Volume hyperbolicity and wildness

TL;DR

This paper constructs examples of volume hyperbolic quasi-attractors that are simultaneously wild on 3-manifolds, challenging the link between volume hyperbolicity and tame dynamical behavior.

Contribution

It introduces flexible periodic points and partially hyperbolic filtrating Markov partitions to build wild volume hyperbolic quasi-attractors on 3-manifolds.

Findings

Existence of volume hyperbolic wild quasi-attractors on 3-manifolds

Generic diffeomorphisms lack attractors or repellers in certain open sets

New tools for controlling topology of quasi-attractors

Abstract

It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence hence for tameness. In this paper, on any 3-manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed 3-manifold $M$, the space $\mathrm{Diff}^1(M)$ admits a non-empty open set where every $C^1$-generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced by the authors. For ejecting the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partition, which we introduce in this paper.