Littlewood conjectures

The Littlewood conjecture in simultaneous approximation is a famous open problem from the 1930s in the border line between Diophantine approximation and homogeneous dynamics. I will recall the conjecture, show some pictures, survey what is known, and discuss an analogue called the p-adic Littlewood conjecture and the positive characteristic counterpart of this story.

Here are the recording and slides of the lecture.

Introduction to measure theoretic entropy

I will define measure theoretic entropy and entropy associated with a measure preserving transformation. I will also prove some basic properties of these notions. Most of the talk will follow the book by Manfred Einsiedler, Elon Lindenstrauss, and Thomas Ward.

Here is the recording of the lecture.

Measure theoretic entropy (cont.)

Continuation of last week's lecture.

Here is the recording of the lecture.

Topological entropy

I will define the topological entropy of a continuous homeomorphism of a compact space and prove it coincides with the supremum of the metric entropies as defined by Yiftach. I will illustrate those notions with concrete computations in the case of the doubling map and the Bernoulli Shifts. I will follow the presentation of Chapter 5 in Einsiedler-Lindenstrauss-Ward.

Here are the notes for Florent's lecture, and here is the recording of the lecture .

More entropy

An introduction to conditional expectation, conditional measures, and the construction and basic properties of conditional entropy.

In these two lectures, we will study a refinement of the notion of conditional entropy from measurable partitions to general countably generated sigma-algebras. In the first lecture, we will give a "crash course" on conditional expectation/measures, and introduce, based on this machinery, the concept of conditional entropy of a sigma-algebra. In the second lecture, we will define the conditional entropy of a measure-preserving transformation and prove the so-called Abramov-Rokhlin formula, which relates the entropy of a system with the one of any of its factors. If time permits, we will say a word about the Pinsker factor of a measure-preserving system, which can be thought of as its maximal factor of zero entropy.

Here is the recording of the first lecture.

More entropy (cont.) Here is the recording of the second lecture.

Here are the notes for both of Carlos's lectures.

Hausdorff dimension bound for exceptions to Littlewood conjecture (1)

During previous talks by Yiftach & Florent we have been introduced to measure theoretic & topological entropy of a Borel probability measure-preserving system, ending with the Variational Principle relating both notions. In the next two talks, Ofek Eliyahu and I will show an application of these ideas in Diophantine approximations. Specifically, we will follow Einsiedler, Katok and Lindenstrauss in order to prove that the set of real pairs which violates Littlewoods conjecture has zero Hausdorff dimension. Proof will go by translating the statement to homogenous dynamics via its relation to the natural action of SLn(R) on the space of lattices of covolume one.

Here are the recording and the notes.

Hausdorff dimension bound for exceptions to Littlewood conjecture (2)

Here are the recording and the notes.

Hausdorff dimension bound for exceptions to Littlewood conjecture (3), and Rudolph's x2, x3 theorem

Ofek's talk will be a continuation of last week's talk. Here are his notes.

Abstract for Michael's talk:

Rudolph's theorem states that the Haar probability measure on the circle R/Z, is the unique measure invariant under the x2 and x3 maps which has positive entropy for x2 and which is ergodic for the joint action of x2 and x3, in contrast to the separate action of x2 and x3 where there is an abundance of ergodic measures for each of the maps.

One of the key points in the proof is that the atoms of a certain sigma algebra are given by orbits of a fixed group G. This allows to obtain a family of measures on G by applying a translation on each of the conditional measures, and the understanding of those measures yields the result. This construction is a special case of a more general construction known as leafwise measures which plays a central role in the measure classification we are trying to understand.

Here are the slides for Michael's talk and the recording of both talks.

Rudolph theorem (cont.)

Continuation of last week's talk. Here are the slides for all three talks and the recording.

Rudolph's theorem (cont.) and Leafwise measures

Michael will conclude his discussion of Rudolph's theorem.

Abstract for Tsivqa's talk: We will discuss the notion of leafwise measures. In the first talk we will talk about the definition and the important properties of leafwise measures, and in the next meetings we will delve into the details of the proofs of these properties. Here is the recording.

Leafwise measures (cont.)

Properties of leafwise measures

We will discuss properties of leafwise measures: Weikun will discuss the shift of the basepoint of a leafwise measures, how to detect invariance, and the notion of recurrence.

Here are the notes for Weikun's lecture and the recording.

Entropy, leafwise measures and invariance

In previous lectures we have developed the basic theories of entropy and leafwise measures. In these lectures we will study the connections between them. In particular we will study how certain entropy assumptions can be used to prove extra invariance of the underlying measure along certain unipotent directions. The main reference is the Clay notes by Einsiedler and Lindenstrauss, and the main results are Theorem 7.6 and 7.9.

Here are slides, the first recording, and the second recording.

