Approximations for Gibbs states of arbitrary Holder potentials on hyperbolic folded sets

TL;DR

This paper develops methods to approximate Gibbs states for hyperbolic non-invertible maps with complex folding behavior, using atomic measures on preimages, applicable to various dynamical systems including Anosov endomorphisms.

Contribution

It introduces a novel approximation technique for Gibbs states in hyperbolic systems with folding, where traditional structures like foliations and Markov partitions may not exist.

Findings

Convergence of atomic measure sums to Gibbs states in complex hyperbolic systems

Applicable to non-expanding endomorphisms with stable directions

Extends to Anosov endomorphisms on infranilmanifolds

Abstract

In the case of smooth non-invertible maps which are hyperbolic on folded basic sets $\Lambda$, we give approximations for the Gibbs states (equilibrium measures) of arbitrary H\"{o}lder potentials, with the help of weighted sums of atomic measures on preimage sets of high order. Our endomorphism may have also stable directions on $\Lambda$, thus it is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (they depend on the whole past and may intersect each other both inside and outside $\Lambda$), and moreover the number of preimages remaining in $\Lambda$ may vary; also Markov partitions do not always exist on $\Lambda$. Our convergence results apply also to Anosov endomorphisms, in particular to Anosov maps on infranilmanifolds.