# 2020-21

##### The seminars are held on Tuesdays at 14:00.
Organizers: Joel Moreira
##### Term 3
• April 27, 2021
Carlos Matheus (Paris)

Title: Hausdorff dimension of Gauss-Cantor sets and their applications to Markov spectrum
Abstract: The classical Lagrange and Markov spectra are closed subsets of the real line associated to certain Diophantine approximation problems introduced by A. Markov in 1879. As it was noticed by O. Perron in 1921, these spectra have nice dynamical characterisations allowing to investigate their structures via the features of the so-called Gauss-Cantor sets (i.e., limit sets of iterated function systems obtained from adequate choices of inverse branches of the Gauss map acting on continued fraction expansions). In particular, it is often the case that some basic questions about the classical spectra can be translated in terms of the Hausdorff dimension of specific Gauss-Cantor sets.

In this talk, we will discuss a joint work in progress with C. Moreira, M. Pollicott and P. Vytnova where (a slightly improved version of) the Pollicott-Vytnova thermodynamical method to compute Hausdorff dimension of Gauss-Cantor sets is explored to find the transition point where the classical spectra acquire full dimension and to give refined bounds on the dimension of their set-theoretical difference.
• May 4, 2021
Nattalie Tamam (San Diego)

Title: Distribution of orbits of geometrically finite groups
Abstract: We often seek to understand a group through the distribution of its orbits on a given space. The distribution of lattice orbits has been studied intensively in a variety of groups. In the talk we will discuss the distribution of orbits of a certain non-elementary discrete hyperbolic groups acting on the light cone, which is not necessarily a lattice, i.e. may have an infinite covolume. We will see what can be said about the distribution, and how it relates to the distribution of the horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.

• May 11, 2021
Anush Tserunyan (McGill University)

Title: Backward and forward ergodic theorems along trees
Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $\{x, T(x), T^2(x), \dots, T^n(x)\}$ in the forward orbit of the point $x$. In joint work with Jenna Zomback, we prove a “backward” ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over arbitrary subtrees of the graph of $T$ that are rooted at $x$ and lie behind $x$ (in the direction of $T^{-1}$). This theorem yields (forward) ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank where the averages are taken along arbitrary subtrees of the standard Cayley graph rooted at the identity. This strengthens results of Grigorchuk (1987), Nevo (1994), and Bufetov (2000).

• May 18 2021
Tuomas Sahlsten (Manchester)

Title: Bourgain’s Lemma 8.43
Abstract: Sometimes I’ve heard a saying that “A good lemma is worth a thousand theorems.”. I am not sure I believe in this in such an extreme, but there may be some truth behind these words. In his paper "The discretized sum-product and projection theorems”, J. Analyse Math. 2010, Jean Bourgain had a slightly hidden Lemma numbered 8.43 on the Fourier transforms of repeated multiplicative convolutions of measures in the real line with a positive local dimension bound in its support. This lemma captures the sum-product phenomenon in additive combinatorics and has turned out to be a very useful tool in bounding Fourier transforms of stationary measures and equilibrium measures for dynamical systems. In this talk we will outline some of our struggles with this Lemma and the recent works with J. Li and C. Stevens that exploit this lemma and its higher dimensional generalisations. We will also briefly discuss how spectral gaps for complex transfer operators help with verifying the assumptions of this lemma in our applications.

• May 25, 2021
Alexandra Skripchenko (Moscow)

Title: Systems of isometries and their relatives
Abstract: Systems of partial isometries of the interval represent a simple combinatorial object which appears in topology in connection with measured foliations on a surface (orientable or non-orientable), in dynamics as a nice model to study billiards in rational polygons and in geometric group theory as a way to describe actions of free groups on R-trees. We will discuss several classes of systems of isometries (interval exchange transformations, interval exchange transformations with flips,interval translation mappings, band complexes) and compare their basic dynamical properties: minimality, ergodicity, invariant measures etc. The talk is mainly based on the joint works with Artur Avila and Pascal Hubert and with Serge Troubetzkoy.

