Pseudo-Anosov flow and dynamics on guts

Abstract

We relate the dynamics on a closed 3-manifold to the topological invariant, homology guts, proposed by Agol and Zhang. Given a closed manifold $M$ that admits a pseudo-Anosov flow $φ$ without perfect fits, we construct a canonical semiflow on the homology guts associated to homology classes carried by $φ$. We show that, up to orbit equivalent outside half annuli, the semiflow on the guts is invariant on each Thurston open cone. As an application, the fundamental group and the sutured structure of homology guts encode closed orbits of $φ$ with particular type. Besides, for a positive class in $H^{1}(M)$, the canonical semiflow on the guts has a well-defined growth rate.