Ergodic Theory Seminar - 21 Fall

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

Schedule for Fall 2021

Sep 1: Lee, Jeong-Yup (Catholic Kwandong University)

Sep 15: Kim, Minsung (Nicolaus Copernicus University, Poland)

Oct 6: Han, Jiyoung (Tata institute of fundamental research, Mumbai)

Oct 20: Ryu, Sieye (University of Sao Paulo)

Nov 3: Lee, Seul Bee (Centro di Ricerca Matematica Ennio de Giorgi, Scuola Normale Superiore di Pisa)

Nov 17: Kim, Taehyeong (Seoul National University)

Dec 1: Lee, Hyunhee (Chungnam National University) (4:00-4:30)

2021.09.01 Lee, Jeong-Yup

Title: Understanding substitution tilings with pure discrete spectrum through a cut-and-project method

Abstract: After the discovery of quasicrystal structures, there has been a lot of study on pure discrete spectrum of tiling dynamics as a characterizing property. There is a general theory that a regular model set, which is a cut and project of a higher dimension lattice, has pure discrete spectrum [Schlottmann 2000]. But the converse is not true in general. We restrict tilings to substitution tilings and study the relation between pure discrete spectrum and regular model sets. Under certain assumptions on expansion maps of substitutions and ‘unimodularity’, the equivalence between the two notions has been shown [Lee-Akiyama-Lee, 2020]. In this talk, we eliminate the unimodularity condition in the earlier result and get the same equivalence. Furthermore, we will discuss some plan on how to extend this result in more general setting of substitution tilings.

2021.09.15 Kim, Minsung

Title: Deviation of ergodic averages of higher rank actions on Heisenberg nilmanifolds and its limit theorem

Abstract: In the work of Bufetov, he studied the limit theorem for translation flows by proving the deviation of ergodic integrals by constructing finitely-additive measures (also called Bufetov functionals or cocycles). This result indicates that his construction has a duality with invariant distributions that appeared in the work of Forni (2002). Following this approach, Bufetov-Forni and Forni-Kanigowski also proved the same type of deviations formula and the limit distributions for horocycle flows and nilflows respectively.

In this talk, we introduce the way of construction of such cocycles and prove the deviation of higher rank actions for Heisenberg nilmanifolds. As a corollary, we obtain its limit distribution as an application.

2021.10.06 Han, Jiyoung

Title: Siegel's and Rogers' integral formulas and their applications

Abstract: Rogers' integral formulas are the moment formulas for the Siegel transform, which sends a bounded and compactly supported function to a function defined on the space of lattices. Using Rogers' formulas, one can obtain random statements in geometry of numbers, related to counting-lattice problems with error bounds.

In this talk, we introduce the regular/primitive versions of Rogers' integral formulas and how to obtain them, and generalization to the S-arithmetic space.

2021.10.20 Ryu, Sieye

Title: Predictability and Entropy for Actions of Amenable Groups

Abstract: In this talk, we consider the following question due to Michael Hochman:

Suppose that a countable amenable group $G$ acts on a compact metric space $X$ and that $S \subset G$ is a semigroup not containing the identity of $G$. If every continuous function $f$ on $X$ is contained in the closed algebra generated by $\{sf : s \in S\}$, then does the action have zero topological entropy?

To provide an affirmative answer, we introduce the notion of an invariant random order.

This is a joint work with Andrei Alpeev and Tom Meyerovitch.

2021.11.03 Lee, Seul Bee

Title: Regularity properties of Brjuno and Wilton functions

Abstract: An irrational number is called a Brjuno number if the series of log(q_{n+1})/q_n converges, where q_n is the denominator of the n-th principal convergent of the regular continued fraction. The importance of Brjuno numbers comes from the study of analytic small divisors problems in dimension one. In 1988, J.-C. Yoccoz introduced the Brjuno function which characterizes the Brjuno numbers to estimate the size of Siegel disks.

In this talk, we first introduce k-Brjuno functions and Wilton function which are related to the classical Brjuno function. We study their BMO(Bounded Mean Oscillation) regularity properties. Then we complexify the functional equations which they fulfill and we construct analytic extensions of the k-Brjuno and of the Wilton function to the upper half plane. This is joint work with Stefano Marmi, Izabela Petrykiewicz, and Tanja I. Schindler.

Secondly, we introduce Brjuno-type functions associated to by-excess(backward), odd, even and odd-odd continued fractions. We see that the Brjuno numbers are characterized by the Brjuno-type functions. Then we deal with their Hölder continuity properties. This is joint work with Stefano Marmi

2021.11.17 Kim, Taehyeong

Title: Hausdorff dimension in inhomogeneous Diophantine approximation

Abstract: In 1985, Dani studied the connection between homogeneous dynamics and Diophantine approximation, which is called Dani's correspondence. The dynamical method under Dani's correspondence has been widely used in the study of metric Diophantine approximation since the breakthrough about Sprindzuk conjecture by Kleinbock and Margulis.

In this talk, we review some recurrence properties of one-parameter flow and corresponding Diophantine properties. Through this connection, we will discuss how dynamical methods give metrical results of inhomogeneous Diophantine approximation.

2021.12.01 Lee, Hyunhee

Title: Inverse limit dynamical systems with Mittag-Leffler condition

Abstract: For each $n\in \mathbb{N}$, let $X_n$ be a compact Hausdorff space and $g_n^m:X_m\to X_n$ be a continuous map. An inverse limit system $g_n^m$ satisfies the {\it Mittag-Leffler condition} if for every $N\in \mathbb{N}$, there exists $k\geq N$ such that for each $l\geq k$, we have $g_N^k(X_k)=g_N^l(X_l)$.

\par In this talk we introduce the concept of factor maps which {\it almost lift pseudo orbits} and discuss the preservation of shadowing theory under factor maps which almost lift pseudo orbits.

Then we study the inverse limit dynamical system $((f_n)_*, \displaystyle\lim_{\longleftarrow} (g_n^m, X_n))$ satisfying the Mittag-Leffler condition.