Out($F_r$) train track automata I: Proper full fold decompositions

TL;DR

This paper introduces train track automata for certain irreducible elements of Out($F_r$) and explores their associated geodesics in Outer Space, enhancing understanding of automorphism dynamics.

Contribution

It presents a new framework for analyzing fully irreducible automorphisms using train track automata and their geodesic representations in Outer Space.

Findings

Construction of train track automata for large classes of automorphisms

Description of associated geodesics in Outer Space

Insights into automorphism dynamics and structure

Abstract

We describe train track automata for large classes of fully irreducible elements of Out($F_r$), and their associated geodesics in Culler-Vogtmann Outer Space.