Dynamics Archives for Fall 2024 to Spring 2025
Symplectic Weyl laws and their applications
When: Thu, September 7, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Dan Cristofaro-Gardiner (UMD) -
Abstract: The classical Weyl law recovers the volume of a domain from the asymptotics of the eigenvalues of the corresponding Laplace operator. I will explain some analogous Weyl laws in symplectic geometry. I plan to emphasize some questions that remain open about the subleading asymptotics. I will also try to show how these formulas can be used to resolve some long standing problems of interest such as the "Simplicity Conjecture", the smooth closing lemma in the conservative surface case, and the packing stability conjecture.
Lyapunov simplicity for the Teichmüller flow
When: Thu, September 14, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Vaibhav Gadre (University of Glasgow) - https://www.maths.gla.ac.uk/~vgadre/
Abstract: A quadratic differential on a Riemann surface is equivalent to a half-translation structure on the surface by complex charts with half-translation transitions. The SL(2,R)-action on the complex plane takes half-translations to half-translations and so descends to moduli spaces of quadratic differentials. The diagonal part of the action is the Teichmuller flow.
Apart from its intrinsic interest, the dynamics of Teichmuller flow is central to many applications in geometry, topology and dynamics. The Kontsevich—Zorich cocycle which records the action of the flow on the absolute homology of the surface, plays a key role.
In this talk, I will explain how the flow detects the topology of moduli spaces. Specifically, we will show that the flow group, namely the subgroup generated by almost flow loops, has finite index in the fundamental group. As a corollary, we will prove that the minus and plus (modular) Rauzy—Veech groups have finite index in the fundamental group, answering a question by Yoccoz.
Using this, and Filip’s results on algebraic hulls and Zariski closures of modular monodromies, we prove that the Konstevich—Zurich cocycle (separately minus and plus pieces) have a simple Lyapunov spectrum, extending the work of Forni from 2002 and Avila—Viana from 2007.
This is joint work with Bell, Delecroix, Gutierrez—Romo and Schleimer.
Two folk theorems in topological dynamics
When: Thu, September 21, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Joe Auslander (UMD) -
Abstract: Let $(Y,T)$ be a factor of $(X,T)$, $f\colon X \to Y$, and suppose whenever $x$ and $x'$ are proximal we have $f(x)=f(x')$. Then $(Y,T)$ is distal.
This theorem is "well known" but there is no simple proof. We will sketch a proof and also an analogous theorem on equicontinuity and regional proximality. We will briefly mention recent work with Anima Nagar on "strong proximality" and "weak distality".
Ergodicity of infinite symmetric extensions of IETs
When: Thu, September 28, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Przemek Berk (Nicolaus Copernicus University, Torun) -
Abstract: We study skew products with base being symmetric interval exchange transformations. We show that if the cocycle is either logarithmic and odd on each of the exchanged intervals or it is given by a shifted indicator of (0,1/2), the the skew product is ergodic. This result is based on a joint work with Frank Trujillo and Corinna Ulcigrai.
Low Complexity Dynamical Systems Workshop
When: Mon, October 2, 2023 - 9:00amWhere: Brin mathematics research center, Computer Science Instructional Center, 8169 Paint Branch Dr, College Park, MD 20742, USA
https://brinmrc.umd.edu/programs/workshops/fall23/fall23-workshop-dynamics.html
Low Complexity Dynamical Systems Workshop
When: Tue, October 3, 2023 - 9:00amWhere: Brin mathematics research center, Computer Science Instructional Center, 8169 Paint Branch Dr, College Park, MD 20742, USA
https://brinmrc.umd.edu/programs/workshops/fall23/fall23-workshop-dynamics.html
Low Complexity Dynamical Systems Workshop
When: Wed, October 4, 2023 - 9:00amWhere: Brin mathematics research center, Computer Science Instructional Center, 8169 Paint Branch Dr, College Park, MD 20742, USA
https://brinmrc.umd.edu/programs/workshops/fall23/fall23-workshop-dynamics.html
Low Complexity Dynamical Systems Workshop
When: Thu, October 5, 2023 - 9:00amWhere: Brin mathematics research center, Computer Science Instructional Center, 8169 Paint Branch Dr, College Park, MD 20742, USA
https://brinmrc.umd.edu/programs/workshops/fall23/fall23-workshop-dynamics.html
Low Complexity Dynamical Systems Workshop
When: Fri, October 6, 2023 - 9:00amWhere: Brin mathematics research center, Computer Science Instructional Center, 8169 Paint Branch Dr, College Park, MD 20742, USA
https://brinmrc.umd.edu/programs/workshops/fall23/fall23-workshop-dynamics.html
A tale of two flows: unipotent-like dynamics on moduli space
When: Thu, October 12, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Aaron Calderon (UChicago) - https://aacalderon.com/
Abstract: The moduli space $M_g$ of Riemann surfaces has many different incarnations, each of which equips it with different geometric structures. Thinking about $M_g$ as the space of complex structures gives rise to the well-studied (Teichmüller) horocycle flow and the field of Teichmüller dynamics, while thinking about it as the space of hyperbolic metrics yields the much more mysterious earthquake flow. In this talk, I’ll discuss to what extent these flows are “the same” and how this connection can be used to transfer results between flat to hyperbolic geometry. No prior experience with Teichmüller theory will be assumed. This represents joint work (some of which is in progress) with James Farre.
