Asymptotic spectral gap for open partially expanding maps

TL;DR

This paper provides a new explicit estimate for the asymptotic spectral gap of open partially expanding maps using a global normal form and semiclassical analysis, extending the diagonal approximation validity.

Contribution

It introduces a novel application of a global normal form and semiclassical techniques to estimate spectral gaps in open dynamical systems.

Findings

Derived explicit bounds for the spectral gap.

Extended the validity of the diagonal approximation.

Applied semiclassical analysis beyond Ehrenfest time.

Abstract

We consider a $\mathbb{R}$-extension of one dimensional uniformly expanding open dynamical systems and prove a new explicit estimate for the asymptotic spectral gap. To get these results, we use a new application of a "global normal form" for the dynamical system, a "semiclassical expression beyond the Ehrenfest time" that expresses the transfer operator at large time as a sum over rank one operators (each is associated to one orbit). In this paper we establish the validity of the so-called "diagonal approximation" up to twice the local Ehrenfest time.