## Dynamical Systems Seminar

**Spring 2019**

**Title**: Billiards in polygons and related dynamical systems

**Speaker:**Robbie Robinson, GWU

**Date and Time:**Friday, March 29, 9:00 to 10:15

**Place:**Phillips 736

**Abstract: **We will continue our discussion about the dynamics of billiards in rational polygons and related dynamical systems.

**Note:**There will be no dynamics seminar on April 5. On April 12 the seminar will be Hana Jang’s specialty exam (details to be published later).

**Title**: Invariant measures, matching and the frequency of 0 for sighted binary expansions.

**Speaker:**Karma Dajani, University of Utrecht, The Netherlands

**Date and Time:**Friday, March 22, 9:00 to 10:15

**Place:**Phillips 736

**Abstract: **Abstract: We introduce a parametrized family of maps S_\alpha, the so called symmetric doubling maps, defined on [-1,1] by S(x) = 2x-d\alpha, where d\in{-1,0,1} and \alpha\in [1,2]. Each map S_\alpha generates binary expansions with digits -1,0 and 1. The transformations S_\alpha have a natural invariant measure that is absolutely continuous with respect to Lebesgue measure. We show that for a set of parameters of full measure, the invariant measure of the symmetric doubling map is piecewise smooth. We also study the frequency of the digit 0 in typical expansions, as a function of the parameter. In particular, we investigate the self similarity displayed by the function \alpha\to\mu_\alpha([-1/2,1/2]) where \mu_\alpha([-1/2,1/2]) denotes the measure of the cylinder where digit zero occurs. This is joint work with Charlene Kalle.

**Title**: Billiards in Polygons and related dynamical systems

**Speaker:**Robbie Robinson & participants

**Date and Time:**Friday, March 1st, 9:00–10:15am(or so)

**Place:**Phillips 736

**Abstract: **We will continue our discussion about the dynamics of billiards in rational polygons. We will discuss and compare to related dynamical systems: rotations, interval exchange transformations, substitutions and geodesic flows on polygons.

**Title**: Introduction to billiards

**Speaker:**Robbie Robinson, GWU Math

**Date and Time:**Friday, February 22nd, 9:00–10:15am

**Place:**Phillips 736

**Abstract: **We will begin our discussion about the topic of the dynamics of billiards: Think of a billiard (i.e, pool) table whose boundary is a piecewise smooth simple closed curve. A ball moves (forever) around the interior, observing the angle of incidence equals angle of reflection rule each time it encounters the edge. Does the motion ever repeat (called periodic motion) or not? Does it go everywhere, or is it confined to a few parts of the table? How much time does it spend in one part of the table compared to another? I will begin with a general survey of what is known about these questions. Then we will switch to the special case of polygonal tables with rational angle corners. This is currently a very active research area related to “Teichmuller” theory. For the rest of the year, we will try to learn a little about it.

**Title**: On Penrose tilings and tiling dynamical systems: VI

**Speaker:**Robbie Robinson, GWU Math

**Date and Time:**Friday, February 15th, 9:00–10:15am

**Place:**Phillips 736

**Abstract: **The set of Penrose tilings is “almost” parameterized by a 4-torus. The R^2 translation action is product of a pair of suspensions of Sturmian dynamical systems. The inflation map is (almost) a hyperbolic total automorphism, with the Markov partition drawing the Penrose tiles.

**Title**: On Penrose tilings and tiling dynamical systems: III

**Speaker:**Robbie Robinson, GWU Math

**Date and Time:**Friday, February 8th, 9:00–10:15am

**Place:**Phillips 736

**Abstract: **This time I discuss de Bruijn’s theorem on the structure of Penrose tilings (as duals of Penrose “pentagrams”) and show how to fit this into a dynamical systems context. This is lecture III in a series of IVTitle: On Penrose tilings and tiling dynamical systems: II

Speaker: Robbie Robinson

Date and time: Friday, February 1, 9-10:15am

Place: Phillips736

Abstract: Continuation of series of talks in which I present the ideas about Penrose tilings due to Penrose, de Bruijn, and Conway, as well as some of my own work on Penrose dynamics. These lectures are based on a series of lectures I gave about 20 years ago at Tsuda College in Tokyo. But work on aperiodic tilings and quasicrystals continues to be active research area under the title of “Aperiodic Order”. Later in the semester, the topic will switch to conformal geometry of billiards*. * As in the Fall, the seminar will feature talks by participants as well as several outside speakers. For more information, see __https://blogs.gwu.edu/ robinson/2019/01/18/dynamical-systems-seminar-spring-2019/__

Title: On Penrose tilings and tiling dynamical systems: I

*The conformal geometry of billiards*, Laura de Marco, BAMS

**48**, 2011,

__http://www.ams.org/__journals/bull/2011-48-01/S0273-0979-2010-01322-7/home.html As before, the seminar will include talks by participants as well as several outside speakers.