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Contribution to the ergodic theory of robustly transitive maps. (English) Zbl 1351.37021

Summary: In this article we intend to contribute in the understanding of the ergodic properties of the set of robustly transitive local diffeomorphisms on a compact manifold without boundary. We prove that \(C^1\) generic robustly transitive local diffeomorphisms have a residual subset of points with dense pre-orbits. Moreover, \(C^1\) generically in the space of local diffeomorphisms with no splitting and all points with dense pre-orbits, there are uncountably many ergodic expanding invariant measures with full support and exhibiting exponential decay of correlations. In particular, these results hold for an important class of robustly transitive maps.

MSC:

37A25 Ergodicity, mixing, rates of mixing
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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