On a class of stable conditional measures for endomorphisms

TL;DR

This paper investigates stable conditional measures for hyperbolic endomorphisms, demonstrating their geometric and maximal stable dimension properties, and characterizing when they are absolutely continuous, with examples of non-reversible systems.

Contribution

It introduces a detailed analysis of stable conditional measures for hyperbolic endomorphisms, including their geometric nature and conditions for absolute continuity.

Findings

Conditional measures are geometric probabilities and of maximal stable dimension.

They are absolutely continuous iff the basic set is a folded repellor.

Examples of non-reversible systems with these measures are provided.

Abstract

We study stable conditional measures for a certain equilibrium measure for hyperbolic endomorphisms, on basic sets with overlaps; we show that these conditional measures are geometric probabilities and measures of maximal stable dimension. They are also proved to be absolutely continuous if and only if the respective basic set is a folded repellor. Examples of such non-reversible systems and their associated measures are given too.