- Ergodic Theory and Dynamical Systems Seminar
- Fridays 11:00 AM -12:00 PM (Learning Seminar) and 3:00 PM - 4:00 PM (ETDS Seminar)
- CUNY Graduate Center, Room 5417
- Organizers: Yunping Jiang, Tamara Kucherenko, Enrique Pujals, and Christian Wolf
No seminar scheduled
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Marco Lopez, CUNY - Graduate Center, New York, NY
Title: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Marco Lopez, CUNY - Graduate Center, New York, NY
Title: TBA
Abstract: TBA
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.
In this introductory talk, we will discuss the aforementioned topics and questions, as well as some applications of Lyapunov exponents for the class of diffeomorphisms with dominated splitting.
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: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Sergiy Merenkov, CUNY - City College, New York, NY
Title: TBA
Abstract: TBA
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Yushan Jiang, CUNY - Graduate Center, New York, NY
Title: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Melkana Brakalova, Fordham University, NY
Title: When the circular dilatation of a quasiconformal mapping at a point equals one.
Abstract: I discuss geometric and analytic conditions implying that the circular dilatation (a.k.a. linear distortion) at a point of a planar quasiconformal mapping equals one, e.g. conformality, \(C^{1+\alpha}\) conformality, asymptotic homogeneity, weak conformality at a point. Many of the results can be viewed as an extension of the Teichmüller-Wittich-Belinskii theorem. Besides being of interest by themselves, they find applications in Nevanlinna theory, modulus of continuity studies, complex dynamics, the theory of \(p\)-integrable Teichmüller spaces, some of which are highlighted in this talk.
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.
Reserved for neural network
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Hao Xing, CUNY - Graduate Center, New York, NY
Title: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 5417
Linda Keen, CUNY - Graduate Center, New York, NY
Title: TBA
Abstract: TBA
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
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.