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 )

We address the problem of necessary conditions and topological obstructions for the existence of robustly transitive endomorphisms on surfaces. Concretely, we show that a weak form of hyperbolicity (namely, partial hyperbolicity) is a necessary condition in order to have robustly transitive displaying critical points, and the only surfaces supporting this class of systems are either the torus or the Klein bottle. Furthermore, we also prove that the induced action by a partially hyperbolic endomorphism in the first homology group has at least one eigenvalue with modulus larger than one.

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We address the problem of giving necessary and sufficient conditions in order to have robustly transitive endomorphisms admitting persistent critical sets. We exhibit different type of open examples of robustly transitive maps in any isotopic class of endomorphisms acting on the two dimensional torus admitting persistent critical points. We also provide some necessary condition for robust transitivity in this setting.

We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of T d with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with controlled geometry) and such that their induced action on H 1 (T d ) is hyperbolic. In absence of the simplicity condition we construct a robustly transitive counter-example.