### Dynamics Archives for Academic Year 2020

#### Period Doubling Cascades - The big unsolved problem

When: Thu, August 29, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jim Yorke (UMD) - https://user.eng.umd.edu/~yorke/

#### Equilibrium measures for some partially hyperbolic systems

When: Thu, September 5, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Agnieszka Zelerowicz (UMD) -
Abstract: In this joint work with Vaughn Climenhaga and Yakov Pesin we study thermodynamic formalism for topologically transitive partially hyperbolic systems in which the center-stable bundle is integrable and nonexpanding, and show that every potential function satisfying the Bowen property has a unique equilibrium measure. Our method is to use tools from geometric measure theory to construct a suitable family of reference measures on unstable leaves as a dynamical analogue of Hausdorff measure, and then show that the averaged pushforwards of these measures converge to a measure that has the Gibbs property and is the unique equilibrium measure.

#### In Teichmuller dynamics - big orbit closures are trivial!

When: Thu, September 12, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Paul Apisa (Yale) -
Abstract - It is a remarkable fact that any holomorphic one-form can be presented as a collection of polygons in the plane with sides identified by translation. Since GL(2, R) acts on the plane (and polygons in it), it follows that there is an action of GL(2, R) on the collection of holomorphic one-forms on Riemann surfaces. This GL(2, R) action is the group action generated by scalar multiplication and Teichmuller geodesic flow on the moduli space of holomorphic one-forms. By work of Eskin, Mirzakhani, and Mohammadi given any holomorphic one-form, the closure of its GL(2, R) orbit is an algebraic variety in the moduli space of holomorphic one-forms and, in a natural coordinate system (called period coordinates), the defining equations are linear.

In the first part of the talk, I will explain how studying the dynamics of GL(2, R) connects to the problem of determining the extent to which two metrics on Teichmuller space - the Teichmuller metric and the Caratheodory metric - agree. This works connects to recent work of Markovic and Gekhtman and to work of Forni on the Lyapunov spectrum of Teichmuller geodesic flow.

In the second part of the talk, I will define the rank of GL(2, R) orbit closures - a measure of size related to (and often agreeing with) dimension. Using the work in the first part of the talk, I will explain why if the GL(2, R) orbit closure of a holomorphic one-form on a genus g Riemann surface has rank at least g/2, then the orbit is dense or its closure is contained in a locus of branched covers of lower genus Riemann surfaces.

No background on Teichmuller theory or dynamics will be assumed for the talk. This work is joint with Alex Wright.

#### Coexistence of attractors and their stability

When: Thu, September 19, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Liviana Palmisano (Uppsala) -

Abstract: In unfoldings of rank-one homoclinic tangencies, there exist codimension 2 laminations of maps with infinitely many sinks. The sinks move simultaneously along the leaves. As consequence, in the space of real polynomial maps, there are examples of: Hénon maps, in any dimension, with infinitely many sinks, quadratic Hénon-like maps with infinitely many sinks and a period doubling attractor, quadratic Hénon-like maps with infinitely many sinks and a strange attractor. The coexistence of non-periodic attractors, namely two period doubling attractors or two strange attractors, and their stability is also discussed.

#### On Cross Sections to the Horocycle and Geodesic Flows on Quotients by Hecke Triangle Groups $G_q$

When: Tue, September 24, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Diaaeldin Taha (University of Washington) -

Abstract: In this talk, we explore explicit cross-sections to the horocycle and geodesic flows on $\operatorname{SL}(2, \mathbb{R})/G_q$, with $q \geq 3$. Our approach relies on extending properties of the primitive integers $\mathbb{Z}_\text{prim}^2 := \{(a, b) \in \mathbb{Z}^2 \mid \gcd(a, b) = 1\}$ to the discrete orbits $\Lambda_q := G_q (1, 0)^T$ of the linear action of $G_q$ on the plane $\mathbb{R}^2$. We present an algorithm for generating the elements of $\Lambda_q$ that extends the classical Stern-Brocot process, and from that derive another algorithm for generating the elements of $\Lambda_q$ in planar strips in increasing order of slope. We parametrize those two algorithms using what we refer to as the \emph{symmetric $G_q$-Farey map}, and \emph{$G_q$-BCZ map}, and demonstrate that they are the first return maps of the geodesic and horocycle flows resp. on $\operatorname{SL}(2, \mathbb{R})/G_q$ to particular cross-sections. Using homogeneous dynamics, we then show how to extend several classical results on the statistics of the Farey fractions, and the symbolic dynamics of the geodesic flow on the modular surface to our setting using the $G_q$-BCZ and symmetric $G_q$-Farey maps. This talk is self-contained and does not assume any prior knowledge of Hecke triangle groups or homogeneous dynamics.