Lyapunov spectral rigidity of expanding circle maps
When: Mon, October 16, 2023 - 11:00amWhere: Brin Center Seminar Room
Speaker: Vadim Kaloshin (Institute of Science and Technology Austria) -
Abstract: Motivated by the question "Can you hear the shape of a drum?" and spectral rigidity for metrics we define Lyapunov spectrum of an expanding circle map of degree at least 2 as the set of all Lyapunov exponents (multipliers) at periodic orbits. This set is analogous to the unmarked length spectrum of negatively curved metrics on surfaces of genus at least 2. Is the following local rigidity holds: every C^r smooth expanding circle map f has a neighborhood (in C^r topology) such that any perturbation of f within this neighborhood that keeps the Lyapunov spectrum must be smoothly conjugate to f (subject to some sparsity assumption on the spectrum on f)? The answer is positive. The proof uses a novel iterative scheme which we will outline in the talk. This is joint work in progress with Kostya Drach
Boundary actions of free semigroups
When: Thu, October 19, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Jenna Zomback (UMD) -
Abstract: We consider the natural action of a free, finitely generated semigroup (the set of all finite words in a finite alphabet $\Sigma$) on its boundary (the space of infinite words in $\Sigma$) by concatenation. While boundary actions of free groups are well-studied, much less is known for semigroups. In joint work with Anush Tserunyan, we completely characterize those Markov measures which make the boundary action ergodic, and those which make it weakly mixing (i.e., the product with an ergodic probability measure preserving action is ergodic). This is an ingredient in the proof of pointwise ergodic theorems for measure preserving actions of free semigroups.
Random walks in quasiperiodic environment
When: Thu, October 26, 2023 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Klaudiusz Czudek (IST) - https://pub.ista.ac.at/~kczudek/
Abstract: Fix an irrational number $\alpha$ and a smooth, positive function $\mathfrak{p}$ on the circle. Fix a point $x$ on the circle, and consider a random walk on integers in which a particle placed at a point $k$ jumps to $k+1$ with probability $\mathfrak{p}(x+k\alpha)$ and to $k-1$ with probability $1-\mathfrak{p}(x+k\alpha)$. This kind of random walk is called a random walk in quasiperiodic environment. To study this Markov chain it is useful to consider so called environment viewed by the particle process. The talk will be devoted to the ergodic properties of that process.
$L^{q}$-spectra of dynamically driven self-similar measures: the multi-dimensional case
When: Thu, November 9, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Emilio Corso (Penn State) - https://www.emiliocorso.com/
Abstract: A great deal of interest in fractal geometry centres on determining the dimensional properties of self-similar sets and measures, as well as of their projections and convolutions. In a seminal contribution dating from nearly a decade ago, Hochman achieved substantial progress towards the celebrated exact overlaps conjecture, establishing that the Hausdorff dimension of self-similar sets and measures on the real line matches the similarity dimension whenever the generating iterated function system satisfies exponential separation. The result was subsequently refined by Shmerkin, who established the analogue for the full $L^q$-spectrum of self-similar measures and successfully applied it to settle long-standing conjectures in dynamics and fractal geometry. In joint work with Shmerkin, we extend the dimensional result to any ambient dimension under an additional unsaturation assumption; as in the one-dimensional case, our framework consists of the class of dynamically driven self-similar measures, which allows for a unified treatment of self-similar and stochastically self-similar measures, their projections and convolutions. The argument relies crucially on an inverse theorem for the $L^q$-norm of convolutions of discrete measures in Euclidean spaces, recently established by Shmerkin.
