Inverse SRB measures for endomorphisms on surfaces

TL;DR

This paper constructs hyperbolic invariant measures with absolutely continuous disintegrations for certain non-invertible endomorphisms on the two torus, revealing multiple inverse SRB measures and criteria for their uniqueness.

Contribution

It extends Burguet's construction to non-invertible maps, producing new examples of inverse SRB measures that maximize folding entropy on the two torus.

Findings

Constructed hyperbolic invariant measures with absolutely continuous disintegrations.

Obtained examples of topologically mixing maps with multiple inverse SRB measures.

Provided criteria for the uniqueness of inverse SRB measures that maximize folding entropy.

Abstract

We extend D. Burguet's construction of SRB measures for the non invertible scenario obtaining hyperbolic invariant measures with absolutely continuous disintegrations on stable manifolds for a certain class of endomorphisms on the two torus. The constructed measures maximize the folding entropy, in particular, one may obtain such SRB measures for conservative perturbations of the examples given by M. Andersson, P. Carrasco and R. Saghin for which the Lebesgue measure does not maximize the folding entropy. This way, we obtain examples of topologically mixing maps with at least two inverse SRB measures. In the case of inverse SRB measures that maximize the folding entropy, we give criteria for uniqueness.