In this talk, we mainly discuss limit sets, particularly omega-limit sets and attractors which are invariant on dynamical systems. We first deal with envelope theory related to IFS and its applications. We also discuss the invariant notions on 3-manifolds with a certain cohomology condition. For surfaces, a topological characterization of the omega-limit sets for analytic flows was solved up to homeomorphisms. In this regard, we give a natural generalization to the higher genus of the solution for the genus 0 case. We derive various applications to the descriptions of dynamic notions...

# Ergodic Theory & Statistical Mechanics

For more information about this seminar, contact Yakov Sinai or Jon Fickenscher

**Please note room location is now Fine Hall 1001.**

**Please click on seminar title for complete abstract.**

##### An invariance on dynamical systems

Chungnam National University

##### Hadamard well-posedness of the gravity water waves equations

The gravity water waves equations consist of the incompressible Euler equations and an evolution equation for the free boundary of the fluid domain. Assuming the flow is irrotational, Alazard-Burq-Zuily (Invent. Math, 2014) proved that for any initial data in Sobolev space $H^s$, the problem has a unique solution lying in the same space, here s is the smallest index required to ensure that the fluid velocity is spatially Lipschitz. We will discuss the strategy of a proof of the fact that the flow map is continuous in the strong topology of H^s.

Princeton University

##### TBA-Joel Moreira

Northwestern University