The product structure

In this lecture, we will concentrate on the product structure among leafwise measures. Suppose a is an element in the diagonalizable group which contracts some nontrivial unipotent group U- but acts isometrically on orbits of some other group T, we will show the leafwise measures for TU- will be the product of the leafwise measures for T and U-. This relation can also be extended to the semi direct product case. The main reference is the Chapter 8 of the Clay notes by Einsiedler and Lindenstrauss.

recording and slides for the talk.

Review + The high entropy method for higher rank torus actions -- the SL(3,R) case

Here are the notes from the review session.

We follow the first half of section 9 of the Pisa notes and derive a measure rigidity result due to Einsiedler and Katok for measures invariant under the full diagonal group with at least one element whose measure theoretical entropy is large enough -- in the SL(3,R) case. This relies on the well earned properties of leaf-wise measures we covered in the last several meetings. In particular, we use a basic upper bound on the a.s. growth rate of leaf-wise measures of balls (Theorem 6.30), the formula for the entropy contribution of leaf-wise measures of stable subgroups (Theorem 7.6) discussed by Brian, and the product structure of such measures (Corollary 8.8) and the "leaf-wise of a product Lemma" (Lemma 8.10) discussed by Daren.

Here are the notes and recording from Erez' lecture.

High entropy method (2).

Here are the notes from Weikun's lecture and recording of both lectures.

High entropy method (3).

Here are the notes and the recording.

Low Entropy Method: The SL2(R) x Something Else case.

We present the low entropy method developed by Lindenstrauss in 2006 for quotients of the form (SL2(R) x Something Else)/Gamma where Gamma is a discrete subgroup intersecting Something Else finitely. The main result is that any probability invariant under the diagonal matrix group A in SL2(R) is both algebraic and SL2(R)-invariant if all ergodic components with respect to A are of positive entropy and Something Else is recurrent. A sketch of the argument can be found in Chapter 10 in the Pisa notes. I will follow Chapter 7 of this paper .

Here are the slides and recording of the talk.

Low Entropy Method: The SL2(R) x Something Else case (continuation).

Here is the recording of the talk.

Low entropy method on a quotient SL(3,R)/Gamma

We will discuss the low entropy method for A action on SL(n, R)/Gamma developed in Einsiedler, Katok, and Lindenstrauss's (EKL) paper. Mostly we will focus on n=3 case for simplicity. Especially, when the leafwise measure with respect to G^{-} is concentrated on one parameter unipotent subgroup, we will prove that the low entropy method gives additional invariance. This method is the one of main ingredients for the measure rigidity theorem in the paper.

Here are the slides for the talk and the link to the video (apologies, no recording of the first hour of the meeting, I pressed "pause" and forgot to unpause).

Combining high and low entropy

I will explain how to combine high and low entropy to complete the classification of A-invariant and ergodic measures with positive entropy on the space of 3-lattices. The main ingredients are:

- Einsiedler-Katok measure rigidity, explained by Erez (high entropy).

- The main theorem of last week's session, explained by Homin (low entropy)

- A version of Ratner's measure classification for SL_2(R)-invariant measures (that is significantly easier than the general version, see here for a relatively short and self contained proof).

I will roughly follow the lines of section section 6 in [EKL]. Here are the notes and recording.

Random walks on homogeneous spaces, Spectral Gaps, and Khintchine's theorem on fractals

Khintchine's theorem in Diophantine approximation gives a zero one law describing the approximability of typical points by rational points. In 1984, Mahler asked how well points on the middle third Cantor set can be approximated. His question fits into an attempt to determine conditions under which subsets of Euclidean space inherit the Diophantine properties of the ambient space. In this sequence of talks, I discuss a complete analogue of the theorem of Khintchine for certain fractal measures which was recently obtained in this paper, in collaboration with Osama Khalil. Our results hold for fractals generated by rational similarities of Euclidean space that have sufficiently small Hausdorff co-dimension. The main ingredient to the proof is an effective equidistribution theorem for associated fractal measures on the space of unimodular lattices. The latter is established using a spectral gap property of a type of Markov operators associated with a random walk related to the generating similarities.

June 6 meeting: I will give an overview of the problem, state the analogue to Khintchine's theorem for fractals and then give an outline of the proof. Then I provide a full proof of the convergence part of Khintchine's theorem for fractal measures assuming effective equidistribution of a sequence of associated measures on expanding horocycle orbits. In the remaining time, I will outline the argument needed for the divergence part, the additional difficulties, and how they can be overcome. Recording of first lecture and notes (if the link to the notes does not work for you send me an email and I will send it to you).

June 10 meeting: For the proof of effective equidistribution of the horocycle measures, we relied on a spectral gap property for certain averaging operators. In this meeting, I will introduce the full setup necessary to define the averaging operators and prove the spectral gap property. Recording of second lecture and notes.

June 13 meeting: I will use the spectral gap to prove effective equidistribution of certain fractal measures on horocycle orbits. If time remains, I will show how this can be used to prove effective equidistribution of spherical averages in the special case of random walks associated to missing digit Cantor sets. Recording of the third lecture and notes.