Relative train tracks and generalized endperiodic graph maps

TL;DR

This paper extends the theory of endperiodic maps to infinite graphs, introducing relative train track maps, and shows these maps can be simplified to canonical forms with well-defined eigenvalues and entropy.

Contribution

It defines generalized endperiodic maps and relative train track maps for infinite graphs, proving their existence and canonical properties, adapting techniques from surface homeomorphisms.

Findings

Any generalized endperiodic map is homotopic to a relative train track map.

The Perron-Frobenius eigenvalue is a canonical quantity with a group theoretic interpretation.

The eigenvalue and entropy are minimized within the homotopy class.

Abstract

Motivated by the work of Cantwell-Conlon-Fenley on endperiodic homeomorphisms of infinite type surfaces, we define and study endperiodic and generalized endperiodic maps of an infinite graph with finitely many ends. Adapting the work of Bestvina-Handel to the infinite type setting, we define endperiodic relative train track maps. We prove that any generalized endperiodic map is homotopic to a generalized endperiodic relative train track map, via a combinatorially bounded homotopy equivalence. We show that the (largest) Perron-Frobenius eigenvalue of a relative train track representation of a generalized endperiodic map $f$ is a canonical quantity associated to $f$ as it admits a canonical group theoretic interpretation. Moreover, the (largest) Perron-Frobenius eigenvalue and the topological entropy of a relative train track map is the smallest among its proper homotopy equivalence class.