• June 1, 2021
Niloofar Kiamari (Rome)

Title: Locating Ruelle-Pollicott resonances
Abstract:In this talk, we will present some new results regarding the spectrum of transfer operators associated to different classes of dynamical systems. Our goal is to obtain precise information on the discrete spectrum. We will describe a general principle which allows us to obtain substantial spectral information. We will then consider several settings where new information can be obtained using this approach, including affine expanding Markov maps, monotone maps, hyperbolic diffeomorphisms. (Joint work with: Oliver Butterley & Carlangelo Liverani.)

• June 8, 2021
Steve Cantrell (Chicago)

Title: Rough similarity and the Manhattan Curve for metrics on hyperbolic groups
Abstract: Consider a hyperbolic group equipped with two hyperbolic metrics. When comparing two such metrics, a natural question to ask is: when are these metrics roughly similar, i.e. when are they within bounded distance after scaling by a positive constant? In this talk we'll discuss rigidity statements that characterise rough similarity in terms of the properties of the Manhattan Curve: a curve that encodes information about the two metrics. We'll see how to study the Manhattan curve using a blend of ideas coming from ergodic theory and geometric group theory. This is based on joint work with Ryokichi Tanaka.

• June 15, 2021
Manuel Stadlbauer (Rio de Janeiro)

Title: Martin boundaries for twisted Ruelle operators
Abstract: In the context of probability theory, the Martin boundary is constructed through possible limits of subharmonic functions of a Markov operator and allows to classify those paths of the associated process which escape to infinity. However, due to its general nature, it is in general non-trivial to obtain an explicit description. On the other hand, in the presence of negative curvature, it is known from Ancona's work on PDEs that one may identify this boundary with the visual boundary of the ambient space. In this talk, we will consider an analogue of Ancona's setting from the viewpoint of thermodynamic formalism. That is, for a subshift of finite type (X,T) and a map g from X to a set of generators of a hyperbolic group G, we consider the skew product defined by S(x,h):= (T(x), hg(x)) defined on the product of X and G. By fixing a potential function defined on X, one then obtains a so-called twisted transfer operator defined on the Hölder functions on X times G with compact support. Under certain natural assumptions on S and the potential function, it is then possible to identify the Martin boundary associated to the twisted transfer operator with the visual boundary of the G. As the method of proof makes use of arguments introduced by Gouezel and Lalley, we are able to include the critical parameter in this setting. This, in particular, allows to extend results by Roblin and Shwartz on the existence of conformal measures for regular covers of CAT(-1)-spaces to the abscissa of convergence of the underlying Kleinian group. This is joint work with Sara Bispo and is available on https://arxiv.org/abs/2005.03723

• June 22, 2021
Paul Colognese (Warwick)

Title: Asymptotic growth for closed geodesics on translation surfaces
Abstract: A famous result by Margulis states that on negatively curved surfaces, the number of closed geodesics of bounded length on the surface grows exponentially as the bound goes to infinity. Translation surfaces are genus g>1 surfaces that admit a flat metric except at finitely many singular points. The Gauss-Bonnet theorem tells us that these "flat surfaces" should have negative curvature on average which hints at the idea that the singularities behave as points of concentrated negative curvature. In this talk, I'll explore this idea by looking at some results for geometric growth on negatively curved surfaces which generalise to translation surfaces, including Margulis' asymptotic formula.

• June 29, 2021
CANCELLED
##### Term 2
• January 19, 2021
Scott Schmieding (University of Denver)

Title: Local P entropy and stabilized automorphism groups.
Abstract: For a homeomorphism of a compact metric space T : X → X, the stabilized automorphism group of this system consists of all self-homeomorphisms of X which commute with some power of T. Motivated by studying this stabilized group in the setting of symbolic systems, we will introduce and discuss a certain entropy for groups called local P entropy. We show that local P entropy can be used to give a complete classification of the stabilized automorphisms groups of full shifts; in particular, we show the stabilized groups for the 2-shift and the 3-shift are not isomorphic.

• February 2, 2021
Tom Ward (Leeds)

Title: Time-changes preserving zeta functions
Abstract: A dynamical system with finitely many periodic orbits of each period has an associated collection of possible time-changes that preserve the property of counting periodic points for some system. Intersecting over all dynamical systems gives a monoid of time-changes that have this property for all such systems. This universal monoid has remarkable properties: the only polynomials it contains are the monomials, and it is uncountable.

