- Ergodic Theory and Dynamical Systems Seminar
- CUNY Graduate Center
- Fridays 11:00 AM -12:00 PM, Room 5417 (Learning Seminar) and 2:30 PM - 3:30 PM, Room 5383 (ETDS Seminar)
- Organizers: Yunping Jiang, Tamara Kucherenko, and Enrique Pujals
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Sudeb Mitra , CUNY - Queens College, New York, NY
Title: Motions: from continuous to holomorphic, via quasiconformal
Abstract: Motions: from continuous to holomorphic, via quasiconformal
Abstract: In their study of the dynamics of rational maps, Mãńe, Sad, and Sullivan introduced the concept of holomorphic motions, where they proved the \(\lambda\)-lemma; see [1]. In this talk, we will discuss some basic properties of continuous and holomorphic motions, and quasiconformal motions, in the sense of Sullivan and Thurston. We will also include some applications in geometric function theory. This talk should be accessible to all graduate students.
[1] R. Mãńe, P. Sad and D. P. Sullivan, On the dynamics of rational maps, Ann. Sci. Ecole Norm. Sup. 16 (1983), 193-217.
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
Axel Kodat , CUNY - Graduate Center, New York, NY
Title: Entropy and local volume growth for \(C^2\) conservative diffeomorphisms
Abstract: Shub's entropy conjecture asserts that the topological entropy of a \(C^1\) diffeomorphism on a closed manifold M is bounded from below by log of the spectral radius of the induced map on homology. While the conjecture was proved for \(C^\infty\) maps by Yomdin in the 80s, it remains largely open in lower regularity. In this talk I will sketch some new partial results in the setting of \(C^2\) volume-preserving diffeomorphisms.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Enrique Pujals, CUNY - Graduate Center, New York, NY
Title: Closing lemmas for surface dynamics
Abstract: It was conjectured that \(C^r\)-generically periodic points are dense in the non-wandering set. We will discuss that problem, origin, interest of that and partial results. At the end, we will show a positive answer for dissipative diffeomorphisms of the disk and recent versions that will be used in the afternoon talks to classify ergodic measures. This is a joint work with Sylvain Crovisier.
Ergodic Theory and Dynamical Systems Seminar 2:30 PM - 3:30 PM, GC 5383
Enrique Pujals, CUNY - Graduate Center, New York, NY
Title: Organizing the non-wandering set of mild dissipative diffeomorphisms of the disk.
Abstract: For mild dissipative of the disk we will show that any ergodic measure is either metric isomorphic to an odometer or it is contained in an homoclinic class . That result will be used to decompose the non-wandering set into different maximal transitive pieces. The main technique used is a (new) closing lemma that we will discuss in the morning seminar. This is a joint work with Sylvain Crovisier.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Tao Chen, CUNY - Graduate Center, New York, NY
Title: Dynamics of a family of real discontinuous functions Part 1
Abstract: In this talk, we will introduce the functional properties of this family and motivation to investigate its dynamics. Furthermore, we will give some preliminary dynamical properties.
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
Tao Chen, CUNY - Graduate Center, New York, NY
Title: Dynamics of a family of real discontinuous functions Part 2
Abstract: In this talk we mainly show the bifurcation diagram of this family, which is different from any known family to our knowledge. Moreover, we associate the family with shift maps on symbolic spaces and show it is a full family.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Yunping Jiang, CUNY - Queens College, New York, NY
Title: Denjoy Counterexamples in Circle Homeomorphisms and Mean Lyapunov Stability
Abstract: In this talk, I will discuss the dynamics of circle homeomorphisms and present a Denjoy counterexample, which is only topologically semi-conjugate to a rigid rotation. We show that a circle homeomorphism is a Denjoy counterexample if and only if it is not an equicontinuous dynamical system. Furthermore, we prove that a Denjoy counterexample is mean Lyapunov stable (MLS). Finally, I will introduce the notion of a-MLS for a sequence ataking value in natural numbers and pose the question: Is a Denjoy counterexample Ω-MLS, where Ω denotes the big-omega function in number theory?
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
Yunping Jiang, CUNY - Queens College, New York, NY
Title: Time-Averages Along Partial Orbits in Dynamical Systems
Abstract: We study the convergence of time-averages of continuous functions along partial orbits for continuous dynamical systems on compact metric spaces. Unlike the classical focus of ergodic theory on almost-everywhere convergence, our work emphasizes convergence at every point, motivated by problems in number theory. To this end, we consider sequences in the natural numbers with a uniform behavior property and introduce two structural notions for dynamical systems along such partial orbits: minimal mean Lyapunov stability (a-MMLS) and minimal mean attractability (a-MMA).Our main result shows that if a continuous dynamical system is both a-MMLS and a-MMA, and is minimally uniquely ergodic, then the time-average of any continuous function along the corresponding partial orbit converges to the space average at every point. This is joint work with Jessica Liu.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Yushan Jiang, CUNY - Graduate Center, New York, NY
Title: The Bowen-Series Coding, Thermodynamics, and the Mostow Rigidity
Abstract: The Bowen-Series coding provides us a method to construct the corresponding circle expanding maps for Fuchsian groups (this could go back to Nielsen). Hence, we can study conformal dynamics by using the theory about expanding maps, like the Ruelle-Perron-Frobenius operator from thermodynamics. Firstly, I will try to illustrate how to construct such piecewise Möbius circle expanding map via a finitely generated Fuchsian group. Then we use the Ruelle-Perron-Frobenius operator to obtain the 2-dimensional Mostow Rigidity: let h be an orientation preserving homeomorphism of circle that conjugates two cocompact Fuchsian group actions, then h is Möbius iff h is absolutely continuous respect to the Lebesgue measure.
