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Volume hyperbolicity and wildness

Published online by Cambridge University Press:  19 September 2016

CHRISTIAN BONATTI  and
KATSUTOSHI SHINOHARA
CHRISTIAN BONATTI
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université de Bourgogne, Dijon 21004, France email bonatti@u-bourgogne.fr
KATSUTOSHI SHINOHARA
Affiliation:
Graduate School of Commerce and Management, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, Japan email ka.shinohara@r.hit-u.ac.jp
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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 and 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 $\text{Diff}^{1}(M)$ 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 in the authors’ previous paper. In order to eject the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partitions, which we introduce in this paper.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 886 - 920
Copyright
© Cambridge University Press, 2016 

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