Relative train tracks and generalized endperiodic graph maps

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.