Texas A&M University, Department of
Mathematics, 216 Milner Hall, 26th of March 2008, 3:00-3:50
Groups and Dynamics Seminar
Combinatorial
models of expanding self-coverings
Volodymyr Nekrashevych of Texas A&M
University
We will present a notion of a "topological automaton", which is a pair of (orbi)spaces M_1, M, with
a covering M_1 -> M and a map M_1 -> M. These objects generalize
post-critically finite rational functions (of one or more variables),
Moore diagrams of group automata, "topological graph" of Katsura,
subdivision rules, etc. One can iterate topological automata and define
their iterate monodromy groups and Cuntz-Pimsner algebras. If the
topological automaton is contracting, then its iterations converge to
the limit space of its iterated monodromy group in a natural sense.
This observation allows to construct simple models of Julia sets of
expanding branched self-coverings (e.g. post-critically finite rational
functions). As an example I will present a combinatorial model of a
class of post-critically finite endomorphisms of CP^n.