Time: Wednesday 4:00 pm (unless otherwise noted)
schedule for the semester (Fall 2024)
September 25: Andreas Koutsogiannis (Aristotle University of Thessaloniki)
October 16: Soonki Hong (POSTECH)
October 30: Sanghoon Kwak (KIAS)
November 6: Bomi Shin (Sungkyunkwan University)
November 20: Keonhee Lee (Chungnam National University)
9. 25: Andreas Koutsogiannis
Title: Limiting behavior of multiple ergodic averages and applications
Abstract: In this talk we will discuss norm-convergence results for multiple ergodic averages along sequences of polynomial growth. Combining these results with Furstenberg’s correspondence principle, we will derive several applications in combinatorics. Namely, we show that subsets of positive integers of positive upper density contain various arbitrarily long combinatorial patterns. We will also obtain far-reaching extensions to Szemerédi’s theorem, which states that any such subset of positive integers contains arbitrarily long arithmetic progressions.
10. 16: Soonki Hong
Title: Local limit theorem of the Brownian motion on simplicial trees
Abstract: In this talk, I will discuss the local limit theorem for Brownian motion on a tree where all edges have length 1, with a discrete group acting isometrically and geometrically on the tree. The local limit theorem describes the asymptotic behavior of the transition probability density (heat kernel) as time tends to infinity. Specifically, I will focus on the uniform mixing property of the discrete geodesic flow under this assumption and explain how this property is utilized in the proof of the theorem.
10. 30: Sanghoon Kwak
Title: Non-unique Ergodicity on the Boundary of Outer Space
Abstract: The Culler--Vogtmann's Outer space CV_n is a space of marked metric graphs, and it compactifies to a set of F_n-trees. Each F_n-tree on the boundary of Outer space is equipped with a length measure, and varying length measures on a topological F_n-tree gives a simplex in the boundary. The extremal points of the simplex correspond to ergodic length measures. By the results of Gabai and Lenzhen-Masur, the maximal simplex of transverse measures on a fixed filling geodesic lamination on a complete hyperbolic surface of genus g has dimension 3g-4. In this talk, we give the maximal simplex of length measures on an arational Fn-tree has dimension in the interval [2n-7, 2n-2]. This is a joint work with Mladen Bestvina, and Elizabeth Field.
11. 6: Bomi Shin
Title: Measure-theoretical spectral decomposition
Abstract: The Spectral Decomposition Property (SDP) plays a central role in understanding the structure of nonwandering sets in dynamical systems.
In this talk, we extend the classical SDP to a measure-theoretic setting for homeomorphisms on compact metric spaces.
We show that a homeomorphism has the SDP if and only if every Borel probability measure satisfies the property.
Furthermore, we study the connection between shadowing properties and spectral decomposition from a measure-theoretic perspective.
This talk is based on reference [1].
[1] Shin, Bomi A measurable spectral decomposition. Monatsh. Math. 204 (2024), no. 2, 311–322.
11. 20: Keonhee Lee
Title: Spectral decomposition and skew product for group actions
Abstract:
Spectral decomposition which is fundamental in the qualitative theory of dynamical systems delineates that the nonwandering set can be decomposed as a finite number of disjoint compact invariant indecomposable sets.
In this talk we establish various types of spectral decomposition for group actions on compact metric spaces. In particular, we use a skew-product associated with a group action to derive the spectral decomposition of the nonwandering set in a given direction. This talk is based on reference [1].
[1] K.Lee, C. Morales and Y. Tang, Spectral decomposition and skew-product for group actions
[2] K. Lee and N. Nguyen, Spectral decomposition and $\Omega$-stability of flows with expanding measures, J. Differential Equations, 269 (2020), 7574-7604.