Invariant measures involving local inverse iterates

We study new invariant probability measures, describing the distribution of multivalued inverse iterates (i.e of different local inverse iterates) for a non-invertible smooth function f which is hyperbolic, but not necessarily expanding on a repellor Λ. The methods for the higher dimensional non-expanding and non-invertible case are different than the ones for diffeomorphisms, due to the lack of a nice unstable foliation (local unstable manifolds depend on prehistories and may intersect each other, both in Λ and outside Λ), and the fact that Markov partitions may not exist on Λ. We obtain that for Lebesgue almost all points z in a neighbourhood V of Λ, the normalized averages of Dirac measures on the consecutive preimage sets of z converge weakly to an equilibrium measure μ− on Λ; this implies that μ− is a physical measure for the local inverse iterates of f . It turns out that μ− is an inverse SRB measure in the sense that it is the only invariant measure satisfying a Pesin type formula for the negative Lyapunov exponents. Also we show that μ− has absolutely continuous conditional measures on local stable manifolds, by using the above convergence of measures. Several classes of examples of hyperbolic non-invertible and non-expanding repellors, with their inverse SRB measures, are given in the end. Mathematics Subject Classification 2000: 37D35, 37A60, 37D20.