The Problem of the Non-Integrability of the Center Distributions of Partially Hyperbolic Diffeomorphisms

Dynamics Student Seminar

Meeting Details

For more information about this meeting, contact Kristin Berrigan, Alp Uzman.

Speaker: Alp Uzman, Penn State

Abstract: A diffeomorphism of a smooth closed manifold is partially hyperbolic if it induces a topological derivative-invariant splitting of the tangent bundle into stable, center and unstable distributions. By well-known theorems all three of these distributions are in fact H\"older, and the stable and unstable distributions always integrate to topological foliations with continuously differentiable leaves. The same is not guaranteed for the center distribution. In this talk I will be discussing the only known explicit family of partially hyperbolic diffeomorphisms whose center distributions are non-integrable. The construction is of algebraic origin, is due to Borel and Smale, and it was presented in Smale's 1967 survey. If time permits I will also discuss the stability of the non-integrability: by a 2003 paper by Brin, Burago and Ivanov non-integrability of the center distribution persists under small homotopic perturbations.

Room Reservation Information

Room Number: 106 McAllister

Date: 02/07/2019

Time: 5:30pm - 8:00pm