Seminar: Ergodic Theory and Dynamcial Systems

Time: 4:00 pm on 1st & 3rd Wednesday (unless otherwise noted)

Schedule for Spring 2022

Mar 2: Kiho Park (KIAS)

Mar 16: Lucas Kaufmann (Center for Complex Geometry of IBS)

April 6: Donald Robertson (The University of Manchester)

April 20: Jaelin Kim (Sorbonne Université)

May 4: Weikun He (Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

May 18: Minsung Kim (Nicolaus Copernicus University)

5. 18 Minsung Kim

Title: Deviation spectrum of ergodic integrals for locally Hamiltonian flows on surfaces

Abstract: The talk will consist of an introduction to the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surfaces and recent related results. These works extend the study of the spectrum of deviations of ergodic integrals beyond the case where the observable vanishes at the singularities. New developments include a better understanding of the asymptotics (in non-degenerate regime) and the appearance of new exponents in the deviation spectrum (in degenerate regime). This is joint work with Krzysztof Frączek.

3. 2 Kiho Park (KIAS)

Title: Limit laws in Dynamical Systems

Abstract: For a class of dynamical systems f \colon X \to X that are considered chaotic, we will look at the validity of various statistical limit laws for additive observables and matrix-valued observables under suitable assumptions (with respect to certain invariant measures on X, including the measure of maximal entropy). Such limit laws include the Law of Large Numbers, Central Limit Theorem, and Large Deviation Principle, and the transfer operator and its spectral properties play key roles in establishing these limit laws. This is joint work with Mark Piraino.

3. 16 Lucas Kaufmann (Center for Complex Geometry of IBS)

Title: Random walks on SL_2(C)

Abstract: Given a sequence of random i.i.d. 2 by 2 complex matrices, it is a classical problem to study the statistical properties of their product. This theory dates back to fundamental works of Furstenberg, Kesten, etc. and is still an active research topic. In this talk, I intend to show how methods from complex analysis and analogies with holomorphic dynamics offer a new point of view to this problem. This is used to obtain several new limit theorems for these random processes, often in their optimal version. This is based on joint works with T.-C. Dinh and H. Wu.

4. 6 Donald Robertson (The University of Manchester)

Title: Sumsets in Dense Sets

Abstract: Erdős asked whether every set of natural numbers having positive density contains a sumset B+C where B and C are infinite sets of natural numbers. In this talk I will outline joint work with J. Moreira and F. Richter on a resolution of this problem using methods from ergodic theory, and discuss related problems in amenable groups.

4. 20 Jaelin Kim ( Sorbonne Université)

Title: Brownian motions and geodesic flows in pinched negative curvature

Abstract: In this talk, I will present asymptotic behaviors of the Brownian motion on finite-volume manifolds with pinched negative curvature. For instance, a central limit theorem for distances and the logarithm of the Green function, the equidistribution of Brownian paths, and the characterization of the asymptotic harmonic manifolds. I will conclude with introducing some related problems.

5. 4 Weikun He

Title: Quantitative equidistribution of linear random walk on the torus.

Abstract: Consider the action of GL_d(Z) on the d-dimensional torus R^d/Z^d. Given a probability measure on GL_d(Z) and a point on the torus, a random walk is defined. In this talk, I will report some recent progress about the equidistribution of such random walk. More precisely, I will talk about quantitative results under the assumption that the acting group has semisimple Zariski closure. This is a generalisation of a theorem of Bourgain, Furman, Lindenstrauss and Mozes. This talk is based on a joint work with Nicolas de Saxcé.