Diffusion for chaotic plane sections of 3-periodic surfaces

TL;DR

This paper investigates the chaotic behavior of plane sections of triply periodic surfaces, connecting Novikov's problem with systems of isometries, and analyzes their diffusion rates and Lyapunov spectra.

Contribution

It introduces a novel approach linking Novikov's problem to systems of isometries and studies the diffusion and Lyapunov properties of these chaotic sections.

Findings

Derived estimations on diffusion rates of plane sections.

Constructed invariant measures for systems of isometries.

Analyzed Lyapunov spectra of the associated suspension flows.

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.