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

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.