Almost sure limit theorems with applications to non-regular continued fraction algorithms
When: Tue, November 14, 2023 - 3:00pmWhere: 3114 EGR
Speaker: Tanja Schindler (Jagiellonian University (Krakow)) -
Abstract: We consider an ergodic measure preserving dynamical system
$(X,T)$ and an observable $f$ mapping $X$ to the reals. By a theorem by
Aaronson there is no strong law of large numbers if either 1) $X$ is a
probability measure space and $f$ is non-integrable or if 2) $X$ has
infinite measure and $f$ is integrable. While in the situation 1) one can
use trimming, i.e. deleting a number of the largest entries, to still
obtain a strong law of large numbers, in case of 2) it is possible to
add additional summands to obtain a strong law of large numbers. In this
talk we will study the situation 2) and also the situation of an
infinite measure space and an observable which is non-integrable on a
finite measure set. Examples of such a situation are different
non-standard continued fraction digits like backwards continued
fractions. This is joint work with Claudio Bonanno.
On Quasi-ergodicity of absorbing Markov Chains
When: Thu, November 16, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Matheus M. Castro (Imperial College London) - https://www.mmcastro.org/
Abstract: In this talk, we study the long-term behaviour of Random Dynamical Systems (RDSs) conditioned upon staying in a region of the space. We use the absorbing Markov chain theory to address this problem and define relevant dynamical systems objects for the analysis of such systems. This approach aims to develop a satisfactory notion of ergodic theory for random systems with escape.
Non existence of small breathers in Klein-Gordon equations
When: Thu, November 30, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Tere Seara (UPC) - https://web.mat.upc.edu/tere.m-seara/index.html
Abstract: Breathers are solutions of evolutionary PDEs, which are periodic in time and spatially localized. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. When the spatial dimension is equal to one, using the so-called spatial dynamics framework (exchanging the roles of time and space variables), breathers can be seen as homoclinic solutions to steady solutions (in an infinite dimensional space of periodic in time solutions).
In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated at the passage of a bifurcation when a pair of pure imaginary eigenvalues collide at the origin and become real when the temporal frequency varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales.We will obtain an asymptotic formula for the distance between the stable and unstable manifolds when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. This formula allows to say that for a wide set of Klein-Gordon equations breathers do not exist.
Renormalization of unicritical diffeos of the disk
When: Thu, December 7, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Enrique Pujals (CUNY) - https://sites.google.com/site/enriquepujalsgc/home
Abstract: In a joint work with S. Crovisier and C. Tresser, it was proved that a Henon map (with Jacobian smaller than 1/4) that is in the boundary of the diffeos with zero entropy, it is infinitely renormalizable. In a joint work with Crovisier, Lyubich and J. Yang, we address the converse. For that, it is generalized the notion of infinitely renormalizable unimodal maps to dissipative diffeomorphisms of the disk. In this new class of dynamical systems, it is shown that under renormalization, maps eventually become Hénon-like, and then converge super-exponentially fast to the space of one-dimensional unimodal maps. These results are based on a quantitative reformulation of Pesin theory, and a new approach analyzing the dynamical effects of 'critical orbits' in a higher dimensional setting.
The first hour of the talk will be about the context of the problem and an outline of the result content of the problem. The second part, it will be focused in the outline of the proof's strategies.
Effective versions of Ratner’s equidistribution theorem
When: Tue, December 12, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Lei Yang (IAS) - https://lively-cat-lover.github.io/lei-yang
Abstract: I will talk about recent progress in the study of quantitative equidistribution of unipotent orbits in homogeneous spaces, namely, effective versions of Ratner’s equidistribution theorem. In particular, I will explain the main idea of my proof for unipotent orbits in $\text{SL}(3,\mathbb{R})/\text{SL}(3,\mathbb{Z})$. The proof combines new ideas from harmonic analysis and incidence geometry. In particular, the quantitative behavior of unipotent orbits is closely related to a Kakeya model.
Spectral Rigidity of Generic Axis-Symmetric Convex Analytic Domains.
