Time

Sep 12 2013 – 3:00pm

Location

MW154

Speaker

Vaughn Climenhaga (University of Houston)

Abstract

When confronted with a smooth dynamical system that appears to possess some sort of non-uniform hyperbolicity, it is useful to find an invariant measure that controls the asymptotic properties of points chosen at random with respect to the natural volume on the phase space.  Such SRB measures have been constructed for systems where it is possible to relate the dynamics to a symbolic system via a Markov partition or Young tower, and also for certain systems with a dominated splitting.  We present a new approach that does not require any Markov structure or uniform geometric structure.  The key is a notion of “effective hyperbolicity”, which can be used to prove a non-uniform version of the Hadamard-Perron theorem on stable and unstable manifolds.  This is joint work with Dmitry Dolgopyat and Yakov Pesin.

This talk will describe recent progress, in joint work with V. Bergelson, on an analog of this problem with the set of positive integers replaced with a countably infinite field. We introduce the notion of double Følner sequence in a field K. This gives rise to an upper density on K that is invariant under both addition and multiplication. Next we show that a set B⊂K with positive upper density contains a configuration of the form {x+y,xy}. We also show that for a finite coloring of K there exist x and y in K such that {x,x+y,xy} is monochromatic.

Time

Oct 24 2013 – 3:00pm – 4:00 pm

Location

MW154

Speaker

Federico Rodriguez Hertz (Penn State)

Abstract

In this talk I shall discuss some recent results about the interaction of topology and dynamics. In particular I will describe a new example joint with A. Gogolev of a simply connected 6 dimensional manifold with a partially hyperbolic system. If time permits I will also survey partially hyperbolic systems in low dimensions and some open problems.

Time

Oct 31 2013 – 3:00pm – 4:00 pm

Location

MW154

Speaker

Aurel Stan (Ohio State University)

Abstract

In this talk, we will discuss recent results on three different fractal billiard tables: the Koch snowflake, a self-similar Sierpinski carpet and (for lack of a better name and none ever given in the literature) the T-fractal. Initially, an investigation of the flow on a fractal billiard table was made on the Koch snowflake.  Results on the Koch snowflake motivated us to investigate the other two billiard tables. Using a result of J. Tyson and E. Durand-Cartagena, we show that there are periodic orbits of a self-similar Sierpinski carpet billiard table.  J. P. Chen and I have begun investigating whether or not it makes sense to discuss the existence of a dense orbit of a self-similar Sierpinski carpet billiard table. Substantial progress has been made in determining periodic orbits of the T-fractal billiard table.  We detail some of the recent results concerning periodic orbits, determined in collaboration with M. L. Lapidus and R. L. Miller.  Less has been done to determine what may constitute a dense orbit of the T-fractal billiard.  We provide substantial experimental and theoretical evidence in support of the existence of an orbit that is dense in the T-fractal billiard table but is not a space-filling curve. We briefly touch on a long-term goal of determining a topological dichotomy for the flow on a fractal billiard table, namely that, in a fixed direction, the flow is either closed or minimal. Parts of this talk will be suitable for an advanced undergraduate/beginning graduate student audience.

A Z^d shift of finite type is the set of all d-dimensional arrays of symbols from a finite set which avoid a finite set of ‘forbidden’ patterns; for example, the set of all ways of assigning a 0 or 1 to every site in Z^2 so that no two 1s are horizontally or vertically adjacent is a Z^2 shift of finite type. Any shift of finite type can be considered as a topological dynamical system, where the dynamics comes from the Z^d-action of ‘shifting’ by vectors in Z^d. A topological dynamical system was defined by Blanchard to have topologically completely positive entropy (or t.c.p.e.) if all of its nontrivial factors have positive topological entropy. (Here, ‘nontrivial’ means not consisting of a single fixed point) In some sense, t.c.p.e. means that there is no ‘hidden’ degenerate behavior within the dynamical system. Though t.c.p.e. is not easy to characterize in general, I will present a surprisingly simple characterization of t.c.p.e. for shifts of finite type.

Time

Nov 14 2013 – 3:00pm

Location

MW154

Speaker

Joseph Rosenblatt (UIUC)

Abstract

The behavior of the norms of ergodic sums can be used to characterize coboundaries. But the behavior of the norms of ergodic sums can be fairly chaotic. Moreover, for a given function, which classes of transformations have that function as a coboundary is a complex issue. These types of things will be considered in some detail for general ergodic transformations of a probability space.