Analysis Seminar
Analysis Seminar
Speaker: Robbie Robinson
Time: Monday 11/23 Monroe Hall 267 4 pm
Title: Entropy and Rank 1
Abstract: I will discuss ergodic group actions, the idea of an amenable group using Folner sequences. The definition of rank 1makes sense even for groups that are not amenable, but most properties we generally associate with rank 1 require amenability and a Folner sequence of towers. Then, getting more specific, I'll talk about directional entropy for Z^d actions, an invariant that John Milnor invented to study cellular automata. Finally I'll discuss directional entropy results for rank 1 Z^d actions, and how they depend on a strengthening of the Folner property.
Speaker: Paul Wright/University of Maryland
Tuesday, Nov. 3, 2009 Monroe Hall #267 4:00pm-5:00pm,
Title: Some rigorous results for the periodic oscillation of an adiabatic piston
Abstract: A simple model of an adiabatic piston consists of a heavy piston of mass M that separates finitely many ideal, unit mass gas particles moving inside two gas containers. Averaging techniques, used to study the motion of the slow-moving piston in the limit where M tends to infinity, suggest that the piston should oscillate periodically. For one-dimensional chambers, the effects of the gas particles are quasi-periodic and can be essentially decoupled, and I will show that we recover a strong law of large numbers that is characteristic of classical averaging over just one fast variable: the deviation of the piston from its averaged behavior is no more than O (M ^ {-1/2}) on a time scale O (M ^ {1/2}) . I will also show that for a very general gas chamber in higher dimensions, the actual motions of the piston converge in probability to the averaged behavior on that time scale, although a strong law is no longer possible. I learned about this problem from the papers of Neishtadt and Sinai, who derived the averaged equations and pointed out that an averaging theorem due to Anosov could be extended to this case.
Speaker: Joseph Herning
Tuesday, Oct 27, 2009 Monroe Hall #267 4:00pm-5:00pm,
Title: In next week's Analysis Seminar, which is Tuesday due to the Colloquium, I will conclude the discussion of the paper by Sadun, and Solomyak on conditions for topological mixing for substitutions on two letters.
Speaker: Joseph Herning
Tuesday, Oct 20, 2009 Monroe Hall #267 4:00pm-5:00pm,
Title: In next week's analysis seminar I will continue Tyler's two-week discussion of the paper by Kenyon, Sadun, and Solomyak on conditions for topological mixing for substitutions on two letters.
Speaker: Robbie Robinson
Monday, Oct 12, Monroe Hall #267 4:00pm-5:00pm,
Title: "The entropy and spectrum of rank-1 actions in one or more dimensions, Part II"
Abstract: After a brief diversion to explain the notion of "the spectrum" as it applies to ergodic theory, I will define rank 1 transformations and show that they have simple spectrum. This will allow us to conclude that the entropy of a rank 1 transformation is zero. But the proof comes at great expense (the spectral theorem & Sinai's theorem). Then I will give a direct proof that rank 1 transformations have entropy zero.
This proof will be a prelude to the main goal of these lectures (that, alas, will come in a later talk): studying the directional entropies for rank 1 actions of Z^d.
Speaker: Robbie Robinson
Date: Tuesday, October 6, 2009, Monroe Hall #267 4:00pm-5:00pm
Title: "The entropy and spectrum of rank-1 actions in one or more dimensions"
Abstract: Rank-1 transformations play a central role in elementary ergodic theory. Two well known facts are that rank-1 transformations have simple spectrum and rank-1 transformations have entropy zero. The first result implies the second. But a more direct proof of entropy zero, while not hard, is difficult (perhaps impossible) to locate in the literature.
In this first talk (of what will likely be series of talks), I will explain these ideas, and present a simple proof of entropy zero for rank 1 transformations. Later we will move on to the Z^d case, where even the definition of rank-1 becomes less clear. But this lack of clarity, once properly understood, helps explain an old construction called "funny rank 1" and also an old example of Dan Rudolph, which shows that directional entropy need not always be zero.
Speaker: Tyler White (I will be presenting part 2 of my talk).
Date: Monday, September 28, Monroe Hall #267
Title: Topological Mixing of Z and R -Actions of Symbolic Dynamical Systems That Arise From Substitutions
Abstract: The goal of this talk is to present results from Kenyon, Sadun, and Solomyak. These results provide necessary and sufficient conditions for flows ( R – actions) and discrete time( Z – actions) symbolic dynamical systems which arise from substitutions on two letters to be topologically mixing. I will begin by providing a brief introduction to subshifts (including their associated topology) that come from substitutions and show how tiling dynamical systems of R can come naturally from such subshifts. Next, I will provide the definition of a topologically mixing dynamical system and provide equivalent definitions of topological mixing in subshifts and tiling spaces. I will conclude by stating and proving a theorem by Kenyon, Sadun, and Solomyak that relates topologically mixing substitutions (resp. tilings) to the lengths of the substitution (resp. tilings).
Speaker: Tyler White
Date: Sept. 21, 2009 4:00-5:00, Monroe Hall #267 (seminar room)
Title: Topological Mixing of Z and R -Actions of Symbolic Dynamical Systems That Arise From Substitutions
Abstract: The goal of this talk is to present results from Kenyon, Sadun, and Solomyak. These results provide necessary and sufficient conditions for flows ( R – actions) and discrete time( Z – actions) symbolic dynamical systems which arise from substitutions on two letters to be topologically mixing. I will begin by providing a brief introduction to subshifts (including their associated topology) that come from substitutions and show how tiling dynamical systems of R can come naturally from such subshifts. Next, I will provide the definition of a topologically mixing dynamical system and provide equivalent definitions of topological mixing in subshifts and tiling spaces. I will conclude by stating and proving a theorem by Kenyon, Sadun, and Solomyak that relates topologically mixing substitutions (resp. tilings) to the lengths of the substitution (resp. tilings).