When: Thu, December 21, 2023 - 12:00pmWhere: Kirwan Hall 3206
Speaker: Vadim Kaloshin (IST Austria) - https://vadimkaloshin.com/
Abstract. During the talk we shall discuss the strategy of proving spectral rigidity of generic Axis-Symmetric Convex Analytic Domains. It involves the combination of dynamical behavior near the boundary and near a KAM curve (caustic). This is a joint work with Corentin Fierobe.
A relation between the Laplace and the length spectrum for convex planar domains close to ellipses.
When: Tue, January 23, 2024 - 12:00pmWhere: Kirwan Hall 1311
Speaker: Vadim Kaloshin (IST (Austria)) - https://vadimkaloshin.com/
Abstract: We shall discuss the relation between the singular support of the wave trace associated to any domain through the Laplace spectrum and how it can differ from the length spectrum. This is a join work with I.Koval and A.Vig.
Cauchy limit laws for linear forms with random coefficients
When: Tue, January 30, 2024 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Bassam Fayad (UMD) -
Abstract: We study the distribution of linear forms with random coefficients evaluated $\mod 1$ on the integers. We obtain a Cauchy limit law for the associated discrepancy function normalized by $\ln^d N$. The key ingredient of the proof is a Poisson limit theorem for the visits to the cusp under the Cartan action of lattices distributed uniformly on a positive codimension leaf of the horospheres. This is a joint work with Dmitry Dolgopyat and Zhiyuan Zhang.
Heat equation from a deterministic dynamics
When: Thu, February 1, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Carlangelo Liverani (U. Roma Tor Vergata) - https://www.mat.uniroma2.it/~liverani/
Abstract: I’ll describe a derivation of the heat equation in the thermodynamics limit, with a diffusive scaling, from purely deterministic dynamics satisfying Newton's equations under an external, time-dependent, external field. (Work in collaboration with G.Canestrari and S. Olla)
Globally coupled maps: Statistical properties and phase transitions
When: Tue, February 6, 2024 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Carlangelo Liverani (U. Roma Tor Vergata) - https://www.mat.uniroma2.it/~liverani/
Abstract: I will discuss infinite systems of globally coupled maps. The goal is to develop a general bifurcation theory that describes what happens when the coupling strength varies.
As an application, we show that phase transitions can occur naturally in a system of globally coupled Anosov maps. (Work in collaboration with Wael Bahsoun)
Higher order mixing for locally uniformly stretching flows
When: Thu, February 8, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Adam Kanigowski (UMD) -
Abstract: The Rokhlin problem asks whether mixing implies higher order mixing. In smooth dynamics the major mechanism that produces mixing is called mixing via shearing. Using this mechanism many systems of zero entropy were shown to be mixing. We show that all such flows are mixing of all orders. Moreover we also show that a quantitative mixing via shearing mechanism implies quantitative higher order mixing. This is joint work with D. Ravotti.
Reeb orbits that force topological entropy
When: Tue, February 13, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Abror Pirnapasov (UMD) - https://sites.google.com/view/abrorpirnapasov/research
Abstract: A transverse link in a contact 3-manifold forces topological entropy if every Reeb flow possessing this link as a set of periodic orbits has positive topological entropy. We will explain how cylindrical contact homology on the complement of transverse links can be used to show that certain transverse links force topological entropy. As an application, we show that on every closed contact 3-manifold exists transverse knots that force topological entropy. We also generalize to the category of Reeb flows a beautiful result due to Denvir and Mackay, which says that if a Riemannian metric on the two-dimensional torus has a contractible closed geodesic then its geodesic flow has positive topological entropy.
Exponential Mixing Via Additive Combinatorics
When: Thu, February 15, 2024 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Osama Khalil (University of Illinois at Chicago) - https://homepages.math.uic.edu/~khalil/research.html
Abstract: The Bowen-Ruelle conjecture predicts that geodesic flows on negatively curved manifolds are exponentially mixing with respect to all their equilibrium states. Dolgopyat pioneered a method rooted in the thermodynamic formalism that settled the conjecture for surfaces. Soon after, Liverani developed an intrinsic functional analytic analog of Dolgopyat's method allowing to settle the case of Liouville measures in higher dimensions, while simultaneously producing more information on rates of mixing. Despite these important breakthroughs, the conjecture remains open in general, even in the fundamental case of measures of maximal entropy. In this talk, we will discuss a method for extending the functional analytic approach to deal with non-smooth invariant measures in a concrete algebraic setting. The key ingredient is a reduction of the problem to one regarding Fourier transforms of dynamically defined measures which we address using new machinery in additive combinatorics. The talk will not assume prior knowledge of these topics.
