Fixed points of endomorphisms of virtually free groups

TL;DR

This paper proves a fixed point theorem for endomorphisms of virtually free groups, showing the fixed point subgroup is finitely generated and analyzing the dynamics of fixed points on the hyperbolic boundary.

Contribution

It introduces an automata-theoretic approach to fixed points in virtually free groups and characterizes the stability and orbit structure of boundary fixed points.

Findings

Fixed point subgroup is finitely generated.

Regular fixed points have finitely many orbits under finite fixed points.

Boundary fixed points are either stable attractors or repellers.

Abstract

A fixed point theorem is proved for inverse transducers, leading to an automata-theoretic proof of the fixed point subgroup of an endomorphism of a finitely generated virtually free group being finitely generated. If the endomorphism is uniformly continuous for the hyperbolic metric, it is proved that the set of regular fixed points in the hyperbolic boundary has finitely many orbits under the action of the finite fixed points. In the automorphism case, it is shown that these regular fixed points are either exponentially stable attractors or exponentially stable repellers.