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

Speaker: Robbie Robinson
Date and time: Friday, January 25, 9-10:15am
Place: Phillips 736
 
Abstract: Penrose tilings were discovered around 1976 by Sir Roger Penrose as he was trying to see how close he could come to tiling the plane by regular pentagons. Penrose tilings are aperiodic tilings that are nevertheless in some sense almost periodic, and they inherit a pentagonal pseudo-symmetry from Penrose’s pentagons.  Penrose tilings were popularized by Martin Gardner, who reported on some work on them by John H. Conway. The big advance in understanding these remarkable tilings, however, came with N. G. de Bruijn’s 1981 papers “Algebraic Theory of non-periodic tilings of the plane I & II”, which showed how to interpret Penrose tilings as a 2-dimensional slice through 5-dimensional space. The mural across from the math office is a piece of Penrose tiling. 
 
In the late 1980’s Penrose tilings were proposed by the U. Penn physicists Levine and Steinhardt as a model for a newly discovered state of matter called quasicrystals. Like quasicrystals, Penrose tilings can have a 5-fold rotational symmetry that is forbidden for ordinary crystals. I learned about Penrose tilings as a postdoc at Penn, and realized many of the ideas in the theory have a dynamical systems interpretation. My 1996 Transactions paper “The dynamical properties of Penrose tilings” showed how to use de Briujn's structure theorem to model Penrose dynamics as a total rotation action.
 
In this talk (and the next few weeks in the seminar) I will present the work of Penrose and de Bruijn, as well as my own work on Penrose dynamics. These talks are based on a series of lectures I gave about 20 years ago at Tsuda College in Tokyo. But work on aperiodic tilings, their dynamics and the relation to quasicrystals continues to be active research area under the title of “aperiodic order”.
 
Once we finish Penrose tilings in a few weeks, the seminar will revert to the subject of the year: Dynamics on surfaces (I will even tell you how to fit Penrose tilings into this context).  We will start working through the paper 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

 

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

Abstract: TBA


TitleQuantization 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.


TitleThe 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.

"Though Borel proved in 1909 that almost all numbers are normal, there exists few explicit numbers of which normality has been proved. We explicitly construct a number using the Kepler tree and prove that it is normal with respect to the Minkowski Question Mark measure. In the talk, I will provide an intuitive and formal definition of (several types of) normality and give an outline of the proof."

 

TitleDynamics 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.


TitleDynamics 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.