TBA
When: Fri, February 16, 2024 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Samuel Kittle (University College London) - https://www.samuelkittle.org/
Abstract: (Talk runs 2:00 to 4:00)
Subshifts with very low word complexity
When: Tue, February 20, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ronnie Pavlov (University of Denver) - https://cs.du.edu/~rpavlov/
Abstract: The word complexity function $p(n)$ of a subshift $X$ measures the number of $n$-letter words appearing in sequences in $X$, and $X$ is said to have linear complexity if $p(n)/n$ is bounded. It’s been known since work of Ferenczi that linear word complexity highly constrains the dynamical behavior of a subshift.
In recent work with Darren Creutz, we show that if $X$ is a transitive subshift with $\limsup p(n)/n < 3/2$, then $X$ is measure-theoretically isomorphic to a compact abelian group rotation. On the other hand, $\limsup p(n)/n = 3/2$ can occur even for $X$ measurably weak mixing. Our proofs rely on a substitutive/$S$-adic decomposition for such subshifts.
I’ll give some background/history on linear complexity, discuss our results, and will describe several ways in which $3/2$ turns out to be a key threshold (for $\limsup p(n)/n$) for several different types of dynamical behavior.
Effective ergodicity of nilflows and bound on Weyl sums
When: Thu, February 22, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Giovanni Forni (UMD) -
Abstract: We will outline a direct approach to bounds on Weyl sums for higher degree polynomials based on ideas from dynamical systems and unitary representation theory for nilpotent Lie groups. This approach originated with Furstenberg derivation of equidistribution of fractional parts of polynomial sequences from unique ergodicity of linear toral skew shifts. In general it is a hard problem in dynamical systems (homogeneous dynamics) to prove ''effective'' counterparts of unique ergodicity results or more general classification results for invariant measures (Ratner theory). For nilflows (and more generally nilsequences) effective results (with power saving) were proved by Green and Tao in 2012 with no precise (and presumably not sharp) information on the exponent. Our approach to effective equidistribution (in joint work with L. Flaminio) is based on ''scaling'' and generalizes the ''renormalization'' approach to effective equidistribution (for instance, for the equidistribution of unstable manifolds of hyperbolic diffeomorphisms). From our method we derive bounds on Weyl sums comparable to the best available ones, derived by J. Bourgain, C. Demeter and L. Guth and later independently by T. Wooley from their proof of the ''Vinogradov Main Conjecture''.
Escape of mass for embedded horospheres
When: Tue, February 27, 2024 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Jens Marklof (Bristol) - https://people.maths.bris.ac.uk/~majm/
Abstract: I will discuss escape of mass estimates for $SL(d,\mathbb{R})$-horospheres embedded in the space of affine lattices, which depend on the Diophantine properties of the shortest affine lattice vector. These estimates can be used, in conjunction with Ratner's theorem, to prove the convergence of moments in natural lattice point problems, including the statistics of directions in lattices, inhomogeneous Farey fractions and the distribution of smallest denominators. Based on joint work with Wooyeon Kim (ETH).
Global smooth rigidity for toral automorphisms
When: Thu, February 29, 2024 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Zhenqi Wang (Michigan State University) -
Abstract: Suppose $f$ is a diffeomorphism on torus whose linearization $A$ is weakly irreducible. Let
$H$ be a conjugacy between $f$ and $A$. We prove the following: 1 if $A$ is hyperbolic and $H$ is weakly differentiable
2. if $A$ is partially hyperbolic and $H$ is $C^{1+\text{holder}}$. Then $H$ is $C^\infty$. Our result shows that the conjugacy in all local and global rigidity results for irreducible $A$ is $C^\infty$. This is a joint work with B. Kalinin and V Sadovskaya.