• February 9, 2021
Amir Algom (Penn State)

Title: On the decay of the Fourier transform of self-conformal measures
Abstract: Let P be a self-conformal measure with respect to an IFS consisting of finitely many smooth contractions of [0,1]. Assuming a mild and natural condition on the derivative cocycle, we prove that the Fourier transform of P decays to zero at infinity. This is related to the highly active study of the properties of the Fourier transform of dynamically defined measures, dating back to the important work of Erdos about Bernoulli convolutions in the late 1930's. This is a joint work with Federico Rodriguez Hertz and Zhiren Wang.

• February 16, 2021
Andrey Gogolyev (Ohio State)

Title: Rigidity for higher dimensional expanding maps
Abstract: Expanding maps are self covers of smooth compact manifolds which expand the lengths of all non-zero tangent vectors. Classification of such maps up to topological conjugacy is known due to work of Shub, Franks and Gromov. Classification up to smooth conjugacy should be quite different because periodic points of expanding maps carry invariants of C^1 conjugacy. Shub and Sullivan classified expanding maps on the circle up to smooth conjugacy on the circle. I will explain smooth classification of expanding maps in higher dimensions on and open dense set in the space of expanding maps. Joint work with F. Rodriguez Hertz.

• February 23, 2021
Anh Le (Ohio State University)

Title: Sets of multiplicative recurrence and applications
Abstract: A set R of positive rationals is a set of multiplicative recurrence if for any finite coloring of $\mathbb N$, there are monochromatic x, y such that x/y is in R. In this talk, we will give criteria for when sets of the form $\{(an+b)/(cn+d): n \in \mathbb N\}$ are sets of multiplicative recurrence. Consequently, we recover two recent results in number theory regarding completely multiplicative functions and the Omega function. This is based on a joint work with S. Donoso, J. Moreira and W. Sun.

• March 16, 2021. Time: 4pm (Note the unusual time!)
Lewis Bowen (U. Texas, Austin)

Title: A New Infinite-Dimensional Multiplicative Ergodic Theorem
Abstract: In 1960, Furstenberg and Kesten introduced the problem of describing the asymptotic behavior of products of random matrices as the number of factors tends to infinity. Oseledets’ proved that such products, after normalization, converge almost surely. This theorem has wide-ranging applications to smooth ergodic theory and rigidity theory. It has been generalized to products of random operators on Banach spaces by Ruelle and others. I will explain a new infinite-dimensional generalization based on von Neumann algebra theory which accommodates continuous Lyapunov distribution. This will be a gentle introductory-style talk; no knowledge of von Neumann algebras will be assumed. This is joint work with Ben Hayes (U. Virginia) and Yuqing Frank Lin (Ben-Gurion U.).

##### Term 1
• October 20, 2020
John Griesmer (Colorado School of Mines) Slides

Title: Single recurrence: recent results, open problems, and mysterious examples
Abstract: The Poincaré Recurrence Theorem says that whenever $(X,\mu,T)$ is a probability measure preserving system and $A\subset X$ has positive measure, there is an $n\in \mathbb N$ such that $\mu(A\cap T^{-n}A)>0$. H. Furstenberg and A. Sárközy independently proved an appealing refinement: under the same hypotheses, one can conclude that $\mu(A\cap T^{-n^2}A)>0$ for some $n\in \mathbb N$. One may ask, in general, for which sets $S\subset \mathbb N$ can one conclude that $\mu(A\cap T^{-n}A)>0$ for some $n\in S$? This is the study of single recurrence in measurable dynamics. Many examples and non-examples are known, but a satisfying description remains elusive. This talk will summarize the current state of knowledge regarding single recurrence, advertise well-known open problems, and introduce some more tractable variations. We will also provide explicit examples of sets whose recurrence properties are unknown.

