Products of random matrices: now with a parameter!

Speaker: 

Victor Kleptsyn

Institution: 

CNRS, Institute of Mathematical Research of Rennes

Time: 

Tuesday, April 5, 2016 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

Consider a random product of i.i.d. matrices, randomly chosen from SL(2,R), satisfying some reasonable nondegeneracy conditions (no finite common invariant set of lines, no common invariant metric).Then a classical Furstenberg theorem implies that the norm of such a random product almost surely grows exponentially.

Now, what happens if each of these matrices depends on an additional parameter? We will discuss the case when the dependence of angle is monotonous w.r.t. the parameter: increasing the parameter «rotates all the directions clockwise».

It turns out that (under some reasonable conditions)
- Almost surely for all the parameter values, except for a zero Hausdorff dimension (random) set, the Lyapunov exponent exists and equals to the Furstenberg one.
- Almost surely for all the parameter values the upper Lyapunov exponent equals to the Furstenberg one
- At the same time, in the no-uniform-hyperbolicity parameter region there exists a dense subset of parameters, for each of which the lower Lyapunov exponent takes any fixed value between 0 and the Furstenberg exponent.

The talk is based on a joint project with A. Gorodetski.

Global Structure Theorems

Speaker: 

Matt Foreman

Institution: 

UC Irvine

Time: 

Tuesday, April 12, 2016 - 1:00pm to 2:00pm

Location: 

440 R

This lecture addresses two central problems in classical ergodic theory: the classification problem and the realization problem. An historical focus of ergodic theory has been the structure and properties of single transformations. Perhaps the most prominent is the Furstenberg-Zimmer structure theorem which describes ergodic transformations in terms of limits of compact and weakly mixing extensions.

This lecture discusses a new phenomenon, Global Structure Theorems. We define two categories: the Odometer Based Systems (finite entropy transformations that have an odometer factor) and Circular Systems (those diffeomorphisms built using a version of the Anosov-Katok technique.) The morphisms  in each category are measure-theoretic joinings.

The main result is that these two categories are isomorphic by a composition-preserving bijection that that takes conjugacies to conjugacies, extensions to extensions, weakly mixing extensions to weakly mixing extensions, compact extensions to compact extensions, distal towers to distal towers (and more). In short, all of the structure present in the odometer based systems is exactly reflected in the Circular Systems and vice versa.

The lecture will conclude with a provocative conjecture.

This is joint work with B. Weiss.

Homemade stable intersections

Speaker: 

Victor Kleptsyn

Institution: 

CNRS

Time: 

Tuesday, March 29, 2016 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

My talk will be related to the stable intersections of dynamically defined Cantor sets (as well as to the finding intervals in their sums). In the famous work of Morreira and Yoccoz it is shown that for a generic pair of dynamically defined Cantor sets with sum of their Hausdorff dimensions greater than one, their intersection (provided that they do intersect) is stable under small perturbations.

However, it would be nice to be able to check this for a particular couple of sets, and up to this moment the only explicitly checkable sufficient condition that is known is that the product of the thicknesses is greater than one.

I will present an approach to finding such a sufficient condition (eventually, in a computer-assisted way) that can be hopefully used to attack the conjecture that claims that F(2)+F(4) contains an interval (that is currently open).

My talk will be based on a joint project will A. Gorodetski, A. Gordenko and E. Nesterova.

stable intersections of regular Cantor sets with large Hausdorff dimensions X

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, March 1, 2016 - 1:00pm to 1:50pm

We will talk about a paper by A. Moreira and J.C. Yoccoz, where they proved a conjecture by Palis according to which the arithmetic sums of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval.

Stable intersections of regular Cantor sets with large Hausdorff dimensions IX

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, February 16, 2016 - 1:00pm to 1:50pm

We will talk about a paper by A. Moreira and J.C. Yoccoz, where they proved a conjecture by Palis according to which the arithmetic sums of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval.

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