**Learning
Seminar on Dynamical Systems and Ergodic Theory and Related Topics**

**Fall
Semester of 2022**

**Organized by
Yunping Jiang and Enrique Pujals**

**Friday, 11:00 pm – 1:00 pm****
Room ****3212**

September
16, Organization Meeting

September
23, Tao Chen, non-ergodicity of some transcendental meromorphic functions

September
30, Conversation with Professor Enrique Pujals (all
students are welcome)

October
7, Enrique Pujals, Axiomatizing unimodality for
surface diffeomorphisms

October
14, Yunping Jiang, Geometry in Furstenberg conjecture

October 21, Yunshan Jiang, Unitary representations of simple Lie groups

Abstract: Let a "special" Lie group G
(like SL(2,R)) acts on a probability space X in a good
way: preserves the measure

and G will have a unitary
representation on the L^2 functions space. Then we can say a lot about the
ergodicity

and mixing properties about this action, like "the ergodicity
implies the action will also be mixing".

This is called the Howe-Moore theorem and it has a lot of
applications. For example, the geodesic flow

and horocycle flow on the unit tangent bundle of a hyperbolic surface
with finite area are ergodic and mixing.

Also by this theorem, we can obtain that the
ergodicity of Teichmüller geodesic flow implies that
the Teichmüller

horocycle flow is ergodic and mixing (the latter one will imply the
ergodicity of mapping class group action

on the space of measured foliations/laminations). We will prove the
Howe-Moore theorem by considering

the unitary representations of simple Lie group G.

October
28, Axel Kodat, Distortion elements in diffeomorphism
groups

November
4, Lucas Furtado, Local stable manifold of a smooth dynamical systems at a
hyperbolic fixed point.

November
11, Yuan (Jessica) Liu, The unique measure of maximum entropy
(MME) on the square-free flow.

November
18, Bryce Gollobit, Classification of infinite-dimension
linear dynamics.

November
25, No event (Thanksgivings)

December
2, Lucas Furtado, Local stable manifold of a smooth dynamical systems at a
hyperbolic fixed point, Part 2

December
9, Yuan (Jessica) Liu, The unique measure of maximum entropy
(MME) on the square-free flow, Part 2

December 16, Final Exam
Period