January 30: Amie Wilkinson (Chicago)
Title: Absolute continuity, exponents and rigidity
Abstract: The geodesics in a compact surface of negative curvature display stability properties originating in the chaotic, hyperbolic nature of the geodesic flow on the associated unit tangent bundle. Considered as a foliation of this bundle, this collection of geodesics persists in a strong way when one perturbs of the Riemannian metric, or the geodesic flow generated by this metric, or even the time-one map of this flow: for any perturbed system there is a corresponding "shadow foliation" with one-dimensional smooth leaves that is homeomorphic to the original geodesic foliation. A counterpart to this foliation stability is a curious rigidity phenomenon that arises when one studies the disintegration of volume along the leaves of this perturbed shadow foliation. I will describe this phenomenon and its underlying causes. This is recent work with Artur Avila and Marcelo Viana.
February 6: Van Cyr (Northwestern)
Title: A Z^2 generalization of the Morse-Hedlund Theorem.
Abstract: Attached
February 20: Vadim Kaloshin (Maryland)
Title: Arnold diffusion via Normally Hyperbolic Invariant Cylinders.
Abstract: In 1964 Arnold constructed an example of instabilities for nearly integrable systems and conjectured that generically this phenomenon takes place. There has been some progress attacking this conjecture in the past decade. Jointly with Ke Zhang we present a new approach to solve this problem. It is based on a construction of crumpled and flower Normally Hyperbolic Invariant Cylinders with kissing property. Then to construct diffusion along these cylinders we apply Mather variational mechanism. A part of the project is also joint with P. Bernard.
February 27: Natalie McGathey (Purdue North Central)
Title: Invariant Measures and Homeomorphisms of Boundaries
Abstract: An important question in ergodic theory is: given an action of a group $G$ on a space $X$, classify all (ergodic) invariant probability measures for this action. We will give such a classification for the case $G = \mathrm{PSL}_2(\mathbb{R}), \mathrm{Isom}_+\mathbb{H}_K^n, where $K$ denotes $\mathbb{R},\ \mathbb{C},\ \mathbb{H},$ or $\mathbb{O}$ with $n=2$, and $X$ is $L/G$ or some large group $L$. In this talk, we will briefly describe how this particular setting is motivated by trying to understand embeddings of Teichmuller space; we will then state the classification results and give some flavor of the proofs.
March 5: Patrick LaVictoire (Wisconsin)
Title: Pointwise ergodic theorems on virtually nilpotent groups
Abstract: Starting with the work of Bourgain, a "quantitative" harmonic analysis approach has proved many nonstandard pointwise ergodic theorems, recently including several results for sparse averages of L^1 functions. While the original methods worked via Fourier analysis, some of them can be reframed in terms of additive combinatorics, and thereby extended to actions of virtually nilpotent discrete groups. This is joint work with J. Rosenblatt and A. Parrish.
March 12: Lin Shu (Peking)
Title: Folding entropy and dimension theory for invariant measures of endomorphisms
Abstract: Consider a $C^2$ non-invertible but non-degenerate endomorphism $f$ on a compact Riemannian manifold without boundary. We are interested in the dimension theories of $f$- invariant measures. By using the notion of folding entropy introduced by Ruelle, we set up an equality relating entropy, folding entropy and negative Lyapunov exponents. Based on this, we establish the exact dimensional property of an ergodic hyperbolic measure. We also give a new formula of Lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms. An application of folding entropy to the theory of entropy production is also addressed. Our results extend several well known results of Ledrappier-Young and Barreira et al. for diffeomorphisms to the case of endomorphisms.