Abstract
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.
Highlights
Introduction and Statement of the MainResultsIn the last two decades many advances have been made to the study of robustly transitive diffeomorphisms, whose geometric properties are very well understood
It follows from Bonatti, Dıaz, Pujals [1] that robustly transitive diffeomorphisms exhibit a weak form of hyperbolicity, namely, dominated splitting
Let us refer to the construction of SRB measures and maximal entropy measures for the class of DA-maps introduced by Mane, and more recently, [12] proved intrinsic ergodicity(unique entropy maximizing measure) for partially hyperbolic diffeomorphisms homotopic to a hyperbolic one on the 3-torus
Summary
In the last two decades many advances have been made to the study of robustly transitive diffeomorphisms, whose geometric properties are very well understood. We denote by RT ∗ ⊂ RT the open subset of C1 robustly transitive local diffeomorphisms that have no splitting in a C1 robust way. Let SRT ∗ ⊂ RT ∗ denote the set of local diffeomorphisms so that all points have dense pre-orbits. There exists a C1 residual subset R1 ⊂ SRT ∗ such that for every f ∈ R1 the set of hyperbolic periodic points is dense and it admits a periodic source with dense pre-orbit. Concerning our results it is an interesting question to understand if there are robustly transitive local diffeomorphisms such that the set of periodic saddle points is dense while the set of periodic sources is non-empty.
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