Hyperbolic pseudo-Anosov maps almost everywhere embed into a toral automorphism

Abstract

Fathi and Franks showed that a pseudo-Anosov diffeomorphism $f$ with orientable foliations and dilation coefficient $\lambda$ with no conjugates (over ${\mathbb Q}$) in the unit circle factors onto a (homologically non-trivial) invariant subset of a hyperbolic toral automorphism. After recounting this result, we show that the factor map is either almost everywhere one-to-one or almost everywhere $m$-to-one for some $m>1$ and the pseudo-Anosov map $f$ is an $m$-to-one ramified covering of another pseudo-Anosov (or Anosov) map on a surface of smaller genus. As a corollary, any pseudo-Anosov diffeomorphism with orientable foliations and hyperbolic action on the first homologies almost everywhere embeds into a hyperbolic toral automorphism.

L-cut splitting of translation surfaces and non-embedding of pseudo-Anosovs (in genus two)

“…As we explain below, the map h is constructed by using the idea of global shadowing and is given by rather explicit power series (see e.g. [3]). It is locally injective on the complement of a finite set [11] and is often a.e.…”

“…It is locally injective on the complement of a finite set [11] and is often a.e. injective [3]. (So many pseudo-Anosov are indeed hiding inside hyperbolic toral automorphisms!…”

“…Again, we conjecture that the hypothesis on the genus can be dropped and abelian-Nielsen non-separation holds for all pseudo-Anosov maps. 3 To get a better grasp of the theorem, let us interpret it in terms of the dynamics lifted to the universal coverM and to the homology coverM . These covers are characterized by having deck groups equal to the fundamental group π 1 (M) and the first homology H 1 (M, Z), respectively.…”

“…(This is provided ψ * is surjective, which can always be arranged via Lemma 3.1). To be specific, we refer to the description of the map h given in [3].…”

“…We follow the path blazed by Band [2] and centered on the global-shadowing characterization of the fibers of the would be embedding ψ : M → T N (i.e., the map in Theorem 1.4). Unlike Band we do not assume at the outset that ψ is the Abel-Franks map, as first constructed by Fathi in [11] (see also [3]). However, the arguments readily adapt to this more general setting.…”