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

Abstract

We give approximations for the Gibbsstates of arbitrary Hölder potentials $\phi$,with the help of weighted sums of atomic measures on preimage sets, in the case of smooth non-invertible maps hyperbolic onfolded basic sets $\Lambda$. The endomorphism may have also stable directions on$\Lambda$ and is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds(instead they depend on the whole past and may intersect each other both inside and outside $\Lambda$). We consider here simultaneously all $n$-preimages in $\Lambda$ of a point, instead of the usual way of taking only the consecutive preimages from some given prehistory. We thus obtain the weighted distribution of consecutive preimage sets, with respect to various equilibrium measures on the saddle-type folded set $\Lambda$. In particular we obtain the distribution of preimage sets on $\Lambda$, with respect to the measure of maximal entropy. Our result is not a direct application of Birkhoff Ergodic Theorem on the inverse limit $\hat \Lambda$, since the set of prehistories of a point is uncountable in general, and the speed of convergence may vary for different prehistories in $\hat \Lambda$. For hyperbolic toral endomorphisms, we obtain the distribution of the consecutive preimage sets towards an inverse SRB measure, for Lebesgue-almost all points.