Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

TL;DR

This paper proves the existence of a spectral gap for certain dynamical systems with contracting fibers and Lorenz-like maps, providing quantitative estimates for their statistical stability under perturbations.

Contribution

It establishes a spectral gap for transfer operators of systems with contracting foliations and Lorenz-like maps, with explicit bounds on statistical stability.

Findings

Spectral gap proven for systems with contracting fibers and Lorenz-like maps.

Quantitative estimate of physical measure variation under perturbations: O(δ log δ).

Asymptotically optimal stability estimate for piecewise smooth maps.

Abstract

We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $\delta$, the physical measure varies continuously, with a modulus of continuity $O(\delta \log \delta )$, which is asymptotically optimal for this kind of piecewise smooth maps.