Texas A&M University, Department of
Mathematics, 216 Milner Hall, 5th of September 2007, 3:00-3:50
Groups and Dynamics Seminar
Symbolic dynamics of tree portraits
Zoran Šunić of Texas A&M
University
We adapt standard notions of symbolic dynamics, such as shift of finite
type and sofic shift, to the case of regular rooted tree portraits. The
connection is rather simple in the sense that everything we say about
the case of a k-regular tree specializes, when k=1, to the usual
(one-sided) symbolic dynamics notion. For instance, closure under the
shift corresponds to self-similarity (closure under the section maps);
a portrait space is closed and self-similar if and only if it is
defined by a set of forbidden patterns; a sofic portrait space is a
space obtained from a portrait space of finite type by a quotient map
and, equivalently, it is a space of tree portraits that can be
recognized by a finite automaton; etc.
We use the above notions to discuss some groups of tree
automorphisms (there is a simple sufficient and necessary condition
that tells when a tree portrait space represents a subgroup of the
group of automorphisms of the k-ary rooted tree). We show that a group
of tree automorphisms is the closure of a self-similar, regular branch
group, branching over its stabilizer of level s-1 if and only if it is
a level transitive group of tree automorphisms defined by forbidden
patterns of size s. Further, we show that a group of tree automorphisms
defined by allowing all tree patterns of some fixed size that appear in
a contracting, self-similar, level transitive group is the closure of a
contracting, self-similar, regular branch group. As an application, we
show that the closed self-similar groups defined by the patterns of
fixed size appearing in the k-ary odometer are not topologically
finitely generated. In particular, we show that there are no
topologically finitely generated, closed, self-similar groups of binary
tree automorphisms defined by forbidden patterns of size at most 2.