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

Abstract

The paper is devoted to the questions connected with the investigation of the S.P. Novikov problem of the description of the geometry of level lines of quasiperiodic functions on a plane with different numbers of quasiperiods. We consider here the history of the question, the current state of research in this field, and a number of applications of this problem to various physical problems. The main attention is paid to the applications of the results obtained in the field under consideration to the theory of transport phenomena in electron systems.

Arnoux-Rauzy interval exchanges

“…This kind of interval exchange transformations was pointed out (in a much more general context, in which Arnoux-Rauzy interval exchange transformations can be considered as a kind of baby example) by S.P. Novikov, see [33] [21] [31], also [19] [20] [17]. Novikov's problem is stated in terms of foliations on surfaces and not in terms of interval exchange transformations.…”

Arnoux-Rauzy interval exchange transformations

“…This kind of interval exchanges was pointed out (in a very different context and language) by S.P. Novikov, see [32][21] [30], also [19] [20][17]. This prompted several authors to make deep studies of the Rauzy gasket in [29] (Lemma 5.9, attributed to J.-C. Yoccoz) [6] [8] [9] [26], partially solving a Date: June 19, 2019.…”