Seminar 10.24.13

Some new constructions in partially hyperbolic dynamics.

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.

Seminar 10.31.13

A family of Hölder inequalities for norms of generalized Gaussian Wick products.

Time

Oct 31 2013 – 3:00pm – 4:00 pm

Location

MW154

Speaker

Aurel Stan (Ohio State University)

Abstract

See http://www.math.osu.edu/files/Columbus_title_and_abstrct.pdf

Colloquium 11.7.13

A pressure metric for the Hitchin component

Time

Nov 7 2013 – 4:30pm – 5:30 pm

Location

CH 240

Speaker

Richard Canary (University of Michigan)

Abstract

If S is a closed surface, its Teichmuller space is the space of all (marked) hyperbolic structures on S. Hitchin showed that there is a component of the space of (conjugacy classes of) representatations of the fundamental group of a closed surface S into PSL(n,R) which is homeomorphic to an open ball. This component contains a copy of the Teichmuller space of S which we call the Fuchsian locus. We will begin by surveying basic facts about Teichmuller space and the Hitchin component. The pressure metric is a analytic Riemannian metric on the Hitchin component which is invariant under the action of the mapping class group and whose restriction to the Fuchsian locus is a multiple of the classical Weil-Petersson metric. We associate to each Hitchin representation a geodesic flow which is a Holder repameterization of the geodesic flow on S. We then use tools from the thermodynamic formalism to construct our metric. (This talk describes joint work with Bridgeman, Labourie and Sambarino.)

Seminar 11.7.13

Convergence of sequences of compatible periodic orbits to nontrivial paths.

Time

Nov 7 2013 – 3:00pm – 4:00 pm

Location

MW154

Speaker

Rob Niemeyer (University of New Mexico)

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.

Seminar 11.21.13

Topologically completely positive entropy for shifts of finite type

Time

Nov 21 2013 – 3:00pm – 4:00 pm

Location

MW154

Speaker

Ronnie Pavlov (University of Denver)

Abstract

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.

Seminar 11.14.13

Coboundaries and ergodic sums

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.