf B)$. As we have also said, the chaotic trajectories of system (2.1) can be divided into two main types: Tsarev-type trajectories and Dynnikov-type trajectories. The former can arise only for partially irrational directions $\mathbf B$ and always have an asymptotic direction in the $\mathbf p$-space [10], [13]. Unlike regular open trajectories, Tsarev-type chaotic trajectories are in general not limited to straight strips of finite width in planes orthogonal to $\mathbf B$ (Fig. 11). The contribution of Tsarev-type trajectories to the magnetic conductivity is, generally speaking, similar to the contribution of topologically regular trajectories, although it also has some special features. A necessary condition for the presence of Tsarev-type trajectories is the existence, on the corresponding level surface of separatrix cycles that are not homologous to zero in the torus $\mathbb{T}^{3}$ (see Fig. 12). As a consequence, the corresponding direction $\mathbf B$ must be orthogonal to some rational direction in the $\mathbf p$-space, but it itself should not be rational. Thus, the directions $\mathbf B$ for which Tsarev-type chaotic trajectories occur are naturally combined into families, each of which is contained in an arc of a great circle lying in some integral plane of the crystal lattice and consists of all partially irrational points on this arc. Rational points on these arcs can be boundary points of stability zones or points of accumulation of stability zones (special rational directions $\mathbf B$); see Fig. 12.A more complex type of chaotic trajectories of system (2.1) are Dynnikov-type trajectories [13]. Trajectories of this type correspond to pronounced chaotic dynamics both in planes orthogonal to $\mathbf B$ (Fig. 13) and on the Fermi surface viewed as a compact surface in $\mathbb T^3$. One consequence of such dynamics is non-trivial regimes of the behaviour of the conductivity in the presence of such trajectories on the Fermi surface [17], [18], particularly noticeable among which is the suppression of conductivity in the direction of the magnetic field, as well as the presence of fractional powers of $B$ in the asymptotics of the conductivity tensor. According to Novikov’s conjecture [49], for a fixed generic dispersion relation the set of directions $\mathbf B$ corresponding to chaotic regimes (of any type) has measure zero and Hausdorff dimension strictly less than 2 on the angular diagram. This conjecture is partially confirmed in the following statement, which is proved in a forthcoming paper by Dynnikov, Hubert, Mercat, Paris-Romaskevich, and Skripchenko. Theorem 2.8. For a generic dispersion relation $\epsilon$ obeying the central symmetry, $\epsilon(\mathbf p)=\epsilon(-\mathbf p)$ and such that all of its level surfaces, as viewed in $\mathbb T^3$, have genus at most 3, the set of directions $\mathbf B$ yielding chaotic regimes has measure zero. The active investigations of chaotic trajectories of Dynnikov type keep going at the present time; below we describe some of the most recent results obtained in this area. Theorem 2.9 (see [29]). Let the surface $\epsilon(\mathbf p)=\epsilon_0$ and the magnetic field $\mathbf B$ be such that the trajectories of system (2.1) are chaotic and the direction $\mathbf B$ is completely irrational. Then in almost all planes orthogonal to $\mathbf B$ the number of open trajectories is the same and equal to either one, or two, or infinity. The least difficult thing is to construct examples where there is exactly one chaotic trajectory in almost every plane orthogonal to $\mathbf B$. For instance, in the case of a surface of genus 3 all ‘self-similar’ examples (see [32]) have this property. In [33] and [34] examples of chaotic regimes in the Novikov problem were constructed, in which each plane orthogonal to $\mathbf B$ contains an infinite number of open trajectories, and these trajectories have an asymptotic direction, but are not contained in straight strips of finite width. This is related to the absence of unique ergodicity of the corresponding foliation on a compact level surface $\epsilon=\epsilon_0$ in $\mathbb T^3$, and requires a very subtle choice of the system parameters. Such a situation is apparently not typical among the chaotic regimes in the Novikov problem. No examples are currently known in which almost every plane orthogonal to $\mathbf B$ contains exactly two chaotic trajectories. We only know