Inverse pressure estimates and the independence of stable dimension for non-invertible maps

TL;DR

This paper investigates the stable dimension of hyperbolic non-invertible maps using inverse pressure, demonstrating its constancy under certain conditions and establishing Lipschitz continuity of the stable distribution.

Contribution

It introduces inverse pressure estimates for stable dimensions in non-invertible maps and proves their constancy and Lipschitz continuity under openness conditions.

Findings

Stable dimension is constant when the map is open on the basic set.

Inverse pressure provides effective estimates for stable dimension.

Stable distribution is Lipschitz continuous in this setting.

Abstract

We use the inverse pressure concept to estimate the stable dimension for hyperbolic non-invertible maps which are conformal in the stable fibers. The non-invertible case is different than the diffeomorphism case. In particular we show that if the map is open on the respective basic set, then the stable dimension is constant everywhere. We prove also in this setting the Lipschitz continuity of the stable distribution.