• October 27, 2020
Anish Ghosh (Tata Institute of Fundamental Research)

Title: Quadratic forms and ergodic theory
Abstract: I will discuss connections between ergodic theory on homogeneous spaces and the study of quadratic forms. This connection began with Margulis's famous resolution of Oppenheim's conjecture. I will talk about some recent work which studies effective versions of Margulis's theorem, especially in the context of inhomogeneous quadratic forms.

• November 3, 2020
Khadim Mbacke War (IMPA)

Title: Closed Geodesics on Surfaces without Conjugate Points
Abstract: We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points. This is based on a join work with Vaughn Climenhaga and Gerhard Knieper.

• November 10, 2020
Brian Chung (U Chicago)

Title: Stationary measure and orbit closure classification for random walks on surfaces
Abstract: We study the problem of classifying stationary measures and orbit closures for non-abelian action on surfaces. Using a result of Brown and Rodriguez Hertz, we show that under a certain average growth condition, the orbit closures are either finite or dense. Moreover, every infinite orbit equidistributes on the surface. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation surfaces. We then consider the problem of verifying this growth condition in concrete settings. In particular, we apply the theorem to two settings, namely discrete perturbations of the standard map and the Out(F2)-action on a certain character variety. We verify the growth condition analytically in the former setting, and verify numerically in the latter setting.

• November 17, 2020
Osama Khalil (Utah)

Title: Random Walks, Spectral Gaps, and Khinchine Theorem on Fractals.
Abstract: In 1984, Mahler asked how well typical points on Cantor’s set can be approximated by rational numbers. His question fits within a program, set out by himself in the 1930s, attempting to determine conditions under which subsets of $\mathbb{R}^n$ inherit the Diophantine properties of the ambient space. Since approximability of typical points in Euclidean space by rational points is governed by Khinchine’s classical theorem, the ultimate form of Mahler’s question asks whether an analogous zero-one law holds for fractal measures. Significant progress has been achieved in recent years, albeit, almost all known results have been of “convergence type”.

In this talk, we will discuss the first instances where a complete analogue of Khinchine’s theorem for fractal measures is obtained. The main new ingredient is an effective equidistribution theorem for certain fractal measures on the space of unimodular lattices. The latter is established via a new technique involving the construction of S-arithmetic Markov operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. This is joint work with Manuel Luethi.
• December 1, 2020
Pankaj Vishe (Durham)

Title: A sparse equidistribution result for $(SL(2,\mathbb{R})/\Gamma_0)^n$
Abstract: Let $G=(SL(2,\mathbb{R})^n$ and $\Gamma=\Gamma_0^n$ where $\Gamma_0$ is a co-compact lattice in $SL(2,\mathbb{R})$. Let $U(x_1,...,x_n)$ denote a standard n-parameter unipotent expanding horospherical subgroup in G. The equidistribution of whole U orbits follow from Ratner's equidistribution theorems. It is believed that sparse arithmetic subsets of these orbits should equidistribute for every orbit, however, results of these type have been extremely rare. Typically one is able to obtain results of metric nature. We prove that the sparse subsets of all U orbits evaluated at integer times which lie on a quadric hypersurface equidistribute in $G/\Gamma$ as long as $n\geq 481$. Key tools here are a combination of bounds for twisted averages of functions over horocycle flow and the Hardy-Littlewood circle method.

• December 8, 2020
Olga Paris-Romaskevich (Marseille) Slides

Title: Introduction to tiling billiards
Abstract: I will tell you about the family of dynamical systems I am passionate about for almost four years now. This family models, in a naive way, a movement of refracted light in a tiling of the plane. The light follows Snell’s law with refraction coefficient equal to -1. This may seem odd as a choice of refraction coefficient... but this definition leads to rich mathematics already in the simplest tilings. In practice, one deals with families of interval exchange transformations with flips and continued fraction algorithms. In the talk I will concentrate myself on the case of periodic triangle tiling, the one that I understand the best. Recently another interpretation of tiling billiards as a physical system has been discovered. The tiling billiard trajectories model, for a large class of tilings, the movement of electrons on the Fermi surface of a metal in the presence of a magnetic field! (a so-called Novikov’s problem) The talk is introductory and elementary. The best way to prepare for it is to watch a short animated movie by Ofir David called Refraction tilings. One image is worth a million words.