Seminar 3.17.16 Fraser

Speaker: Jon Fraser (Manchester)

Title: Inhomogeneous iterated function systems

Abstract: Inhomogeneous iterated function systems are natural generalisations of the classic iterated function systems, commonly used to generate examples of fractal sets. The key difference is that one begins with a fixed ”condensation” set which is then dragged into the construction by the iterates of mappings in the IFS. Such systems have applications in image compression in situations where one wants to produce an image with lots of similar objects appearing at different scales, like a flock of seagulls or a forest. I will review some structural properties of the attractors of such systems and go on to discuss their dimension theory. Some of this talk will be joint work with Simon Baker (Reading) and Andras Mathe (Warwick).

Seminar 2.25.16 Bunimovich

Speaker: Leonid Bunimovich (Georgia Institute of Technology)

Title: Finite time properties of transport in chaotic systems.

Abstract: We are used to the idea that only asymptotic in time properties of systems with “complex” dynamics should (and could) be understood rigorously. Therefore basically always asymptotics when time goes to infinity or integration over an infinite time interval are involved. I will discuss  finite time properties of survival, first passage and recurrence probabilities for some classes of dynamical and stochastic systems.

Seminar 2.18.16 Shah

Speaker: Nimish Shah (Ohio State)

Title: Equidistribution of stretching translates of curves on homogeneous spaces

Abstract: We consider a finite piece C of an analytic curve on a minimal expanding (abelian) horospherical subgroup of G=SL(n,R) associated to a certain diagonal element g in G. We consider the subgroup action of G on a finte volume homogeneous space X, and consider the trajectory of C from some point x in X. We want to understand algebraic conditions on C which ensure that in the limit, the translates of the curve Cx by powers of g get equidistributed in the (homogeneous) closure of the G-orbit of x. In this talk we describe some recent joint work with Lei Yang on this problem.

Such results have applications to metric properties of Diophantine approximation- namely, to show non-improvability of Dirichlet’s approximation on curves.

Seminar 2.11.16 Todd

Speaker: Mike Todd (St. Andrews)

Title: Continuity of measures

Abstract: Given a convergent family of interval maps and the associated family of SRB measures, one might hope that the measures would converge to the SRB measure of the limit map.  In non-uniformly hyperbolic systems, this naive approach can fail. I’ll give sharp conditions on precisely when this failure occurs for a very general class of maps. This is part of a wider study of continuity of thermodynamic quantities in collaboration with Neil Dobbs.

Seminars for Spring 2016

Currently our schedule for the Spring is as follows.

Feb 11: Mike Todd (St. Andrews)

Feb 18: Nimish Shah (Ohio State)

Feb 25: Leonid Bunimovich (Georgia Tech)

Mar 17:  Jon Fraser (Manchester)

Mar 24: Joel Moreira (Ohio State)

Apr 7: Ben Webb (Brigham Young)

Apr 21: Kostya Medynets (United States Naval Academy)

Apr 30: Michal Misiurewicz (IUPUI)

May 12: COLLOQUIUM: Mark Pollicott (Warwick)

Seminar 12.3.15 Tiozzo

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.

Seminar 11.12.15 Constantine

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).

Seminar 10.29.15 Son

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.

Seminar 9.24.15 Zheng

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.

Seminar 9.17.15 Brown

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.”