Smooth Models for Fibered Partially Hyperbolic Systems
When: Tue, March 5, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Meg Doucette (University of Chicago) - https://math.uchicago.edu/~doucette/
Abstract: We discuss the existence and construction of smooth models for certain fibered partially hyperbolic systems. Fibered partially hyperbolic systems are partially hyperbolic diffeomorphisms that have an integrable center bundle, tangent to a continuous invariant fibration by invariant submanifolds. Under certain restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. We then discuss some preliminary work on removing or weakening the assumptions on the fiber. (The talk will run 2:00 to 3:30.)
Spectral Rigidity near KAM curves for convex billiards
When: Wed, March 6, 2024 - 11:00amWhere: Kirwan Hall 3206
Speaker: Vadim Kaloshin (Institute of Science and Technology Austria) -
Abstract: During the talk I will discuss information about dynamics near KAM curve contained in the Length Spectrum and its application to spectral rigidity for convex billiards. This is based on a joint work with C. Fierobe (Talk is 11:00 to 12:20)
Thick Arnold tongues
When: Thu, March 7, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Alexey Okunev (Penn State) -
Abstract: This talk will explore a physically motivated problem that exhibits interesting and perhaps unexpected mathematical features. Consider a periodic two-dimensional Hamiltonian flow with $H(x, y) = \cos(x) \cos(y)$. After adding a small constant forcing, some trajectories become unbounded with the same asymptotic direction as the forcing. We consider a simple model for the movement of a particle put in a fluid that moves according to this flow: the particle has inertia of its own but is subject to the drag force proportional to the particle’s velocity relative to the surrounding fluid.
The particle no longer follows the trajectories of the Hamiltonian system. Moreover, while the asymptotic direction of the Hamiltonian trajectories coincides with the direction of the forcing, that of the particle does not, the particle veers in a direction it "prefers". Particle drift direction depends on the forcing direction in a nontrivial way, determined by a Cantor-like function, but with an unexpected feature: the plateaus of this function occupy a set of full measure. In a two-parameter representation (one parameter is the forcing slope, the other is the drag coefficient), this gives rise to Arnold tongues, where the tongues correspond to rational slopes of the particle trajectories. However, unlike Arnold's example, the complement to the union of all tongues has zero measure. We will also describe how this problem reduces to the study of the rotation number for monotone families of non-decreasing circle maps with flat spots. This is a joint work in progress with Mark Levi.
The subleading asymptotics of symplectic Weyl laws
When: Tue, March 12, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Oliver Edtmair (Berkeley) -
Abstract: The classical Weyl law states that the eigenvalues of the Laplace operator asymptotically recover Riemannian volume. Certain sequences of symplectic spectral invariants of dynamical origin satisfy a similar property: The leading term of their asymptotics is governed by symplectic volume. Such symplectic Weyl laws have recently led to striking applications in dynamics (e.g. smooth closing lemma) and C^0 symplectic geometry (e.g. simplicity conjecture). In this talk, I will report on work in progress concerning the subleading asymptotics of symplectic Weyl laws. I will highlight close connections to symplectic packing problems and questions about the algebraic structure of groups of Hamiltonian diffeomorphisms and homeomorphisms. (Talk runs 2:00 to 3:30)
Random dynamics, SL(2,R) actions and measure rigidity
When: Thu, March 28, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Alex Eskin (University of Chicago) -
Abstract: It became clear recently that some version of the measure rigidity phenomenon as seen in the theorems of Ratner and Benoist-Quint occurs in a much more general setting. I will state some theorems and conjectures. I will also begin to discuss the connection to the dynamics of a single hyperbolic diffeomorphism or flow, in particular to the relation between u-Gibbs states and SRB measures. (Talk runs 2:00 to 3:50)
Ergodic averages along $\Omega(n)$
When: Tue, April 2, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Katy Loyd (Northwestern) -
Abstract: Following Birkhoff's proof of the Pointwise Ergodic Theorem, it is natural to consider whether convergence still holds along various subsequences of the integers. In 2020, Bergelson and Richter showed that in uniquely ergodic systems, pointwise convergence holds along the number theoretic sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors of $n$ with multiplicities. In this talk, we will see that by removing this assumption, a pointwise ergodic theorem does not hold along $\Omega(n)$. In fact, $\Omega(n)$ satisfies a rather strong non-convergence property. We will further classify the strength of this non-convergence behavior by considering weaker notions of averaging (based on current joint work with S. Mondal). (Talk runs 2:00 to 3:20)
Equidistribution of Discrepancy Sequences (joint with Dolgopyat)
When: Thu, April 11, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Omri Sarig (Weizmann) - https://www.weizmann.ac.il/math/sarigo/
Abstract: Suppose $\alpha$ is an irrational number, and let
$D_N:=\#\{1\leq n\leq N: n\alpha\, \text{mod}\, 1\in [0,1/2]\} - \frac{N}{2}.$
This is a sequence of half-integers. We will characterize the $\alpha$ of bounded type so that $D_N$ is equidistributed in $\tfrac{1}{2}\mathbb Z$
in the sense of the ratio ergodic theorem. It turns out that this is the case for $\alpha=\sqrt{3}$, but not for $\alpha=\sqrt{2}$.
