Diffusion for chaotic plane sections of 3-periodic surfaces

Abstract

We study chaotic plane sections of some particular family of triply periodic surfaces. The question about possible behavior of such sections was posed by S. P. Novikov. We prove some estimations on the diffusion rate of these sections using the connection between Novikov's problem and systems of isometries - some natural generalization of interval exchange transformations. Using thermodynamical formalism, we construct an invariant measure for systems of isometries of a special class called the Rauzy gasket, and investigate the main properties of the Lyapunov spectrum of the corresponding suspension flow.Comment: 31 pages, 11 figures. arXiv admin note: text overlap with arXiv:1311.5361. in Inventiones mathematicae, 201

Topological integrability, classical and quantum chaos, and the theory of dynamical systems in the physics of condensed matter

“…In a more precise formulation, it follows from the papers [50,14] that the properties (1)- (2) hold for open trajectories of (1.1) for the directions of B , sufficiently close to rational directions, while it follows from the paper [15] that properties (1)- (2) hold for open trajectories that are stable with respect to variations of the energy level ǫ(p) = ǫ 0 . It is not difficult to see here that to satisfy conditions (1)-(2) it is sufficient to require either the stability of trajectories with respect to small rotations of the direction of B , or stability with respect to variations of the energy level.…”

“…Let us also note here that for the dynamical system (1.1) the existence of the Zorich -Kontsevich -Forni indices, strictly speaking, requires additional justification and does not follow automatically from the general theory based on certain generic requirements. As an example of such a justification, we can indicate the work [2], in which the construction and investigation of Dynnikov type trajectories was carried out, and the existence of the indicated indices for the Fermi surface of a rather general form was established. We give here a general description of the Zorich -Kontsevich -Forni indices defined in the general case for foliations generated by closed 1-forms on compact surfaces M 2 g .…”

On the Hausdorff dimension of minimal interval exchange transformations with flips

“…The scheme of the proof is as follows: the measure of a subsimplex is proportional to the inverse of the Jacobian, thus one begins by replacing the measures of subsimplices in the sum of equation by the corresponding Jacobians; using the above formula the Jacobians are then replaced by the exponential of the roof function; so we only need to evaluate the following sum: where by we denoted a point of since the roof function is locally constant. The last sum can be evaluated using the exponential tails of the roof function (namely, the convergence of the corresponding integral): first, the exponential tail implies that , where is the set of partition subsets for which is between and (see [, Lemma 17]); then the sum we are interested in it can be estimated from above by a geometric series with ratio .…”

“…where by a we denoted a point of Δ (l) since the roof function is locally constant. The last sum can be evaluated using the exponential tails of the roof function (namely, the convergence of the corresponding integral): first, the exponential tail implies that Card(Y (N )) Ce (n−σ)N , where Y (N ) is the set of partition subsets for which r(a) is between N and N + 1 (see [7,Lemma 17]); then the sum we are interested in it can be estimated from above by a geometric series with ratio e −σ .…”

The Theory of Closed 1-Forms, Levels of Quasiperiodic Functions and Transport Phenomena in Electron Systems

“…For a given value of the magnetic field, we can introduce the characteristic length l (1) B of the trajectory in the p -space that separates two sections that approach at the distance ≤ δp(B) to other sections of the trajectory (l (1) B decreases with the growth of B). At the same time, we can introduce the characteristic length l (2) B passed by the electron along a trajectory in the pspace between two acts of scattering by impurities (l (2) B ∼ Bτ ). It is easy to see that the intraband magnetic breakdown has small effect on the quasiclassical regimes of conductivity that we describe, if we have the condition l B the magnetic breakdown should have a significant effect on the conductivity behavior.…”