Combinatorial models of expanding dynamical systems

TL;DR

This paper introduces a generalized framework for expanding dynamical systems using iterated monodromy groups, showing that their Julia sets can be modeled as inverse limits of simplicial complexes.

Contribution

It extends the concept of iterated monodromy groups to broader structures and establishes a combinatorial model for their Julia sets.

Findings

Julia sets depend only on the associated iterated monodromy group

Julia sets are inverse limits of simplicial complexes

Generalized models apply to a wider class of dynamical systems

Abstract

We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined "Julia set" of the generalized dynamical systems depends only on the associated iterated monodromy group. We show then that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules.