Collapsed Anosov flows and self orbit equivalences

Abstract

We propose a generalization of the concept of discretized Anosov flows that covers a wide class of partially hyperbolic diffeomorphisms in 3manifolds, and that we call collapsed Anosov flows. They are related with Anosov flows via a self orbit equivalence of the flow. We show that all the examples from [BGHP] belong to this class, and that it is an open and closed class among partially hyperbolic diffeomorphisms. We provide some equivalent definitions which may be more amenable to analysis and are useful in different situations. Conversely we describe the class of partially hyperbolic diffeomorphisms that are collapsed Anosov flows associated with certain types of Anosov flows.

Partial hyperbolicity and pseudo-Anosov dynamics

“…In this section we introduce some preliminaries and fix notations which will be used later and relate with the objects introduced in the previous section. The reader familiar with [BFFP 3 ,BFP] can safely skip this section, except for Subsection 2.5 where the notion of super attracting fixed point in the universal circle is introduced.…”

“…We remark also that the intersection of Ą W cs and Ą W cu gives rise to a onedimensional branching foliation Ă W c which also has a well defined leaf space (see [BFP,§2.3]). By one dimensional branching foliation T which subfoliates a foliation F we mean a collection of C 1 -curves such that in the universal cover, for every L P r F the curves of r T contained in L have the same properties (i)-(iv) defining two dimensional branching foliations (of course, in (i) one needs to change properly embedded plane to properly embedded line).…”

“…It follows from [FP 2 , Theorem 1.1] (see also [FP 2 , §6]) that a uniform branching foliation is R-covered (here we use the hypothesis that the approximating foliations are Reebless). We will say that a branching foliation is by hyperbolic leaves if the metric on M makes all leaves uniformly Gromov hyperbolic (see [BFP,§A.3]).…”

“…One can compactify L to L " L Y S 1 pLq making it homeomorphic to the closed disk (see [BH,§III.H.3] For several reasons, we will choose a metric in M whose restriction to leaves of F is CATpκq for some κ ă 0 which can be done using the Candel metric, where we approximate by an actual foliation, as opposed to a branching foliation (see [BFP,Proposition A.5]). This is not so important, since there are canonical ways to define visual metrics, but it is convenient to have a geometric sense of what is happening.…”

“…This has challenged our understanding of the topological structure of these systems. This paper aims to solve the classification problem (as formulated in [BFP,Question 1]) completely for some particularly relevant classes of manifolds and isotopy classes of maps.…”