Robust Transitivity for Endomorphisms
Cristina Lizana Araneda
robustness | transitivity | endomorphisms | volume expanding
The main goal of this work is to give some necessary and some sufficient conditions for endomorphisms on compact manifolds without boundary to be robustly transitive. More concretely, under what conditions a differentiable map, not necessarily invertible, having a dense orbit, verifies that a sufficiently close perturbed map also exhibits a dense orbit. In the case of robustly transitive diffeomorphisms is known that a necessary condition is that the tangent bundle admits a dominated splitting. For the case of endomorphisms, that is no longer true. In consequence, conditions that guarantee robustness for transitive endomorphisms cannot depend on the existence of decomposition of the tangent bundle. For local diffeomorphisms, we show that a necessary condition for robust transitivity is to be volume expanding. Although volume expanding is not a sufficient condition to have endomorphisms robustly transitive. Because of this, we must ask for more hypothesis that guarantee robustness. Indeed the additional hypothesis that we ask is: given any arc in a certain region with a large enough diameter to have a point that its future orbit remains in the expanding region, which implies the existence of a locally maximal expanding invariant set for the original system that intersects every arc big enough.