that they do not exist in the case of genus 3 (see [29]). In considerations of galvanomagnetic phenomena in metals we must take only those trajectories of (2.1) into account that lie on the Fermi level. Accordingly, it is natural to introduce angular diagrams showing the presence of open trajectories, as well as their type, on the Fermi surface $\epsilon(\mathbf p) = \epsilon_{\mathrm F}$ for different directions of the magnetic field. Such diagrams are, of course, poorer than the diagrams for the total dispersion relation; they contain domains consisting of directions $\mathbf B$ for which all trajectories on the Fermi level are closed (see Fig. 14, for example). The stability zones on such diagrams are defined as the closures of connected components of the set of directions $\mathbf B$ for which system (2.1) has stable topologically regular open trajectories at the level $\epsilon=\epsilon_{\mathrm{F}}$.  Figure 14.The Fermi surface of gold and the corresponding exact mathematical stability zones on the angular diagram [56], [57]. As before, each stability zone $\Omega^{*}_{\alpha}$ on such a diagram is a domain with piecewise smooth boundary on the unit sphere. It is characterized by certain values of the topological invariants $(m^{1}_{\alpha}, m^{2}_{\alpha}, m^{3}_{\alpha})$ and is a subdomain of the corresponding zone $\Omega_{ \alpha}$ defined for the total dispersion relation. However, if we are talking about the full family of open trajectories on the Fermi surface that are associated with a given stability zone $\Omega_{\alpha}$, then the corresponding set of directions $\mathbf B$ goes beyond the zone $\Omega^*_\alpha$. The reason for this is the presence of unstable periodic trajectories for some partially irrational directions $\mathbf B$ located near the boundaries of all zones $\Omega^{*}_{\alpha}$. Such directions form extensions, beyond the boundary of $\Omega^{*}_{\alpha}$, of arcs of partially irrational directions $\mathbf B$ which correspond to the presence of stable periodic trajectories in this zone (Fig. 15). The adjoining segments form an everywhere dense set on the boundary of the stability zone, and their lengths tend to zero as the periods of the corresponding trajectories increase. As shown in [58], such a structure results in rather complex analytic properties of the magnetic conductivity tensor both in the interiors of the zones $\Omega^{*}_{\alpha}$ and near their boundaries. As observed in [59], to determine the exact boundaries of the zones $\Omega^{*}_{\alpha}$ experimentally one should perhaps use methods other than direct measurements of the conductivity in strong magnetic fields. Angular diagrams for particular Fermi surfaces can certainly be very simple. For example, they can entirely consist of directions $\mathbf B$ for which all trajectories of (2.1) are closed, or they can admit only unstable periodic trajectories for some $\mathbf B$ (Fig. 16). The topology of the Fermi surface and its embedding in $\mathbb T^3$ impose some restrictions on the structure of an angular diagram. Such a characteristic as the dimension of the image of the first homology of the Fermi surface in the first homology of $\mathbb{T}^{3}$, plays an important role here. We call it the topological rank of the Fermi surface. It can obviously take values 0, 1, 2, and 3. Angular diagrams containing stability zones can only arise for Fermi surfaces of topological rank 2 or larger, and more than one stability zone can exist only if the topological rank is 3. Note that Fermi surfaces of rank 3 must have genus $g \geqslant 3$.As concerns the existence of directions $\mathbf B$ on angular diagrams for Fermi surfaces that are not contained in any stability zone, but are limit points of the union of all stability zones, such a situation is possible only when the diagram of the total dispersion relation contains an infinite number of stability zones. Such directions $\mathbf B$ will be called singular for the Fermi surface in question. As noted in [60], for the existence of singular directions it is necessary that the Fermi energy $\epsilon_{\mathrm F}$ falls in a rather narrow energy interval determined by the dispersion relation $\epsilon(\mathbf p)$. Namely, consider a fixed periodic function (dispersion relation) $\epsilon(\mathbf p)$ taking values in some interval:
$$
\begin{equation*}
\epsilon_{\min} \leqslant \epsilon (\mathbf p)\leqslant \epsilon_{\max}.
