Sums of two homogeneous Cantor sets II

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, May 3, 2016 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets).

Sums of two homogeneous Cantor sets I

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, April 26, 2016 - 1:00pm to 2:00pm

We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets). We will also discuss the connection of this problem with the question on properties of one dimensional self-similar sets with overlaps.

Random metrics on hierarchical graph models

Speaker: 

Victor Kleptsyn

Institution: 

CNRS, Institute of Mathematical Research of Rennes

Time: 

Tuesday, April 19, 2016 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

In the domain of quantum gravity, people often consider random metrics on surfaces, defined as Riemannian ones with the factor being an exponent of the Gaussian Free Field. Though, GFF is only a distribution, not a function, and its exponent is not well-defined. This leaves open the problem of giving a mathematical sense to this definition (or, to be more precise, of showing rigorously that one of the known regularization procedures indeed converges).

In a joint work with M. Khristoforov and M. Triestino, we approach a « baby version » of this problem, constructing random metrics on hierarchical graphs (like Benjamini's eight graph, dihedral hierarchical lattice, etc.). This situation is still accessible due to the graph structure, but already shares with the planar case the complexity of high non-uniqueness of candidates for the geodesic lines. The behavior of some (pivotal, bridge-type) graphs seems to be a good model for the behavior of the full planar case.

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.

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