Rigidity - Week 10 - 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 3
00:00:11 recap: example 1 (nonresonant example): Gamma = SL(3,Z) acting on 3-torus linearly
00:05:14 example 2 (resonant example): boundary action of G = SL(3,R) on RP^2
00:37:53 theorem (Brown-Fisher-Hurtado): If Gamma is a lattice in SL(n,R) acting smoothly on a manifold of dimension at most n-2 (n at least two), then the action is trivial
00:39:17 theorem (Brown-FRH-Wang): If alpha is a smooth Gamma action on M, where Gamma is a lattice in G = SL(n,R) and mu is an A-invariant ergodic measure projecting to haar on G/Gamma, where A is R-split Cartan in G, then for any ij such that chi_{ij} is nonresonant, mu is ij-unipotent invariant
00:42:53 recap: higher rank Abramov-Rokhlin formula
00:47:47 proof of higher rank Abramov-Rokhlin
00:54:50 2 of 3
00:55:00 recap: setup for higher rank Abramov-Rokhlin
00:59:28 recap: key lemma: the entropy along a coarse Lyapunov foliation is bounded from above by the sum of the entropies along the horizontal and vertical subfoliations
01:00:47 lemma 6.1: entropy of an ergodic T relative to a common refinement of xi and eta is bounded from above by the sum of entropies relative to eta and the common refinement of xi and eta_T, where eta_T is the common refinement of eta and all its iterates under T
01:05:32 observation: basic inequality: the entropy relative to the common refinement of xi and eta is bounded from above by the sum of the entropies relative to xi and eta
01:06:00 heuristics for lemma
01:18:32 proof of key lemma
01:40:10 main properties of entropy
01:50:52 3 of 3
01:51:00 recap: lemma 6.1
01:51:50 recap: main properties of entropy
02:01:15 proof of lemma 6.1
02:16:39 corollary: If (Y,S,nu) is a measurable factor of (X,T,mu), alpha is the point partition of Y and xi is a partition of X refining the pullback of alpha, then the entropy of T rel xi is bounded from above by the entropy of S rel alpha and the entropy of T rel xi along the fibers
02:21:47 heuristics for corollary
02:25:17 corollary: in the context of the previous corollary, if T and S are C^{1+Holder} diffeos, the factor map is Holder, and xi is Sinai partition, then the entropy of S is entropy of S rel alpha
02:27:08 proof of corollary
02:27:41 heuristics for corollary
02:28:25 corollary: if alpha and rho are smooth Z^k actions with rho a Holder factor, chi is a coarse Lyapunov exponent for rho, and the entropy of rho along W^chi is positive, then chi is also a coarse Lyapunov exponent for alpha
02:32:20 fake example: unstables not projecting onto unstables
02:37:21 more interesting example

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Links:

Mentioned paper:
https://www.aimsciences.org/article/d...
https://arxiv.org/abs/1001.2473

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License:

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