Fall 2022

I will show that groups with purely non-Hausdorff singularities (along with some other assumptions) have subexponential growth with bounded exponents. 

In this talk, I am going to talk about some important classical definitions, theorems etc. about dynamical systems on the circle. We are going to discuss the following questions.

* When a circle diffeomorphism $f : S^1 \to S^1$ is conjugate to a pure rotation? In other words, when does there exist a homeomorphism $h : S^1 \to S^1$ and a number $a$ such that $h(f(x))=h(x)+a$?

* When the conjugating homeomorphism $h$ is known to be a diffeomorphism? How smooth is it?

* What if $f$ has a critical point and/or a break point?

* What if a group with more than one generator acts on the circle?

We will introduce a class of odometers (variations on the standard dyadic adding machine), discuss some of their dynamical properties, and show how all odometers in this class are orbit equivalent to the "universal odometer" via methods from a paper by Kerr-Li : 1912.02764.pdf (arxiv.org).

Consider a rational function ϕ over a global field K whose degree is d and take successive preimages of  0. If the number of preimages of ϕ-n(0) is dn for every n, we can construct a d-ary tree by putting the n-th preimages of 0 in the level n. It turns out that the Galois group G∞(ϕ) of a certain field extension of K acts on the d-ary tree, so it can be seen as a subgroup of { Aut}(T∞).

When can we say anything about the index [{ Aut}(T∞): G∞(ϕ)]? What can it be said?

This time I will continue showing some examples of infinite index and if I have enough time, an application to Number Theory.

Consider a rational function ϕ over a global field K whose degree is d and take successive preimages of  0. If the number of preimages of ϕ-n(0) is dn for every n, we can construct a d-ary tree by putting the n-th preimages of 0 in the level n. It turns out that the Galois group G∞(ϕ) of a certain field extension of K acts on the d-ary tree, so it can be seen as a subgroup of { Aut}(T∞).

When can we say anything about the index [{ Aut}(T∞): G∞(ϕ)]? What can it be said?

In my talk, I will mention some known results about this question besides some applications to Arithmetic problems. 

We introduce entropy, first used by Kolmogorov to distinguish Bernoulli shifts, as an isomorphism invariant for probability-measure-preserving dynamical systems.  We also introduce the f-invariant for measure-preserving dynamical systems of free groups as an extension of classical entropy.  If we have time we will also introduce the notion of orbit equivalence and its relationship with entropy. 

The study of groupoids is deeply connected with the study of dynamical systems. In this analogy, discrete topological dynamics, i.e. those topological systems over a discrete (rather than continuous) time parameter, correspond to what are known as étale groupoids. The first goal of this talk is to illustrate the connection between these concepts, especially in the discrete-étale setting. After this, we shall discuss how to associate, to each étale groupoid, a C*-algebra, an algebraic/analytic object, and discuss how dynamical properties of the groupoid are associated to algebraic properties of the corresponding C*-algebra. After this we shall discuss, time permitting, the generalization of these topics to those not-necessarily discrete dynamical systems, and/or other connections between groupoids and the study of C*-algebras.

I will provide an introduction to iterated monodromy groups and their standard actions, first in the case of finite degree partial self-coverings of path connected spaces, then its generalization to an expanding self-covering map on a connected space and how this case relates to the theory of Ruelle-Smale systems, hyperbolic groupoids, and hyperbolic duality.

I will give a moderate introduction to the interplay between groups and topological dynamics. Especially, I will introduce some group theoretical problems such as simplicity, torsion, growth, amenability, etc., and how topological dynamics can be used to solve them.