(Joint work with Dolgopyat)
Absolute Continuity of Furstenberg Measures
When: Tue, April 16, 2024 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Samuel Kittle (University College London) - https://www.samuelkittle.org/
Abstract: Given a probability measure on $\textrm{PGL}_2(\mathbb{R})$ under relatively weak conditions there is a unique probability measure on the projective line which is stationary under convolution with this measure. This is known as the Furstenberg measure. When the measure on $\textrm{PGL}_2(\mathbb{R})$ has finite support this measure may have fractional dimension. In this talk I will discuss some background on finding the dimensions of stationary measures and Furstenberg measures as well as sufficient conditions for them to be absolutely continuous. (Talk runs 2:00-4:00)
Why period-doubling Cascades exist: using topological degree theory without renormalization
When: Thu, April 18, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Jim Yorke (UMD) -
Abstract: The talk focuses on one-parameter families of maps in the plane and their topology. A period-doubling cascade for a map depending on a real parameter is a path of periodic orbits, a path on which there are infinitely many period doublings, a path along which the period goes to infinity. I will begin by describing the nature of topological fixed point theorems which were long thought to be non constructive. Ideas that altered them so that they became constructive can be applied to showing the existence of period-doubling cascades in planar maps. This is smooth topology and has nothing to do with Feigenbaum's renormalization. (Talk runs 2:00 to 3:20)
Optimal spectral gaps
When: Thu, April 25, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Michael Magee (Durham) - https://www.mmagee.net/
Abstract: I'll discuss spectral gaps in the context of graphs, hyperbolic surfaces, and unitary representations of discrete groups. The main focus will be on spectral gaps that are (asymptotically) optimal. One famous example of this phenomenon is a theorem of Friedman stating that random d-regular graphs on a large number of vertices are almost Ramanujan. Now, analogs of this result are known for hyperbolic surfaces.
What underpins some of the recent progress in this area are notions of strong spectral gaps arising from operator algebras that I'll also explain. My talk will also contain some very basic open questions that I hope will be of broad appeal.(Talk runs 2:00-4:00PM)
Local Limit Theorems for non stationary subshifts of finite type
When: Thu, May 2, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Dmitry Dolgopyat (UMD) -
Abstract: Local Limit Theorem is a powerful tool for investigating properties of dynamical systems. In this talk we describe obstructions to Local Limit Theorem and present the tools suitable for analyzingthese obstructions. Based on joint papers with Omri Sarig and with Yeor Hafouta.
The Ruelle spectrum for self-similar tilings
When: Thu, May 9, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Rodrigo Treviño (UMD) -
Abstract: In this talk I will discuss recent results about the speed of mixing of systems derived from self-similar tilings. I will show how one can define anisotropic spaces of smooth functions where the action of the transfer operator can be understood in part through the induced action on cohomology of the tiling space. No background on tilings or anisotropic Banach spaces will be assumed.
Lyapunov Spectral Rigidity and Expanding Circle Maps
When: Tue, May 14, 2024 - 11:00amWhere: Kirwan Hall 3206
Speaker: Vadim Kaloshin (Institute of Science and Technology Austria) -
Abstract: (Joint with K. Drach) We shall discuss smoothness of conjugacy of expanding circle maps satisfying sparsity conditions. The method involves an iterative scheme of coordinate changes and the Whitney extension theorem. (Talk runs 11:00 to 12:20)
Relations in Topological Dynamics
When: Tue, May 21, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Anima Nagar (Indian Institute of Technology Delhi) -
Abstract: This is an ongoing work with Joseph Auslander. We talk about various relations in topological dynamics and the properties they display. (Talk runs 2:00 to 3:30)