The talks from online working seminar in ergodic theory session 3
August 2 Title: Multiple mixing for SL(2,R)
Speaker: Jon Chaika
Abstract: We present a special case of an argument of Mozes that mixing implies mixing of all orders for certain Lie groups.
Slides
Video
August 16 Title: Furstenberg's topological x2 x3 result
Speaker: Barak Weiss
Abstract: In 1967 Furstenberg proved that any
closed subset of the one dimensional
torus R/Z, invariant under the two maps
x -> 2x mod 1, x -> 3x mod 1, is either
finite or the entire torus. I will explain a
proof of this result due to Boshernitzan (1994).
Furstenberg's proof is slightly longer
but perhaps more conceptual. I will explain the main steps
in Furstenberg's approach and their connection to joinings.
Lecture Notes
Video
August 30 Title: No IET is mixing
Speaker: Gianluca Faraco
Abstract: In 1980, Katok proved that no interval exchange transformation (IET) is mixing for any Borel invariant measure. The same holds for any special flow constructed by mean of any IET and a roof function of bounded variation. In this talk, I aim to explain to you the proof of these results.
Notes
Video
September 13 Title: Fubini foiled: pathological foliations from symbolic codings
Speaker:Aaron Calderon
Abstract: In this talk,
I'll present a counterintuitive construction of A. Katok (exposited by Milnor) which at
first glance seems to contradict Fubini's theorem.
In particular, one can build a
full-measure set E in the unit
square and a foliation of the square by smooth curves such that
any leaf of the foliation meets E in exactly one point.
If time permits,
I'll also mention work of
Ruelle-Wilkinson and Shub-Wilkinson that shows
these sorts of pathological examples
are common in non-uniformly hyperbolic dynamics.
Notes
Video
September 27 Title: Gaps in the sequence √n (mod 1)
Speaker: Keivan Mallahi Karai
Abstract: In this talk I will present some of aspects of the proof
of a theorem of Elkies and McMullen (Duke Math Journal, 2004)
on the asymptotic distribution of the gap sizes for the finite sequence
( √n (mod 1) : 1 ≤ n ≤ N ) as N goes to infinity.
The proof relies, among other things, on tools from homogenous dynamics and
relates the problem to one in the geometry of numbers.
Notes
October 11 Title: Siegel-Veech transform
Speaker: Sayantan Khan
Abstract: In this talk, I will talk about the Siegel-Veech transform, and how it can be used to count the number of cylinders (of bounded length) on a translation surface. This counting result relies amongst other tools, on the ergodicity of the SL(2, R) action on the moduli space of translation surfaces. This talk will not assume prior knowledge of translation surfaces, and most of the techniques used will be techniques coming from homogeneous dynamics.
Notes
Video
October 25 Title: Pointwise ergodic theorem
along a subsequence of integers
Speaker:
Sovanlal Mondal
Abstract
Notes
Video Note: I (Jon) started the recording a bit late. Sorry!
November 8 Title: The question of q, a look at the interplay of number theory and ergodic theory in continued fractions
Speaker:
Joseph Vandehey
Abstract: In the theory of continued fractions, the denominator of the truncated fraction (often denoted q) contains a great deal of information important in applications. However, q is a surprisingly complicated object from the point of view of ergodic theory. We will look at a few problems related to q and see how different techniques have overcome these difficulties, including modular properties (Moeckel, Fisher-Schmidt), renewal-type theorems (Sinai-Ulcigrai, Ustinov), and "nonstandard" arrangements of points (Avdeeva-Bykovskii).
Slides
Video
November 22
Title: A construction of Bowen-Margulis measure
Speaker: Pouya Honaryar
Pre-talk at 8:30 am Utah time (same link)
Abstract: In this talk we try to understand the Bowen-Margulis measure for
geodesic flow on manifolds of (variable) negative curvature.
In the first half-hour (pre-talk), I will discuss the necessary backgrounds
from hyperbolic geometry and dynamics, and during the next hour (main talk),
I will explain a construction due to Hamenstädt, which relates
Bowen-Margulis measure to the Hausdorff measure with respect to a
certain metric.
Slides
Pre-talk video
Talk video
March 2 Title: Ratner/Masur equidistribution by orbit matching
Speaker: Jon Chaika
Abstract: In a 1992 article where she surveyed her recent breakthrough on unipotent flows on homogeneous spaces, Ratner presented an argument
for the equidistribution of horospherical orbits in the context of horocycle
flow on SL(2,R)/Lattice. This idea is separate from the ideas in her celebrated work on unipotent flows and I will present her argument for horospherical equidistribution in the simplest situation I can think of: proving the ergodicity of a particular directional flow on the flat two torus. Ratner's argument has similarities to Masur's criterion for unique ergodicity of translation flows, proven around the same time. Time permitting I will comment on Masur's criterion as well.
Ratner's paper
Video
April 13 Title: Regularity of the cohomological equation for circle rotations
Speaker: Carlos Ospina
Abstract: Given an smooth function h, this talk will focus on solving the equation \psi(Rz)-\psi(z) = h(z) for circle rotations. We will see how the Diophantine condition on the rotation implies smooth solutions.