Learning Seminar 11:00 AM-12:00 PM, GC 5417
Enrique Pujals, CUNY - Graduate Center, New York, NY
Title: A mechanisms/phenomenon counterpart, with focus on homoclinic tangencies.
Abstract: By a mechanism, we mean a simple dynamical configuration (involving few periodic points and their invariant manifolds) that it “generates itself” (meaning that produces a cascade of diffeomorphisms sharing the same configuration) and it “creates or destroys” rich and different dynamics for nearby systems (for instance horseshoes, cascade of bifurcations, entropy’s variations). By a dynamical phenomenon, we mean any dynamical property which provides a good global description of the system (like hyperbolicity,> transitivity, minimality, zero entropy, spectral decomposition) and which occurs on a “rather large” subset of systems. We are going to dwell on the relation between these two concepts and we will focus on the particular case of homoclinic tangencies for surface diffeomorphisms and its relation with the "Newhouse phenomenon".

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Enrique Pujals, CUNY - Graduate Center, New York, NY
Title: Prevalence of the coexistence of infinitely many attractors and the role of Blenders and Parablenders.
Abstract: following the talk in the morning, we will discuss homoclinic tangencies in higher dimensions and the prevalence (in terms of parametric families) of the Newhouse phenomenon. To understand this problem, we will introduce the concept of blenders (a higher dimensional horseshoes with some funny properties) and the parablender (a parametric version of the blenders).

