Rigidity - Week 11 - Federico Rodriguez Hertz

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This is a graduate level topics course in mathematics given by Prof. Federico Rodriguez Hertz in Spring 2021 at Penn State. The focus of the course was rigidity phenomena from the point of view of dynamics.

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Table of Contents:

00:00:00 1 of 2
00:00:53 Cartan action of d-1 commuting matrices on d-torus, d at least 3
00:07:11 theorem (Katok-Spatzier): for a Cartan action, the only ergodic invariant measure with positive entropy is Lebesgue
00:08:10 setup for a result of Kalinin-Katok: Cartan homotopy data
00:10:53 main lemma: If alpha is a Z^{d-1} action on d-torus with Cartan homotopy data, then up to finite index alpha is bi-Holder conjugate to the action on homology
00:14:29 observation: there are large alpha-invariant measures
00:17:19 theorem (Kalinin-Katok): If alpha is a Z^{d-1} action on d-torus with Cartan homotopy data, then any large alpha-invariant measure is absolutely continuous rel Lebesgue, and the Lyapunov exponents of alpha match with the Lyapunov exponents of the action on homology
00:18:58 addendum (Katok-FRH): in the context of the Kalinin-Katok theorem, there is a unique large alpha-invariant measure
00:19:36 discussion of main lemma
00:22:54 lemma (Hartman-Grobman): if f is a continuous self-map of d-torus and the induced action f_# on pi_1 is injective and hyperbolic, then f is topologically semiconjugate to f_# by a map h homotopic to id, and any two such semiconjugacies differ by a constant
00:28:11 proof of lemma
00:43:45 lemma: if f and h are as in the previous lemma and g commutes f, then h semiconjugates g to an affine automorphism with g_# as the linear part
00:45:46 proof of lemma
00:50:39 proof of main lemma
00:55:47 2 of 2
00:55:57 recap: Katok-Kalinin theorem
00:57:50 Cartan action
00:59:30 recap continued
00:59:40 discussion of theorem
01:02:39 lemma 1: in the context of Katok-Kalinin theorem, if mu is a large alpha invariant measure, then the Lyapunov exponents rel mu are proportional to the Lyapunov exponents of the action on homology
01:04:01 corollary: for any large alpha invariant measure mu, the Lyapunov exponents are simple
01:04:27 proof of lemma 1
01:15:55 second proof of lemma 1
01:52:00 discussion of proof

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Links:

Mentioned paper:
https://www.aimsciences.org/article/d...
https://arxiv.org/abs/math/0602176

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License:

CC BY-NC-SA 4.0
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Public License
https://creativecommons.org/licenses/...

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