Topological obstructions for robustly transitive endomorphisms on surfaces

Abstract

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.

Robust transitivity and domination for endomorphisms displaying critical points

“…It is feasible that the dominated splitting E ⊕ F provided by Theorem A admits the finest dominated splitting 1 such as E ⊕ k i=1 F i which the derivative Df restricted to the extremal subbundle F k is volume expanding. In our setting, it was proved for surface endomorphism in [LR17]. That result would be as the one obtained for diffeomorphisms in [BDP03].…”

“…i and τ + i are slightly different from the ones in [LR17] (recall definition in (1)). However, it should be noted that m f is the time that ker(Df n ) have maximal dimension in U 0 and Cr κ (f ) is the set such that the kernel of Df m f has maximal dimension.…”

“…Two-dimensional endormorphisms with critical points. Let us quickly comment about the approach in [LR17]. For a robustly transitive surface endomorphism f displaying critical points having nonempty interior, one can define the set Λ consisting of all the orbits (x i ) i which hits on the interior of the critical set infinitely many times for the past and the future.…”

“…Furthermore, it is used the definition of τ − i together with the fact that dim ker(Df n xi ) ≤ 1, for all n ≥ 1, to show that F is Df -invariant. Then, to get that E ⊕ F is a dominated splitting, it is proved in [LR17] that the absence of domination property allows to create a point having full-dimensional kernel for some C 1 -perturbed of f which is incompatible with robust transitivity.…”

“…Although these examples exhibit some form of weak hyperbolicity and are homotopic to a hyperbolic linear endomorphism on T 2 , it was recently that any result about necessary conditions were established. In [LR17], it was proved for surface endomorphisms that a weak form of hyperbolicity is needed for robust transitivity, so-called partially hyperbolic. Furthermore, it was given also several topological obstructions for the existence of robustly transitive in this setting as the only surfaces supporting these kind of maps should be the Torus or the Klein bottle; and the action of a robustly transitive endomorphism on the first homology group has at least an eigenvalue with modulus greater than one, and consequently, together with Mañé's result in [Mañ82], one concludes that there is no robustly transitive endomorphisms homotopic to the identity.…”