Seminar on homogeneous dynamics and applications

Tel Aviv University, Fall 2018

	Time and place	Wednesday 13:10-15 Schreiber 209
	About	We will amuse ourselves by reading some books and papers about random matrix products. This is an important topic, which among other applications, is used in the analysis (Benoist-Quint, Eskin-Mirzakhani, Eskin-Lindenstrauss) of stationary measures on homogeneous dynamics and moduli spaces of translation surfaces. After covering the main results on random matrix products, we will study random walks on homogeneous spaces and see how this tool gets used.

	Schedule	October 25, Ilya Gringlaz Tools from (commutative) ergodic theory The lecture will present basic results in ergodic theory. These form the background for the (noncommutative) analysis of random matrix products, and are also used in the proofs. Results to be covered are ergodicity, the Birkhoff pointwiseergodic theorem, von-Neumann mean ergodic theorem, and the "Lemma that Alex loves" (a result saying that for an ergodic system, a.s. the differences between time and space averages change sign infinitely many times). Reading material for the ergodic theorems is Einsiedler and Ward, Ergodic Theory with a view towards Number Theory. The "lemma that Alex loves" is proved as Lemma 2.18 in the book Random Walks on Reductive Groups by Benoist and Quint and in the paper G. Halasz, Remarks on the remainder in Birkhoff's ergodic theorem, Acta Math. Hung. 28(3) 1976. October 31, Ilya Gringlaz and Amit Wolecki Oseledec theorem Ilya will complete last week's presentation. Amit will begin the discussion of Oseledec's theorem which is based on this paper by Simion Filip. For more information about cocycles and bundle maps, a good reference is the book by R. J. Zimmer, Ergodic theory and semisimple groups. NOTE: NO MEETING ON NOVEMBER 7 November 14, Amit Wolecki Oseledec theorem (continued). November 21, Amit Wolecki Oseledec theorem (continued). Amit will continue the proof of the Oseledec theorem. Here are Amit's notes. November 28, Amit Wolecki and Itamar Cohen-Matalon Oseledec theorem (conclusion) + unique ergodicity + Breiman theorem. Amit will complete the proof of the Oseledec theorem. Itamar will discuss the space of measures on a compact metric space, and the implication of unique ergodicity. December 5, Itamar Cohen-Matalon Breiman theorem. Itamar will discuss Breiman's theorem about random walks with a unique stationary measure. One place to read about it is the book of Benoist and Quint, Random walks on reductive groups, Part 1, Chap. 2. Here are Itamar's notes. December 12, Uriya Pumerantz Stationary measures, limit measures, Martingale theorem Uriya will present the decomposition of a stationary measure into its limit measures. See the book of Benoist and Quint, Random walks on reductive groups, section 1.5, and see Uriya's notes. December 19, Roie Salama Stationary measures on projective space Roie will introduce stationary measures on projective spaces for irreducible linear random walks, following chapter 3.2 of the book of Benoist and Quint. Here are Roie's notes. December 26, Arijit Ganguly Positivity of the first Lyapunov exponent Is the Oseledec filtration trivial? Arijit Ganguly will discuss a well-studied situation when it is not. Namely, Furstenberg showed that if mu is a compactly supported measure on SL(V) such that the semigroup generated by supp(mu) is unbounded and strongly irreducible, then the norms of the matrices g_n ... g_1 grow exponentially almost surely. The proof that Arijit will present is in the book of Benoist and Quint, section 3.7, and is based on the "Lemma that Alex loves" (lecture of Oct 25) and on work of Guivarch and Raugi. Wednesday January 2, no seminar Action Now meeting Weizmann institute For more information on the meeting please see the Action Now webpage. Wednesday January 9, Arijit Ganguly Positivity of the first Lyapunov exponent (continued) Arijit will complete the proof of some lemmas used in the proof from two weeks ago. Then he will present some preparatory material for Yiftach's lecture. Wednesday January 16, Arijit Ganguly Positivity of the first Lyapunov exponent (continued) Arijit will present the proof of a result required for Yiftach's lecture. See theorem 2.2 of this paper. Monday January 21, Yiftach Dayan 13-15, Schreiber room 210 PLEASE NOTE SPECIAL TIME Normal numbers on fractals Yiftach will present a number-theoretic application of the results on random walks proved during the semester: let mu be the natural measure supported on the dilate of the middle thirds set Cantor set, dilated by an irrational number. Then mu-almost every point is normal in base 3. This will follow from a more general result about certain semigroups of circle maps with a unique stationary measure. Work in progress of Yiftach, Arijit and Barak.
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