\end{equation*}
\notag
$$
It is easy to see that for values of $\epsilon_{\mathrm F}$ close to $\epsilon_{\min}$ or $\epsilon_{\max}$ the Fermi surfaces are very simple, and the angular diagrams corresponding to them are trivial (all trajectories of system (2.1) are closed). One can introduce values $\epsilon^{\mathcal A}_{1}$ and $\epsilon^{\mathcal A}_{2}$,
$$
\begin{equation*}
\epsilon_{\min}<\epsilon^{\mathcal A}_{1}< \epsilon^{\mathcal A}_{2}<\epsilon_{\max},
\end{equation*}
\notag
$$
such that for the Fermi energy $\epsilon_{\mathrm F}$ in the interval $(\epsilon^{\mathcal A}_{1}, \epsilon^{\mathcal A}_{2})$ the corresponding angular diagrams contains stability zones. The resulting angular diagrams can in turn also be divided into two classes (diagrams of type A and diagrams of type B), which are qualitatively different. Namely, 1) generic diagrams of type A contain only a finite number of stability zones, and throughout the domain corresponding to the presence of only closed trajectories on the Fermi surface the Hall (transverse) conductivity is of the same (electron or hole) type (Fig. 17, (a)); 2) generic diagrams of type B contain an infinite number of stability zones, and the domain corresponding to the presence of only closed trajectories on the Fermi surface has both parts corresponding to electronic Hall conductivity and parts corresponding to hole Hall conductivity (Fig. 17, (b)). For generic dispersion relations we can define a finite energy interval $(\epsilon^{\mathcal B}_{1} , \epsilon^{\mathcal B}_{2})$,
$$
\begin{equation*}
\epsilon^{\mathcal A}_{1} <\epsilon^{\mathcal B}_{1} < \epsilon^{\mathcal B}_{2} <\epsilon^{\mathcal A}_{2},
\end{equation*}
\notag
$$
such that for all values of $\epsilon_{\mathrm F}$ in this interval the corresponding angular diagrams are of type B, while values of $\epsilon_{\mathrm F}$ in the intervals $(\epsilon^{\mathcal A}_{1}, \epsilon^{\mathcal B}_{1})$ and $(\epsilon^{\mathcal B}_{2}, \epsilon^{\mathcal A}_{2})$ correspond to angular diagrams of type A. Generic diagrams of type A do not contain singular directions ${\mathbf B}$. By contrast, generic diagrams of type B must contain such directions. We also note that, since the Fermi levels at which special rational directions $\mathbf B$ can arise have measure zero, for generic values $\epsilon_{\mathrm F}$ in the interval $(\epsilon^{\mathcal B}_{1}, \epsilon^{\mathcal B}_{2})$ singular directions $\mathbf B$ correspond to the existence of chaotic trajectories of Tsarev or Dynnikov type. It was shown in [14] that the measure of the set of directions $\mathbf B$ corresponding to the presence of chaotic trajectories on a generic Fermi surface is zero. According to Novikov’s conjecture [50], [51], for generic surfaces the Hausdorff dimension of this set is strictly less than 1. In conclusion, we note that for actual dispersion relations the length of the interval $(\epsilon^{\mathcal B}_{1}, \epsilon^{\mathcal B}_{2})$ is apparently rather small, and this perhaps explains the current lack of clear evidence of chaotic trajectories of system (2.1) in experiments with real conductors. Of course, it is also possible that since trajectories of this type were unknown until very recently, certain experimental data could not be interpreted appropriately. However, we hope that trajectories of this type and the behaviour of magnetic conductivity corresponding to them will anyway be discovered in the future for suitable classes of conductors among the huge variety of new materials produced currently. 3. General Novikov problem. Statement and resultsAs we said in the introduction, the general Novikov problem is to describe the geometry of open level curves of a quasiperiodic function with an arbitrary number of quasiperiods on the plane. One of the most natural formulations is as follows: describe the level curves of a family of functions obtained as the compositions of a fixed generic $N$-periodic function $F$ on the space $\mathbb{R}^{N}$ with all possible affine embeddings $\iota\colon\mathbb{R}^{2}\to \mathbb{R}^{N}$. The global properties of level curves of interest to us depend mainly on the direction of the plane $\iota(\mathbb R^2)$, which is a point in the Grassmann manifold $G_{N,2}$, and can also depend on the parameters of the shift. The change of affine coordinates in $\mathbb R^2$ does not play any role to us. The most significant result in the Novikov problem for $N>3$ is as follows. Theorem 3.1 (see [47] and [48]). There is an open everywhere dense subset $S \subset C^{\infty}(\mathbb{T}^{4})$ of 4-periodic functions $F$, and for each $F\in S$ there exists an open everywhere dense subset $X_{F} \subset G_{4, 2}$ such that for any $\xi \in X_{F}$ every level curve of the restriction of the function $F$ to any two-dimensional plane having direction $\xi$ in $\mathbb R^4$ is contained in a straight strip of finite width. The widths of these strips and the diameters of compact level curves are bounded above by a constant which depends only on the pair $(F,\xi)$, and any open non-singular level curve traverses the corresponding strip (Fig. 4). The directions of strips containing open level curves are orthogonal to some integer vector $(m^1,m^2,m^3,m^4)$, which is a locally constant function of $(F,\xi)$. It is easy to see that the situation of stable ‘topologically regular’ behaviour of the level curves of a quasiperiodic function on the plane is possible for any number of quasiperiods. For example, it takes