Скачать или смотреть Rigidity - Week 9 - 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
00:00:31 recap: entropy of a diffeo along an unstable partition
00:07:37 observation: entropy along unstables is well defined (i.e. it is independent of the subordinate partition)
00:08:52 lemma 1: the entropy of a finer partition is not less
00:09:59 lemma 2: the entropy is invariant under the induced action on partitions
00:10:16 observation continued
00:13:06 theorem (higher rank Pesin Stable Manifold Theorem): If alpha is a Z^k action by diffeos and mu is an ergodic measure, then tangent to each coarse Lyapunov direction E^i there is a manifold W^i mu almost everywhere; and the entropy along W^i is well defined
00:15:58 recap: coarse Lyapunov directions
00:20:22 theorem continued
00:24:27 proof of theorem
00:25:41 R^k case
00:27:24 corollary (Huyi Hu): on the half-space where chi^i is positive, entropy along W^i is linear as a function of time
00:28:54 proof of corollary
00:30:10 fibered version of corollary
00:30:55 corollary: on the half-space where chi^i is positive, entropy along W^i is a constant multiple of chi^i
00:32:44 example: rank one case
00:37:33 theorem (Brown-FRH-Wang): entropy of time-n map is the sum of entropies along W^i, where i is such that chi^i(n) is positive
00:39:18 proof of theorem
00:42:40 theorem (Pesin entropy formula): if mu is absolutely continuous, then the entropy is the sum of all dim(E^i) chi^i, or equivalently half of the sum of all |chi^i|
00:44:54 corollary: entropy is a seminorm on Z^k; the unit ball relative to it is a polyhedron
00:47:06 reference: Katok-Katok-FRH
00:47:30 preview: fiber exponents and base exponents
00:50:11 2 of 3
00:50:20 recap: theorem: entropy of time-n map is the sum of entropies along W^i, where chi^i(n) is positive
00:52:05 setup for the suspension case of theorem, for a Gamma action, Gamma a lattice in a higher rank semisimple Lie group
01:00:45 corollary: Oseledets for suspension
01:09:04 vertical and horizontal subfoliations of coarse foliations in the case of suspensions
01:17:27 heuristics for Abramov-Rokhlin in the case of suspensions
01:20:38 theorem (higher rank Abramov-Rokhlin): If mu is an measure invariant under the restriction to a Cartan subgroup of the suspension of a Gamma action by diffeos, if mu projects to Haar, and if W is a coarse Lyapunov foliation projecting to the ij-direcion, then the entropy of time-t map along W is the sum of chi^{ij}(t) and the entropy of time-t map along the vertical subfoliation of W
01:25:42 fiber Lyapunov exponents (aka weights, in the linear case), base Lyapunov exponents (aka roots)
01:27:45 resonant root, non-resonant root
01:28:56 resonant/nonresonant roots in terms of the Abramov-Rokhlin formula
01:29:36 corollary: if chi_{ij} is a non-resonant root, then mu is invariant under the ij-unipotent subgroup
01:31:00 proof of corollary
01:34:49 lemma: a measure is invariant under a G-action admitting long tubular neighborhoods iff its conditionals along orbits are proportional to Haar on G. Similarly the measure is quasi-invariant iff conditionals along orbits are of Haar class
01:38:44 proof of lemma
01:39:11 relation of lemma to corollary
01:40:21 example 1: Gamma = SL(3,Z) acting on 3-torus linearly
01:50:37 preview: example 2: G = SL(3,R), Gamma a lattice in G, the standard action of Gamma on 2-sphere or RP^2
01:52:50 3 of 3
01:52:55 example 1 continued
02:03:56 Dirichlet Unit Theorem
02:05:39 example 1 continued
02:39:55 example 1, cocompact lattice case
02:44:47 Log-L^(k,1) assumption in Oseledets

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