#### On the mixing local limit theorem and applications

When: Thu, October 3, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Peter Nandori (Yeshiva University) - https://www.math.umd.edu/~pnandori/
Abstract: We consider a joint generalization of mixing and the local limit theorem (MLLT). First we verify that the MLLT holds for several examples of chaotic dynamical systems. Then we discuss two applications. The first application is in the study of global observables. Let us consider a periodic or almost periodic mechanical system preserving an infinite physical measure. Then global functions are uniformly continuous functions that admit average values over large boxes. We show how the MLLT for a periodic system can be used to prove the mixing of such global functions. We also study the Birkhoff theorem for global observables in some toy models. The second application is an extension of the local limit theorem to higher order expansions (also known as Edgeworth expansion).

This talk is based on joint work with Dmitry Dolgopyat, Marco Lenci and Francoise Pene.

#### Kakutani Equivalence of Unipotent Flows

When: Thu, October 10, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Kurt Vinhage (Penn State) -

Abstract: Algebraic dynamical systems are good representatives and case studies for associated problems in a more general study. In the positive entropy setting, the usual canonical representatives of such systems have a unique invariant, the entropy. For zero entropy flows, however, the equivalence problem is much richer: two unipotent flows on homogeneous spaces of semisimple Lie groups are measurably conjugate if and only if they are equivalent in a much stronger algebraic fashion, giving a huge number of distinct flows and transformations. Weakening conjugacy to orbit equivalence trivializes the problem due to Dye's theorem. I will describe a notion of equivalence (called Kakutani equivalence) which weakens measurable conjugacy without trivializing it, discuss our recent work toward classifying zero entropy algebraic flows up to this notion. Joint with Adam Kanigowski and Daren Wei.

#### Translation surfaces with multiple short saddle connections

When: Thu, October 17, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Ben Dozier (Stony Brook) - https://www.math.stonybrook.edu/~bdozier/
Abstract: Questions about billiards on rational polygons can be converted into questions about the straight-line flow on translation surfaces. These in turn can be converted (via renormalization) into questions about the dynamics of the SL_2(R) action on strata of translation surfaces. By the pioneering work of Eskin-Mirzakhani, to understand dynamics on strata, one is led to study affine measures (these are supported on affine invariant manifolds and are locally Lebesgue in period coordinates).

It is natural to ask about the interaction between measures of certain subsets of surfaces and the geometric properties of the surfaces. I will discuss a proof of a bound on the volume, with respect to any affine measure, of the locus of surfaces that have multiple independent short saddle connections. This is a strengthening of the regularity result proved by Avila-Matheus-Yoccoz. A key tool is the new compactification of strata due to Bainbridge-Chen-Gendron-Grushevsky-Moller, which gives a good picture of how a translation surface can degenerate.

#### Solving the cohomological equation

When: Thu, October 31, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Federico Rodriguez-Hertz (Penn State) -

Abstract: There are different methods to show vanishing of the cohomological equation u-uoT=w. We plan to discuss some of this methods, especially in the case of systems with some hyperbolicity as well as discuss higher rank versions of them. Different degrees of regularity will as well be discussed.

#### A prime system with many and big self-joinings

When: Thu, November 7, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Jon Chaika (University of Utah) - https://www.math.utah.edu/~chaika/
Abstract: Let (X,mu,T) be a measure preserving system. A factor is a system (Y,nu,S) so that there exists F with SF=FT and so that F pushes mu forward to nu. A measurable dynamical system is prime if it has no non-trivial factors. A classical way to prove a system is prime is to show it has few self-joinings, that is, few T x T invariant measures that on X x X that project to mu. We show that there exists a prime transformation that has many self-joinings which are also large. In particular, its ergodic self-joinings are dense in its self- joinings and it has a
self-joining that is not a distal extension of itself. As a consequence we show that being quasi-distal is a meager property in the set of measure preserving transformations, which answers a question of Danilenko. This is joint work with Bryna Kra.