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Lucas Furtado, CUNY - Graduate Center, New York, NY
Title: Newhouse theorem and some of its consequences: describing a simple mechanism on the two-sphere for the creation of infinitely many sinks
Abstract: Newhouse 1968 proved that hyperbolic dynamical systems are not dense on the two-sphere. Ten years later, Newhouse 1979 identified a mechanism, namely the homoclinic tangencies, which gives rise to this robustly non-hyperbolic behavior that was discovered ten years before. In this talk, we will describe Newhouse's 1968 construction of a diffeomorphism on the 2-sphere which exhibits robustly non-hyperbolic behavior. Time permitting, we will discuss how, arbitrarily close to these diffeomorphisms, there are dynamics exhibiting infinitely-many sinks.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Yunping Jiang, CUNY - Queens College, New York, NY
Title: Convergence of Time-Average along Uniformly Behaved in \(N\) Sequences on Every Point
Abstract: Following the morning talk, we investigate the convergence of time-averages along sequences in $N$ that exhibit uniform behavior. Specifically, we define a uniformly behaved arithmetic sequence \({\bf a}\) in \(N\) and an a-mean Lyapunov stable dynamical system \(f\). We consider the time-averages of a continuous function \(\phi\) along the \({\bf a}\)-orbit of \(f\) up to \(N\). We demonstrate that this partial time-average converges for every point in the space if \({\bf a}\) is uniformly behaved in \(N\) and \(f\) is minimal, uniquely ergodic, and \({\bf a}\)-mean Lyapunov stable. An example of a non-trivial \({\bf a}\)-mean Lyapunov stable dynamical system is a Denjoy counterexample, provided that the sequence \({\bf a}\) has a positive lower density. From number theory, we know that the counting function of the prime factors in natural numbers is uniformly behaved in \(N\). Additionally, we have identified several examples of uniformly behaved sequences in \(N\), including the subsequence of natural numbers indexed by the Thue-Morse (or Rudin-Shapiro) sequence, leveraging the oscillatory properties of their characteristic sequences. This is a joint work with Jessica Liu.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Gabriel Lacerda, Universidade Federal do Rio de Janeiro (UFRJ)
Title: An Introduction to (Metric) mean dimension
Abstract: In this introductory talk, we will explore Mean dimension, a relatively new method for measuring the complexity of infinite-dimensional dynamical systems. Introduced by M. Gromov in the late 1990s and developed further by E. Lindenstrauss and B. Weiss in the early 2000s, Mean dimension is a dynamical version of the Lebesgue covering dimension. We will also discuss the Metric mean dimension and how it relates to the Mean dimension. Finally, recent findings and open questions in the field will be covered.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Gabriel Lacerda, Universidade Federal do Rio de Janeiro (UFRJ)
Title: Mean dimension explosion of induced homeomorphisms
Abstract: Given a dynamical system, which we will call the base system, a natural way to extend this system to an infinite-dimensional setting is to consider the induced dynamics acting on the hyperspace of closed and nonempty subsets of the phase space. In this talk, we will discuss the Mean dimension explosion phenomenon: when the base system has zero topological entropy, but the Mean dimension of the induced map is infinite. In particular, this phenomenon is attained for Morse-Smale diffeomorphisms. Furthermore, for an orientation-preserving circle homeomorphism H, Mean dimension explosion does not occur if and only if H is conjugated to a rotation.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Bruno Nussenzveig, CUNY - Graduate Center, New York, NY
Title: Introduction to critical circle maps
Abstract: In this talk, we will introduce (multi-)critical circle maps, a class of one-dimensional dynamical systems which frequently appear at the so-called boundary of chaos and exhibit rigidity phenomena that have been a topic of intense research in the past three decades. We will define the sequence of dynamical partitions associated to a critical point and state the celebrated Real A-Priori Bounds for critical circle maps. As a toy application demonstrating the power of the Real Bounds, we will show that, for a critical circle map \(f\) with irrational rotation number, the sequence of closest return maps \(f^{q_n}\) has uniformly bounded derivatives.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Bruno Nussenzveig, CUNY- Graduate Center, New York, NY
Title: Invariant distributions and automorphic measures for critical circle maps
Abstract: Following the morning talk, we shall discuss the recent proof that, for a minimal critical circle map \(f\) and any given \(s > 0\), there exists a unique {\it automorphic measure} of exponent \(s\) for \(f\). This is an expansion upon the work of Douady and Yoccoz, who proved in the eighties that the same holds when \(f\) is a \(C^2\) circle diffeomorphism with irrational rotation number. We shall also briefly discuss two applications of this result. The first one is that these maps admit no invariant distributions of order 1 independent of the unique invariant measure. The second one is an improvement of the Denjoy-Koksma inequality for absolutely continuous observables.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Tao Chen, CUNY - Graduate Center, New York, NY
Title: Functional properties of Nevanlinna functions
Abstract: McMullen and Lyubich's dichotomy theorem states that a holomorphic map is either ergodic (the Julia set is the whole sphere, and the map is ergodic) or attracting (almost every point on the sphere is attracted to the post-critical set). However, it may be challenging to determine which case applied to a given map, of which the Julia set is the sphere. Recently, joint with Linda Keen and Yunping Jiang, we give a criterion to determine whether a family of Nevanlinna functions are ergodic or not. In this talk, I will give an introduction to the functional properties of these maps.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Canceled.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Marco Lopez, CUNY - Graduate Center, New York, NY
Title: Khintchine's Dichotomy law in Diophantine approximation
Abstract: In this talk we will present a Dichotomy law proven by Khinchin: The set of \psi-well-approximable real numbers has either full Lebesgue measure or zero Lebesgue measure according to whether a series of \psi-values diverges or converges. Although Khinchin proved this result by other methods, it can also be deduced from a Theorem proved by Duffin and Schaefer in 1941 (their theorem led to the famous Duffin-Schaeffer Conjecture). If time allows we will also touch upon the Mass Transference Principle, a result by Beresnevich and Velani from 2006 and related to the Duffin-Schaeffer conjecture.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Marco Lopez, CUNY - Graduate Center, New York, NY
Title: A dichotomy law for shrinking target sets under non-autonomous systems
Abstract: In this talk we will continue with the theme on Khinchin's dichotomy law, now in the context of shrinking target sets (analogous to well approximable numbers). The main result in this talk, due to Sun & Cao from 2017, is a dichotomy law under Cantor-series expansions. Such expansions are described by a nonautonomous iterated function system (IFS). If time remains we will discuss the same theme on more general nonautonomous IFSs.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Carlos Vasquez , Pontificia Universidad Católica de Valparaíso
Title: Lyapunov exponents for diffeomorphisms with dominated splitting: regularity and applications
