Robust transitivity for endomorphisms

Abstract

We address the problem about under what conditions an endomorphism having a dense orbit, verifies that a sufficiently close perturbed map also exhibits a dense orbit. In this direction, we give sufficient conditions, that cover a large class of examples, for endomorphisms on the n−dimensional torus to be robustly transitive: the endomorphism must be volume expanding and any large connected arc must contain a point such that its future orbit belong to an expanding region.Date: September 21, 2018. † This work was partially supported by TWAS-CNPq and Universidad de Los Andes. 1 2 C. LIZANA AND E. PUJALSextra hypothesis that we require can be formulated as follows: any arc of diameter large enough have a point such that its forward iterates remain in some expanding region (see Main Theorem below).Before introducing the Main Theorem, we recall some definitions and we introduce some notation that we use throughout this work.An endomorphism of a differentiable manifold M is a differentiable function f : M → M of class C r with r ≥ 1. Let us denote by E r (M ) (r ≥ 1) the space of C r −endomorphisms of M endowed with the usual C r topology. A local diffeomorphism is an endomorphism f : M → M such that given any point x ∈ M, there exists an open set V in M containing x such that f from V to f (V ) is a diffeomorphism.Definition 1.1. We say that a map f ∈ E 1 (M ) is volume expanding if there exists σ > 1 such that |det(Df )| > σ.Observe that volume expanding endomorphisms are local diffeomorphisms. If L : V → W is a linear isomorphism between normed vector spaces, we denote by m{L} the minimum norm of L, i.e. m{L} = L −1 −1 .Definition 1.2. We say that a set Λ ⊂ M is a forward invariant set for f ∈ E r (M )

Contribution to the ergodic theory of robustly transitive maps

“…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 considered in [7].…”

“…Note that Lemma 4.1 proves the robustness of IRG property, which is fundamental to prove the density of the pre-orbit of any point under the perturbed map. For further details, see [7]. After the discussion above, we are now in condition to present a large class of examples that illustrate our main results.…”

“…In particular all periodic points have in fact dense pre-orbits. In fact, it follows from [7] that hypotheses (2)…”

“…Observe that every map satisfying the assumptions of Theorem 4.1 belong to F . Indeed, property (4) can be shown to be a consequence of the hypotheses (ii) and (iii) of Theorem 4.1 ( we refer the reader to [7] for details), while by property (2) all periodic points have dense pre-orbit. Moreover, since the periodic points are dense, there are plenty of them in the expanding region.…”

Robust transitivity and domination for endomorphisms displaying critical points

“…Based on the examples of robustly transitive diffeomorphisms were constructed robustly transitive non-expanding endomorphisms. In [LP13] were obtained necessary and sufficient conditions for robustly transitive local diffeomorphisms. In particular, it is shown that it is not necessary any weak form of hyperbolicity for robustly transitive; a trivial example is an expanding linear endomorphism with complex eigenvalues, which does not admit a dominated splitting.…”