#### The mapping class group of minimal subshifts

When: Thu, November 14, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Kitty Yang (Northwestern University) -

Abstract: Let $(X, \sigma)$ be a subshift and $\textrm{Aut}(X)$ be the automorphism group, the group of self conjugacies of $(X, \sigma)$. The mapping class group, denoted $\mathcal{M})(\sigma)$, is the group of self flow equivalences up to isotopy. We show that $\mathcal{M}(\sigma)$ is constrained in the case of low-complexity minimalsubshifts, similar to constraints on $\textrm{Aut}(X)$. We classify mapping class group for Sturmians, and also a generic set of codings of IETs.

#### A Local limit theorem for random distance expanding maps

When: Thu, November 21, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Yeor Hafouta (Ohio State) -

Abstract: In the first part of the talk I will review several limit theorems for stationary processes such as hyperbolic and expanding dynamical systems.
In the second part I will discuss recent results for random, uniform and non uniform, distance expanding dynamical systems. The main focus will be the local central limit theorem, and if time permits I will also discuss additional results such as the Berry-Esseen theorem (optimal convergence rate in the central limit theorem) and an almost sure central limit theorem. This part is partially based on joint work with Yuri Kifer.

#### Statistical properties of some partially hyperbolic maps

When: Thu, December 5, 2019 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Carlangelo Liverani ( U. Roma Tor Vergata) - https://www.mat.uniroma2.it/~liverani/
Abstract: The study of the spectrum of the transfer operator has proven to be an extremely powerful method to study the statistical properties of hyperbolic dynamical systems (both in discrete and continuous time). The next natural step is to try to apply such a strategy to partially hyperbolic systems. Unfortunately, at the moment, almost no result is available, with the notable exception of some skew-products and group extensions. I will discuss an attempt to progress in this direction by investigating a simple (but not of skew-product type) two dimensional system.

#### Local rigidity of parabolic actions

When: Thu, January 30, 2020 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Zhenqi Wang (Michigan State) -
Abstract:

We show $C^\infty$ local rigidity for a broad class of abelian unipotent algebraic actions on homogeneous spaces of semisimple Lie groups.
The method of proof is a combination of KAM type iteration scheme and representation theory. This is the first time in literature (strong) local rigidity for parabolic actions is addressed.

#### Ergodic properties of low complexity symbolic systems

When: Thu, February 6, 2020 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Van Cyr (Bucknell University) - https://vancyr.scholar.bucknell.edu/
Abstract: The topological entropy of a subshift is the exponential growth rate of the number
of words of different lengths in its language. For subshifts of entropy zero, finer
growth invariants constrain their dynamical properties. In this talk we will survey
how the complexity of a subshift affects properties of the ergodic measures it carries.
In particular, we will see some recent results (joint with A. Johnson, B. Kra, and
A. Sahin) relating the word complexity of a subshift to its set of ergodic measures
as well as some applications.

#### Dynamical random walks with internal states

When: Thu, February 20, 2020 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Davit Karagulyan (UMD) -
Abstract: By dynamical random walk (DRW) we mean a map $F$ acting on $M\times \Z^d$, where $M$ is the internal state of the particle. This means that at each point in time $n\geq 0$ we have a pair $(x_n, z_n)\in M\times \Z^d$, where $z_n$ is the position of the particle and $x_n$ is a latent variable which changes by the local dynamics $F$ and makes the particle move from one site to another. One is then interested in the statistical properties of the position $z_n$ of the particle as $n$ goes to infinity.

I will discuss different examples of (DRW) and then introduce a one dimensional model of (DRW), whose local dynamics is driven by expanding maps. I will show that
under certain conditions, we will have the central limit theorem.

#### On the spectral properties of translation flows and substitution systems

When: Thu, February 27, 2020 - 2:00pm
Where: Kirwan Hall 1311

Speaker: Boris Solomyak (Bar-Ilan University) - http://u.math.biu.ac.il/~solomyb/personal.html
Abstract: I will describe some recent results by A. Bufetov and myself on the spectral properties of typical translation flows on flat surfaces of genus $g\geq 2$ and substitution dynamical systems. The questions considered concern the dimension of spectral measures and conditions for singularity of the spectrum.

#### TBA

When: Thu, May 7, 2020 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Marcel Guardia (Universitat Politènica de Catalunya) - https://mat-web.upc.edu/people/marcel.guardia/