Seminar: Ergodic Theory and Dynamcial Systems
Time: 4:00 pm (unless otherwise noted)
Current schedule for the semester
Sep 7: Jungwon Lee (University of Warwick)
Sep 21: Joel Moreira (University of Warwick)
Oct 5: Jonguk Yang (University of Zurich)
Oct 19: Jamerson Bezerra (Nicolaus Copernicus University, Poland)
Nov 2: Seonhee Lim (Seoul National University)
Nov 16: Sangjin Lee (IBS-CGP)
Nov 30: Sanghoon Kwon (Catholic Kwandong University)
9. 7 Jungwon Lee
Title: Another view of Ferrero-Washington Theorem.
Abstract: Ferrero and Washington observed the joint equidistribution of digits of p-adic integers, which describes the growth of class number in a certain tower of number fields. We reprove the main equidistribution instance, based on the ergodicity of a p-adic skew extension dynamical system that can be identified with two-sided Bernoulli shift (joint with Bharathwaj Palvannan).
9. 21 Joel Moreira
Title: Infinite Ergodic Ramsey theory
Abstract: In the 1970's Furstenberg obtained an ergodic theoretic proof of Szemeredi's theorem, stating that any set of natural numbers with positive upper density contains arbitrarily long (but finite) arithmetic progressions. Since then, Furstenbeg's approach has been extended to many other combinatorial applications, giving rise to the field of Ergodic Ramsey Theory. However, until recently, ergodic theoretic methods were unable to handle infinite configurations. I will present recent joint work where we overcame this difficulty to answer some Ramsey theoretical questions raised by Erdos in the 1980's using methods from ergodic theory.
10. 5 Jonguk Yang
Title: Rigidity of Dynamics from Small Scale Geometry
Abstract: A class of dynamical systems is said to exhibit rigidity if whenever two systems satisfy a certain notion of equivalence, it implies that they must actually be equivalent in some stronger sense. One of the earliest examples of this phenomenon appears in the work of Denjoy in 1932. He showed that any C^2-diffeomorphism of the circle without periodic points must be topologically equivalent to an irrational rotation. In this talk, we give an outline of this theory with emphasis on how the study of rigidity naturally leads to the analysis of small scale geometry and renormalization. We will then state some recent results about rigidity of dynamical systems in higher dimensions.
10. 19 Jamerson Bezerra
Title: Regularity of the Lyapunov exponent of random product of matrices
Abstract: The study of the regularity of the Lyapunov exponent of random products of SL2(R) matrices is a rich subject with many important contri- butions in the past years. It is well established in the literature that the function which associates each finite supported measure $\mu$ its Lyapunov exponents $L(\mu)$ is continuous, however, in general, it can have really poor modulus of continuity.
The purpose of this talk is to present a quantitative result on the control of the modulus of continuity for generic finitely supported measures $\mu$. More specifically, we provide an explicit upper bound on the local Holder regularity of the Lyapunov for this generic class.
11. 2 Seonhee Lim
Title: Complex continued fractions and finite range structure
Abstract: In this talk, we will introduce Hurwitz complex continued fraction and Nakada's work on its finite range structure. We will then explain how to obtain the Gaussian distribution for the Euclidean algorithm, which is a generalization of the work by Baladi and Vallee. (This is a joint work with Jungwon Lee and Dohyeong Kim.)
11. 16 Sangjin Lee
Title: Topological and categorical entropies in symplectic topology.
Abstract: Topological entropy is a well-known invariant of a topological dynamical system. Motivated by topological entropy, Dimitrov, Haiden, Katzarkov, and Kontsevich defined an invariant of a categorical dynamical system. The new invariant is called "categorical entropy." Moreover, one can connect two entropies in symplectic topology.
In this talk, I will introduce the notion of categorical entropy. And I will discuss how symplectic topology associates two entropies. As the result of the discussion, in a symplectic topological setup, the categorical entropy bounds the topological entropy from below.
11. 30 Sanghoon Kwon
Title: Zeta function of graphs of groups from cuspidal tree-lattices
Abstract: We investigate the behavior of zeta functions of infinite graphs of groups which are quotients of cuspidal tree-lattices. This includes every non-uniform arithmetic quotients of the tree of $PGL_2$ over local fields. Several examples of zeta functions are provided. In particular, we give pairs of non-isomorphic cuspidal tree-lattices which have the same Ihara zeta function. We also discuss the spectral behavior of a sequence of graphs of groups constructed in Efrat (1990) in a zeta function point of view.