Combinatorial models of expanding 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.

The new version differs substantially from the first one. Many parts are moved to other (mostly future) papers, the main open question of the first version is solved

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• Finitely presented groups associated with expanding maps
• From rubber bands to rational maps: A research report
• Post-Critically Finite Maps on ${\mathbb{P}}^n$ for $n\ge 2$ are Sparse
• The Julia set of a post-critically finite endomorphism of PC^2
• A solution to the degree-d twisted rabbit problem
• An algebraic characterization of expanding Thurston maps
• Mating, paper folding, and an endomorphism of PC^2
• The Polyhedral Tree Complex