Abstract: In smooth dynamics, Lyapunov exponents play a key role in understanding the behavior of a dynamical system. When the Lyapunov exponents are nonzero, the theory initiated by Pesin provides detailed geometric information on the dynamics. On the other hand, vanishing exponents are an exceptional situation associated with some rigidity of the system. An interesting question is how the Lyapunov exponents depend on parameters.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Carlos Vasquez , Pontificia Universidad Católica de Valparaíso
Title: Regularity with respect to the parameter of integrated Lyapunov exponents for diffeomorphisms with dominated splitting
Abstract: We consider families of diffeomorphisms with dominated splittings, preserving a Borel probability measure and we study the regularity of the integrated Lyapunov exponent associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in $[0,1]$), and under suitable conditions, we obtain formulas for the derivatives. We also discuss some applications. This is a joint work with Radu Saghin and Francisco Valenzuela-Henríquez from PUCV.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Mathew Grote, CUNY - Graduate Center, New York, NY
Title: Parameter identifiability by differential algebra
Abstract: In this talk I will introduce the problem of parameter identifiability, a topic in differential equations often encountered in dynamics as the "inverse problem." This applied math question has been attacked from many fields of study. I will draw some connections to dynamics while detailing my current work on structural identfiability via differential algebra, drawing a connection between symbolic computation and manifolds.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Sergiy Merenkov, CUNY - City College, New York, NY
Title: No bounded geometry wandering domains for \(C^1\)-surface diffeomorphisms
Abstract: A classical problem investigated by Denjoy and others of finding assumptions, such as regularity, for circle diffeomorphisms to have dense orbits has a natural counterpart for surfaces. Particularly, Norton—Sullivan in 1996 and  Kwakkel—Markovic in 2010 provided various conditions for torus, respectively, higher genus surface diffeomorphisms to not have sufficiently regular wandering domains. For example, Kwakkel and Markovic proved that a positive entropy \(C^{1+\alpha}\)-diffeomorphism of a closed surface of genus at least 1 cannot have wandering domains of bounded geometry. In the same paper they raised a question on whether the regularity can be lowered to \(C^1\). I will present a positive answer to that question, without the entropy assumption.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Yushan Jiang, CUNY - Graduate Center, New York, NY
Title: Marked Length Spectrum Rigidity: A Crossroad of Geometry, Dynamics and Group Theory
Abstract: K. Burns and A. Katok asked a question: if we know every length of every closed geodesic on a Riemannian manifold (it's called the marked length spectrum, not like length spectrum is only a subset of real number), can we determine the Riemannian metric up to isometry? This famous question remains open in general, but has been solved completely for negatively curved closed surfaces by J-P. Otal and independently by C. Croke (slightly later but in greater generality). In these cases, the answer is YES. I will follow A. Wilkinson's lecture note, firstly, to explain how this geometric problem involves with the geodesic flows by a Livsič's theorem from smooth dynamics. Then I will sketch the proof and illustrate the difficulties for the general cases (properties of Liouville current, smoothness and volume preserving properties of the conjugacy between Anosov flows). At last, if time permits, after mentioning some recent groundbreaking results, I'll also introduce the reformulations of this question in geometric group theory.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Christian Wolf, CUNY - City College, New York, NY
Title: Computable Markov partitions
Abstract: Markov partitions are a powerful tool to relate the dynamics of hyperbolic systems (Axiom A, Anosov, hyperbolic Julia sets, limit sets of iterated function systems, etc.) to a conjugate symbolic system. In this talk we present a computational analysis version of this relation. We introduce the notion of a computable Markov partition and show its existence for various classes of hyperbolic systems. This allows us to obtain algorithms delivering "exact" computations for various dynamical invariants including entropy, Hausdorff dimension, dimension of measures, etc. The results of this talk are part of an ongoing project with Michael Burr and Tamara Kucherenko.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Yuri Lima, Universidade Federal do Ceará, Brasil.
Title: Measures of maximal entropy for non-uniformly hyperbolic maps
Abstract: For \(C^{1+}\) maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We also present an application: finite horizon dispersing billiards have at most one adapted measure of maximal entropy, which is hyperbolic, Bernoulli, and fully supported. Moreover, it is the Liouville measure if and only if all multipliers of non grazing periodic orbits coincide with the topological entropy. Joint work with Davi Obata and Mauricio Poletti.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Yuri Lima, Universidade Federal do Ceará, Brasil.
Title: Measures of maximal entropy for non-uniformly hyperbolic maps
Abstract: For \(C^{1+}\) maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We also present an application: finite horizon dispersing billiards have at most one adapted measure of maximal entropy, which is hyperbolic, Bernoulli, and fully supported. Moreover, it is the Liouville measure if and only if all multipliers of non grazing periodic orbits coincide with the topological entropy. Joint work with Davi Obata and Mauricio Poletti.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Emma Dinowitz, CUNY - Graduate Center, New York, NY
Title: Dimension theory via symbolic dynamics
Abstract: Multifractal analysis of lyapunov exponents is the study of the Hausdorff dimension of the set of points with a fixed lyapunov exponent. Recent work of Sarig, Lima, and others have constructed countable state markov partitions modeling a set of points with recurrent hyperbolicity properties. Using this framework we prove hausdorff dimension upper bounds on the set of points with exponent \(\alpha\) which are symbolically represented.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417

Cancelled to accomodate AWM Thanksgiving Potluck

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Hao Xing, CUNY - Graduate Center, New York, NY
Title: Volume Computation in Homogeneous Dynamics
Abstract: In homogeneous dynamics, oftentimes we need to consider the "high dimensional time average" --- namely instead of averaging over \{[0,T]\), we average over the volume of a "ball" in a matrix group whose "radius" is T. This volume estimate poses challenges in proving equidistributional results in some settings. In this presentation, I will present some aspects/techniques of such computations. The talk will be made accessible to graduate students (and perhaps even advanced undergraduates).

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Linda Keen, CUNY - Graduate Center, New York, NY
Title: Ergodicity Properties of Nevanlinna functions
Abstract: Nevanlinna functions are a class of transcendental meromorphic functions characterized by their covering properties: they have no critical points and a finite number of asymptotic values. The orbits of the asymptotic values determine the dynamics. Their dynamics share many of the properties of rational functions. In particular, they have no wandering domains and the orbits of the asymptotic values completely determine the dynamics. In this talk we will describe Nevanlinna functions and for those Nevanlinna functions whose Julia set is the whole sphere, we will determine when the action is ergodic and when it is not. This is joint work with Tao Chen and Yunping Jiang.