Speaker: Giulio Tiozzo (Yale University)

Title: The core entropy of quadratic polynomials

Abstract: The core entropy of quadratic polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in combinatorial terms, and provides a useful tool to study parameter spaces of polynomials.

A classical tool to compute the entropy of a dynamical system is the clique polynomial (recently used by McMullen to study the entropy of pseudo-Anosov maps). We will develop an infinite version of the clique polynomial for infinite graphs, and use it to study the symbolic dynamics of Hubbard trees.

Using these methods we will prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle, answering a question of Thurston.

Speaker: Dave Constantine (Wesleyan)

Title: Circle rotations and conditionally convergent series

Abstract: Take the harmonic series and assign to its terms a pattern of + and – signs by a fair coin flip. One can show that this process produces a convergent series almost surely, although there are (of course) sequences of flips which give divergence. Now, instead of a random process, let’s produce these signs deterministically. In particular, let $f$ be a function on $X$ taking values in $\{+1,-1\}$ and let $T:X\to X$. We can similarly investigate the convergence of $\sum f(T^nx)/n$.

In this talk I’ll discuss this problem for an irrational circle rotation and a function assigning +1 to the top half of the circle and -1 to the bottom half. I’ll show that while convergence is the almost sure behavior, there are irrational rotations which give divergence, that all of them are Liouville numbers, but that not all Liouville numbers give divergent series. We’ll also take a quick look at what seems to happen when the series converges.

This is joint work with Joanna Furno (IUPUI).

Speaker: Younghwan Son (Korea Institute of Advanced Study)

Title: Ergodic sum fluctuations in substitution dynamical systems
Abstract: In this talk we will discuss deviation of ergodic sums for substitution dynamical systems with an incidence matrix having eigenvalues of modulus 1. Especially we will present central limit theorem for fixed points of substitution. This is a joint work with E. Paquette.

Speaker: Cheng Zheng (Ohio State)

Title: Sparse equidistribution of unipotent orbits in finite-volume quotients of PSL(2,R)

Abstract: We consider the orbits {pu(n1+γ)|n ∈ N} in Γ\PSL(2,R), where Γ is a non-uniform lattice in PSL(2,R) and {u(t)} is the standard unipotent one-parameter subgroup in PSL(2, R). Under a Diophantine condition on the intial point p, we can prove that the trajectory {pu(n1+γ )|n ∈ N} is equidistributed in Γ\PSL(2, R) for small γ > 0, which generalizes a result of Venkatesh for cocompact lattices Γ.

Speaker: Aaron Brown (University of Chicago)

Title: From entropy to rigidity: applications to lattice actions

Abstract: Consider a smooth action of a lattice in a higher-rank, simple Lie group G on a compact manifold M.  We show that if the dimension of M is sufficiently small relative to the rank of G, then there always exists an invariant probability measure for the action.  If the dimension of M falls in an intermediate range (relative to the rank of G) we show there exists a quasi-invariant measure such that the action is isomorphic to a relatively measure-preserving extension over a standard boundary action.  The proofs of these results follow from existing measure rigidity techniques combined with a new entropy formula for measures invariant under smooth actions of higher-rank Abelian groups.  This formula establishes a “product structure” (along coarse Lyapunov foliations) of entropy for measures invariant under a smooth action of a higher-rank abelian group. The product structure of entropy follows, in turn, from a generalization of the Ledrappier-Young entropy formula to “entropy subordinated to a foliation.”

Speaker: Vitaly Bergelson (Ohio State)

Title: Potpourri of Open Problems and Conjectures