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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 $Λ$, we give approximations for the Gibbs states (equilibrium measures) of arbitrary Hölder potentials, with the help of weighted sums of atomic measures on preimage sets of high order. Our endomorphism may have also stable directions on $Λ$, 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 $Λ$), and moreover the number of preimages remaining in $Λ$ may vary; also Markov partitions do not always exist on $Λ$. Our convergence results apply also to Anosov endomorphisms, in particular to Anosov maps on infranilmanifolds.