Robust transitivity for endomorphisms (Q2842228)

Given a \(C^r\) endomorphism \(f\) of a \(n\)-dimensional torus having a dense orbit, the paper addresses the question of finding sufficient conditions under which a perturbed endomorphism \(g\), close to \(f\) in the \(C^r\) norm, still presents a dense orbit.NEWLINENEWLINEThe sufficient conditions are that the subset of the torus where \(f\) fails to be volume expanding is, roughly speaking, not too big in diameter and all sufficiently large arcs in the expanding region have a point \(x\) for which all iterates \(f^n(x)\) still lie in the expanding region. As the authors remark, it is conjectured that such hypotheses could be weakened, however it is well known that being only volume expanding is not sufficient to imply robust transitivity. On the other hand, the proof shows that robust transitivity is obtained also in the absence of a dominated splitting (a weak form of hyperbolic splitting, see [\textit{C. Bonatti} et al., Ann. Math. (2) 158, No. 2, 355--418 (2003; Zbl 1049.37011)]).NEWLINENEWLINEThe required hypotheses imply that, given \(f\), there exists a constant \(R_f > 0 \), such that given any open set \(U\) there exist \(x \in U\) and a natural number \(k\) such that \(f^k(U)\) contains a ball of radius \(R_f\) centered at \(f^k(x)\). The proof relies on showing that such property is robust in the \(C^r\) sense. To do so, given the expanding region \(V\), it is possible to do a fine analysis of the arcs contained in \( \Lambda = \bigcup_{n \in N} f^{-n}(V)\). By a shadowing-lemma type of argument, the properties of \(\Lambda\) are shared by functions \(g\) \(C^r\)-close to \(f\). The proof is completed by showing that the diameter of the arcs grows as needed under the iterates of \(g\).NEWLINENEWLINEThe paper is completed by four examples of robustly transitive endomorphisms: two of them correspond to partially hyperbolic endomorphisms and two of them, which can be called ``derived from expanding'' show in which sense robust transitivity does not need to rely on the existence of a dominated splitting.