Robust transitivity for endomorphisms admitting critical points

Abstract

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.

Topological obstructions for robustly transitive endomorphisms on surfaces

“…However, it was not known, until now, if the existence of robustly transitive endomorphisms displaying critical points implies topological obstructions. Although some examples of robustly transitive local diffeomorphisms in the torus T 2 and the Klein bottle K 2 (as expanding endomorphisms 2 for instance) are well-known, examples of robustly transitive endomorphisms admitting critical points first appear in [BR13] and [ILP16] on the torus T 2 ; and recently in [LR19], we built a new class of robustly transitive endomorphisms exhibiting persistent critical points on the 2-torus and on the Klein bottle, based on the geometric construction developed by Bonatti-Díaz in [BD96] to produce robustly transitive diffeomorphisms. The proof of Theorem B for endomorphisms displaying critical points use a classical argument provided the existence of continuous subbundles over M , which follows from Theorem A, and it will be presented in Section §4.…”

Robust transitivity and domination for endomorphisms displaying critical points

“…In the endomorphism displaying critical points setting, the first examples were given in [BR13] and [ILP16]. 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.…”

New classes of $C^1$ robustly transitive maps with persistent critical points

“…Some examples of robustly transitive maps admitting persistent critical points are exhibit in [BR13] and [ILP16]. However, these examples are in the homotopy class of a linear endormorphism with real eigenvalues such that |µ| ≤ 1 < |λ|.…”

“…Some comments are in order. In [ILP16] was constructed some examples that are isotopic to expanding linear maps, but there is a mistake in the proof when showing that the new map is robustly transitive. The usual argument for proving transitivity is to take any two open sets and to show there is an iterate of one of them such that intersects the other one.…”