Uniqueness of minimal unstable lamination for discretized Anosov flows

TL;DR

This paper proves the uniqueness of minimal unstable laminations for discretized Anosov flows under certain conditions, leading to results on quasi-attractors and extending to some skew-product systems.

Contribution

It establishes the first uniqueness results for minimal unstable laminations in discretized Anosov flows, including non-transitive cases and related skew-product systems.

Findings

Uniqueness of minimal unstable lamination for transitive discretized Anosov flows.

Finiteness of quasi-attractors in non-transitive cases under certain conditions.

Extension of results to specific one-dimensional center skew-products.

Abstract

We consider the class of partially hyperbolic diffeomorphisms $f:M\to M$ obtained as the discretization of topological Anosov flows. We show uniqueness of minimal unstable lamination for these systems provided that the underlying Anosov flow is transitive and not orbit equivalent to a suspension. As a consequence, uniqueness of quasi-attractors is obtained. If the underlying Anosov flow is not transitive we get an analogous finiteness result provided that the restriction of the flow to any of its attracting basic pieces is not a suspension. A similar uniqueness result is also obtained for certain one-dimensional center skew-products.