# Dynamical Systems Seminar Archives

**Spring 2019**

**Title**: a proof of Champernowne's theorem.

**Speaker:**Jingyi Xiao, GWU

**Date and Time:**Friday, April 26, 9:00 to 10:15

**Place:**Phillips 736

**Please Note:**Last meeting for Dynamics Seminar

**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.

**Fall 2018**

**Title**: TBA**Speaker:** Robbie Robinson & possibly some participants**Date and Time:** Friday, November 30, 9:00–10:15am**Place: **Phillips 736

**Abstract: **TBA

**Title**: Quantization and Shrinking Targets for Dynamical Systems**Speaker:** Joe Rosenblatt, Indiana University-Purdue University Indianapolis and the University of Illinois Urbana-Champaign**Date and Time:** Friday, November 16, 9:00–10:15am**Reschedule Date and Time: **Monday November 19, 11:00am-12:15pm**Place: **Phillips 730

**Abstract: **We describe how to use dynamical systems to create good quantization of the underlying measure in the dynamical system. Measuring quantization involves considering various ways to analyze how well a set of points approximates a distribution. So we consider different quantization error measurements for various methods of generating sets of points that approximate a fixed distribution. These considerations lead naturallyto questions about shrinking targets in dynamical systems.

**Title**: The Furstenberg Structure Theorem**Speaker:** Joe Auslander, University of Maryland**Date and Time:** Friday, November 9, 9:00–10:15am**Place: **Phillips 736

**Abstract: **Hillel Furstenberg's theorem on the structure of distal minimal flows is more than fifty years old, but it still dominates topological dynamics. We will indicate the main steps in the proof, note some consequences, and discuss some related theorems in topological dynamics and ergodic theory.

**Title**: “On the construction of a Minkowski number.”**Speaker****:** Mathijs de Lepper, Netherlands**Date and Time:** Friday, November 2, 9:00–10:15am**Place: **Phillips 736

**Abstract: **Mathijs de Lepper, our visiting student from Utrecht University, will speak on the work that he has been doing with professor Robinson. During the past few weeks they have been working on his thesis: Constructing a ?-normal number.

**Title**: Dynamics of the geodesic and horocycle flows. **Speaker****:** Robbie Robinson & possibly some participants **Date and Time:** Friday, October 26, 9:00–10:15am**Place(New Room):** Phillips 736

**Abstract: **I am presenting the chapter by Anthony Manning in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, ed. C. Series, T. Bedford and M. Keane, Oxford University Press, 1991.

**Title**: Dynamics of the geodesic and horocycle flows. __ CANCELLED__.

**Speaker**

**:**Robbie Robinson & possibly some participants

**Date and Time:**Friday, October 19, 9:00–10:15am

**Place(New Room):**Phillips 736

**Abstract: **I am presenting the chapter by Anthony Manning in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, ed. C. Series, T. Bedford and M. Keane, Oxford University Press, 1991.

**Title**: Geodesics and horocycles on a surface of constant negative curvature; the geodesic and horocycle flows.**Speaker:** Robbie Robinson & possibly some participants **Date and Time:** Friday, October 12, 9:00–10:00am**Place:** Rome 771

**Abstract:**The geodesic flow (for a compact hyperbolic surface) is an Anosov flow. In particular it has a Markov partition and positive topological entropy (so it is rather chaotic). The horocycle flow is minimal and uniquely ergodic. It has entropy zero (so not especially chaotic), yet is strongly mixing and even has Lebesgue spectrum. The geodesic and horocycle flows satisfy a famous commutation relation h^(s e^t) g^t = g^t h^s. Come find out what all this means (or at least I will begin to explain…to be continued later).

**Title**: Geodesic flows on flat and hyperbolic surfaces**Speaker:** Robbie Robinson, GWU Math **Date and Time:** Friday, September 28, 9:00–10:00am**Place:** Rome 771

**Abstract:**Robbie Robinson will speak on the Caroline Series paper “The Geometry of Markov numbers”. Also, some participants may speak on topics from the study questions list.

This seminar will not meet October 5.

**Title**: Non-Euclidean Geometry, Continued Fractions and Ergodic Theory**Speaker:** Robbie Robinson, GWU Math **Date and Time:** Friday, September 21, 9:00–10:00am**Place:** Rome 771

**Abstract: **I will continue to speak on Caroline Series paper by the same title, and if time permits move on to describing how continued fractions relate to cutting sequences: the sequences that describe how a line cuts through the square grid in the square.

**Title**: Hyperbolic Geometry, Dynamics and Continued Fractions.

** **

**Speaker:** Robbie Robinson, GWU Math

**Date and Time:** Friday, September 14, 9:00–10:00am

**Place:** Rome 771

**Abstract: **I will discuss the relation between regular continued fractions and a tessellation of hyperbolic space corresponding to the action of the modular group. The corresponding hyperbolic surface is a punctured sphere with two cone points. This talk will follow the paper of Caroline Series in the Math Intelligencer, vol 4, 1982.

**Moderator:** John Conway

**Date and Time:** Friday, September 7, 9:00–10:00am

**Place:** Rome 771

**Abstract: **It will consist of three short presentations on background material by participants: (1) Topological Classification of surfaces, by Colin Walker, (2) Basic Properties of Mobius Transformations by Josh Sparks and (3) An introduction to Continued Fractions by Hana Jang and Jingyi Xiao. The moderator will be John Conway.

**Title**: Introduction geodesic flows on hyperbolic and translation surfaces.

**Speaker:** Robbie Robinson, GWU Math

**Date and Time:** Friday, August 31, 9:00–10:00am

**Place:** Rome 771

**Abstract: **Hyperbolic surfaces are quotients of the hyperbolic plane by a Fuchsian group, which is a discrete group of Mobius transformations. The geodesic flow moves with unit speed along geodesics in the unit tangent bundle. For example, a hyperbolic octagon is the fundamental domain for a hyperbolic surface of genus 2.