place if, as the corresponding $N$-periodic function in $\mathbb R^N$, we take a small perturbation of a periodic function depending on just one coordinate. Then, for the family of quasiperiodic functions obtained as the restrictions of a fixed $N$-periodic function to all possible planes we can also define stability zones, which are open domains in $G_{N,2}$. It should be noted, however, that with the increase of the number of quasiperiods, the shape of topologically regular level curves often becomes more complicated and approaches chaotic behaviour on finite scales. In general, as the number of quasiperiods increases, the Novikov problem approaches the problem of random potentials on the two-dimensional plane. Below we present a number of topological results related to the Novikov problem with an arbitrary number of quasiperiods and generalizing the previously known results for $N=3$. Lemma 3.2. Let $F(z^{1},\dots, z^{N})$ be an $N$-periodic function with respect to some integer lattice in $\mathbb{R}^{N}$, and let $\xi\in G_{N,2}$ and $c\in\mathbb R$ be such that for any two-dimensional affine plane $\Pi$ having direction $\xi$ all level curves $F\big|_\Pi=c$ are compact. Then the diameters of all these level curves are bounded above by the same constant for all planes of direction $\xi$. This fact follows from the compactness of the image of the surface $F=c$ in the torus $\mathbb T^N$: each compact level curve (singular or non-singular alike) has a neighbourhood of finite diameter such that all other level curves $F =c$ in parallel planes that intersect this neighbourhood lie entirely in it. We can choose a finite family of such neighbourhoods so that their images cover the whole image of the surface $F=c$ in the torus $\mathbb T^N$. Note that in the case $N=3$ we do not need the assumption that in all planes having a fixed direction all level curves $F=c$ are compact (see Lemma 2.1). Theorem 3.3. Let $F(z^{1}, \dots, z^{N})$ be an $N$-periodic function, and let $\xi\in G_{N,2}$ be a fixed direction of two-dimensional planes in $\mathbb R^N$. Then the set of values $c\in\mathbb R$ such that for some plane $\Pi$ having direction $\xi$ the level set $F\big|_\Pi=c$ has unbounded components forms a closed interval $[c_1,c_2]$ or consists of one point $c_0$. The proof follows mostly the lines of the proof of a similar assertion for $N=3$ which was given in [13] and [14]. Note that, again, there is some difference from $N= 3$: we consider the whole set of planes having the same direction simultaneously, while in the case of three quasiperiods the statement holds for each individual plane having this direction. In the case of more than three quasiperiods we cannot rule out the situation when all level curves $F\big|_\Pi=c\in[c_1,c_2]$ in some planes $\Pi$ having direction $\xi$ are compact, but their diameters are not bounded above. Note, however, that if $\xi$ is not contained in a hyperplane of rational direction, then for any $c\in[c_1,c_2]$ (or $c=c_0$) and any plane $\Pi$ in direction $\xi$ the level set $F\big|_\Pi=c$ has either unbounded or arbitrarily large closed components. Theorem 3.4. Let $F(z^{1}, \dots, z^{N})$ be an $N$-periodic function, and let $\xi\in G_{N,2}$ be a fixed direction of two-dimensional planes that contains a unique, up to a coefficient, non-zero integer vector and does not lie in a rational hyperplane. Then the following is true for all planes $\Pi$ having direction $\xi$: The proof of the first part follows the proof of a similar assertion for $N=3$ presented in [13], § 6. Now, we take as $\gamma$ a path in $\mathbb R^N$ that lies entirely in some plane having direction $\xi$ and such that the endpoint of $\gamma$ is obtained from its initial point by a shift by $\mathbf w$, an irreducible non-zero integer vector parallel to $\xi$. Among all such paths we choose the one that has fewest points of intersection with the surface ${F=c}$. Next, in the orthogonal complement $\xi^\perp$ we choose a small neighbourhood $W$ of the origin so that a shift of $\gamma$ by a vector in $W$ cannot increase the number of points of intersection with the surface $F=c$. The union $\Gamma$ of all shifts of $\gamma$ by all possible vectors of the form $\mathbf u + \mathbf v$, where $\mathbf u\in W$ and $\mathbf v\in\mathbb Z^N$, cuts each plane having direction $\xi$ into strips of a finite number of shapes, arranged in a ‘quasiperiodic’ order, and the pattern of level curves in each strip is periodic. The rest of the argument does not differ from the case $N=3$. The second and third assertions of the theorem are proved using the same construction. Compact level curves $F=c$ in planes having direction $\xi$ do not intersect $\Gamma$, so that they are contained in strips of bounded width in which the pattern of lines of intersection is invariant under the shift by $\mathbf w$. Hence there is a general upper bound for the diameters of these level curves. The presence of non-periodic unbounded level curves $F=c$ in all planes having direction $\xi$ is equivalent to the non-emptiness of the intersection of the surface $F=c$ with $\Gamma$. This implies the third assertion of the theorem. Note that, similarly to the case $N=3$, the existence of an asymptotic direction of the level curves in the last theorem does not mean that these curves lie in straight strips of finite width.
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\Bibitem{DynMalNov22} |
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