Texas A&M University, Department of Mathematics

Special Day on

Groups and Dynamics

Fall Workshop, December 5, 2007

Schedule

628 Blocker Building

627 Blocker Building

Organizers: Rostislav Grigorchuk, Volodymyr Nekrashevych, Gilles Pisier, and Zoran Šunić of Texas A&M University

Abstracts

Vitaly Sushchansky of Silesia Technical University, Gliwice, Poland
Semigrops of automatic transformations

The talk is an overview of different constructions and results on semigroups (and inverse semigroups) of transformations defined by finite automata. In particular, some examples of semigroups generated by 2-state automata over a 2-letter alphabet will be discussed, including semigroups of intermediate growth.


Yaroslav Vorobets of Texas A&M University
Stability of periodic billiard trajectories in polygons and polyhedra

A periodic billiard trajectory in a polygon (polyhedron) is called stable if it survives an arbitrary small perturbation of the polygon (resp. polyhedron).  This notion was introduced in relation to the outstanding conjecture that any polygon admits a periodic billiard trajectory.

First I will review the existing results on stability of periodic billiard trajectories in polygons, including recent results by Schwarz and Hooper.  Then I will consider in more detail billiards in polyhedra.

Any periodic billiard trajectory in a polyhedron can be isolated or else it is included into a family of periodic trajectories of  the same length and combinatorial type.  The family can be one-dimensional (a band) or two-dimensional (a beam).  It turns out that neither bands nor beams can survive small perturbations of a polyhedron.  As a consequence, generic polyhedra can have only isolated periodic billiard trajectories.


Sergey Bezuglyi of Institute for Low Temperature Physics, Kharkov, Ukraine and University of Washington
Bratteli diagrams in Cantor dynamics.

Bratteli diagrams play an important role in various areas of mathematics. They were first used in the theory of operator algebras for the classification of some classes of C*-algebras. A.M. Vershik proved that any ergodic automorphism of a measure space is isomorphic to an adic transformation acting on the path space of a Bratteli diagram. The most recent applications of Bratteli diagrams are related to Cantor dynamics. The starting point was the remarkable result of Herman, Putnam, and Skau about conjugacy of any minimal homeomorphism of a Cantor set to the Vershik map defined on a simple Bratteli diagram.

The talk will be focused on the study of aperiodic homeomorphisms of a Cantor set. It turns out that they also admit a realization as Vershik maps of Bratteli diagrams. The case of stationary diagrams is completely studied. It is proved that they are the diagrams arising from the so called substitutional systems. This realization allows us to find an explicit description of ergodic invariant measures for a substitutional dynamical system.


Todor Tsankov of California Institute of Technology
Measured equivalence relations and their full groups

I will give a brief overview of the theory of countable, measure-preserving equivalence relations and their connections with different branches of mathematics. Then I will explain how full groups can be used to differentiate equivalence relations and sketch some recent results in that direction. The talk is based on joint work with John Kittrell.


Rostyslav Kravchenko of Texas A&M University
Images of Bernoulli measures under automata transformations

We study images of Bernoulli measures under automata transformations and consider two cases: the case of automata of polynomial growth and the case of strongly connected automata. We prove that automata of polynomial growth preserve the class of any Bernoulli measure. Moreover, we supply examples of automata of exponential growth which also preserve the class of Bernoulli measure, though thos is not true for the automata of exponential growth in general. In the case of the strongly connected automata we clarify the statement and the proof of a theorem on the frequency of zeros in the image of a sequence of zeros and ones under the action of the strongly connected automaton.