A one day meeting on dynamical systems and ergodic theory. There will be connections to number theory via Diophantine approximation. The three speakers will be
Anish Ghosh (Tata institute) Title of talk: Levy-Khintchine theorems: a brief history and recent progress
Abstract: I will describe some beautiful limiting theorems on continued fractions. I will then introduce a method of interpreting these theorems using dynamical systems on homogeneous spaces. This interpretation allows us to provide a new perspective on this classical question, and to prove new results.
Dmitry Kleinbock (Brandeis) title of talk: Averaging over dilated submanifolds and Diophantine approximation
Abstract: Consider an ergodic R^d-action on a probability space, and take a smooth k-dimensional submanifold M of R^d. When can one prove an ergodic theorem for averages over dilated copies of M? A well-studied special case is that of spherical averages. We prove a quantitative theorem of this kind assuming exponential multiple mixing of the action. This is applied to the diagonal action on the space of unimodular lattices in R^{d+1} to produce a zero-one law for uniform multiplicative Diophantine approximation. In particular we will discuss a uniform version of Littlewood’s conjecture (stronger than the original one) and conclude that its set of exceptions has measure zero. Joint work with Prasuna Bandi and Reynold Fregoli.
Mike Todd (St. Andrews) Title of talk: Almost sure orbits closeness
Abstract:
Given a dynamical system $f$ with initial conditions $x, y$ and a sequence $(r_n)_n$, we define the set $E_n$ as the pairs $x, y$ where there is some pair $1\leq i, j\leq n$ such that the distance between the iterates $f^i(x)$ and $f^j(y)$ is less than $r_n$. If $(r_n)_n$ shrinks sufficiently slowly, almost every pair $x, y$ will meet this condition for all large enough $n$, i.e., $(\mu\times \mu)(\limsup_n E_n)=1$. On the other hand, if $(r_n)_n$ shrinks too quickly then the measure of this set is zero. We are interested in the transition between these behaviours: for simple maps we have a condition on $(r_n)_n$ which gives a dichotomy on the measure of $\limsup_n E_n$ being 0 or 1, depending on the condition being satisfied or not. For more general systems, we get close to such a dichotomy, depending on the system’s properties. In this talk I will outline these and related results, joint work with Kirsebom, Kunde and Persson.
The talks will be from 13:30 to 17:30 in LG02 (Fry building) (including breaks), schedule:
12:00 Lunch
13:30-14:30 Anish Ghosh
15:00-16:00 Mike Todd
16:15-17:15 Dmitry Kleinbock
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