Activities This Week

An old question of Poincaré concerns creating periodic points via perturbations of a diffeomorphism. While this question was initially studied in the 60s, various facets of it remain largely open. Recently, several advances were made in the context of Hamiltonian diffeomorphisms, which are diffeomorphisms arising from classical mechanics. I will discuss a variant of this problem in the presence of certain invariant sets, as well as an application to invariant measures. This is based on a joint work with Erman Cineli and Sobhan Seyfaddini.

כל יריעה חלקה ”נראית מקומית כמו $\mathbb{R}^N$“. במקרה זה גאומטריה, טופולוגיה, ואלגברה הן כולן מקומית טריויאליות. אבל לא הכל חלק בעולם שלנו, למשל העקומים הנתונים על-ידי המשוואות $xy=0$ או $y^2=x^d$.

כל נקודה סינגולרית כוללת בתוכה:

• חוגים, אידאלים, מודולים;
• חבורה יסודית, הומולוגיות הומוטופיות;
• מרחבי “מודולים”

וכו.

אני אציג מבוא נגיש לתחום זה.

The study of Fourier decay for stationary measures is a classical problem, with roots in Erdős’ 1939 work on Bernoulli convolutions. Over the years, tools and ideas from number theory, smooth dynamics, and arithmetic combinatorics have led to substantial progress, yet several fundamental questions remain open. After introducing the problem and its history, I will present recent joint work with Federico Rodriguez Hertz and Zhiren Wang, establishing essentially optimal results for a large class of stationary measures with respect to smooth nonlinear maps.