Classification of Sets of Periodic Points of Solenoidal Automorphisms

<strong>Mathematics Seminar of the School of Physical Sciences -------------------------------------------------------------</strong> Title: <strong>Classification of Sets of Periodic Points of Solenoidal Automorphisms</strong> Speaker:<strong> Sharan Gopal</strong> (Indian Statistical Institute, Bengaluru) Date: <strong>April 9, 2015 </strong> <strong>Abstract: </strong>I start with an introduction to Topological Dynamics and then describe the problem of characterising the sets of periods as well as sets of periodic points of a family of dynamical systems. Characterisations for some families are stated. Then follows the main part of the lecture, where the above problem is considered for the family of automorphisms on a solenoid. By definition, a solenoid is a compact connected finite- dimensional abelian group. Equivalently, a solenoid is a topological group G, whose dual group Gˆ satisfies: Z^n = Gˆ = Q^n for some n. Here, a characterisation of sets of periodic points is given for the family of automorphisms on a one-dimensional solenoid G (i.e., n = 1). For this, a one-dimensional solenoid is first described as a quotient of the additive group of adeles. In the higher dimensions also, the characterisation is given in two particular cases: Gˆ = Z^n and Gˆ = Q^n. Moreover, when Gˆ = Z^n, G is an n-dimensional torus; thus this extends the result given by [Kannan et al., 2010] for a two-dimensional torus.