Ergodic Theory and Dynamical Systems Seminar 2:30 PM - 3:30 PM, GC 5383
Tamara Kucherenko, CUNY - City College, New York, NY
Title: Thermodynamic Formalizm for Coded Shift Spaces
Abstract: I will discuss thermodynamic formalism for coded shifts and present results concerning the uniqueness of measures of maximal entropy and equilibrium states for Hölder potentials, as well as their properties.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Arshiya Farhath Gulam Dasthagir, CUNY - Graduate Center, New York, NY
Title: Quasiconformal mappings and holomorphic motions — an Introduction
Abstract: We will discuss some basic properties of quasiconformal mappings and holomorphic motions. In particular, we will deliberate on the \(\lambda\)-lemma and Slodkowski’s extension theorem. We will give some explicit examples.
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
Chenxi Wu, University of Wisconsin – Madison, Madison WI
Title: Sublinearly Morse Geodesics in Non Uniquely Ergodic Directions
Abstract: This is a joint work with Matthew Durham and Kejia Zhu. It is known that a “generic” geodesic ray in Teichmuller space (in the sense of random walks on mapping class groups) would be sublinearly Morse and also point at a uniquely ergodic direction (i.e. hits a point in the Thurston’s boundary where the measured foliation is uniquely ergodic). We found out that, however, by making use of the non uniquely ergodic Teichmuller geodesics constructed by Veech, one can have sublinearly Morse geodesic rays that do not converge to a single uniquely ergodic point on Thurston’s boundary. The paper is arXiv: 2504.17986.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Bruno Nussenzveig, CUNY - Graduate Center, New York, NY
Title: Finite time blow-up for a special class of ODEs
Abstract: It is a well-known fact that the solutions to \(x’ = x^2\) on \([0, \infty)\) exhibit finite time blow-up: we have \(x(t) \to \infty\) infinity as \(t \to\) some finite positive \(t_0\), and solutions cannot be extended infinitely far into the future. In this talk, we will describe a program to study the question of finite time blow-up for solutions of a class of autonomous ordinary differential equations in \(\mathbb{R}^n\) which have a specific kind of projective symmetry. For concreteness, we will only allude to the general program and will instead focus on carrying out this program in the specific case of ODEs given by homogeneous quadratic polynomial functions.
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
Christian Wolf, Mississippi State University, Starkville, MS
Title: Computability for Skew products in two complex dimensions
Abstract: The computability of Julia sets of rational maps on the Riemann sphere has been intensively studied in recent years. For example, Braverman established that hyperbolic and parabolic Julia sets are computable in polynomial time. In this talk, we present the first results on computability related to maps of more than one complex dimension.
We consider certain polynomial endomorphisms of \(\mathbb{C}^2\), the polynomial skew products; i.e., maps of the form
\(f(z,w) = (p(z), q(z,w)),\)
where \(p\) and \(q\) are complex polynomials of the same degree \(d\geq 2\).
We show that if a polynomial skew product is Axiom A, then its chain recurrent set, which is equal to its non-wandering set and also equal to the closure of the periodic orbits, is computable. Our algorithm also identifies the various hyperbolic sets of different types, i.e., expanding, attracting, and hyperbolic sets of saddle-type.
One consequence of our results is that Axiom A is a semi-decidable property on the closure of the Axiom A polynomial skew product locus. Finally, we introduce an algorithm that establishes the lower semi-computability of the hyperbolicity locus of polynomial skew products of a fixed degree. The results presented in this talk are joint work with Suzanna Boyd.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Dan Thompson , Ohio State University, Columbus, OH
Title: Equilibrium States for Geodesic Flows
Abstract: We discuss some classic and recent in the smooth ergodic theory of geodesic flows. We will start with an intuitive overview of some classic results developed by luminaries such as Anosov, Bowen and Ruelle in the well understood setting of compact surfaces with variable negative curvature. This setting gives good motivation for why the equilibrium state theory was developed. We’ll talk about how Bowen’s specification techniques work in this setting. These methods have proved useful in recent years for extending the theory into settings including geodesic flow for compact spaces of non-positive curvature, and geodesic flow for negative curvature manifolds which are not compact.
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
Dan Thompson , Ohio State University, Columbus, OH
Title: Specification and strong positive recurrence for flows on complete metric spaces
Abstract: We extend Bowen’s specification approach to thermodynamic formalism to flows on complete separable metric spaces. The key point, particularly for the existence of a finite equilibrium state, is a Strong Positive Recurrence (SPR) assumption. As one application, we establish that for a sufficiently regular potential with SPR for the geodesic flow on a geometrically finite locally CAT(-1) space, there exists a unique equilibrium state. Examples of CAT(-1) spaces range from manifolds with negative curvature bounded above by -1, and at the other extreme, graphs and trees equipped with a notion of length. This is joint work with Vaughn Climenhaga and Tianyu Wang.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Nikoloz Devdariani , CUNY - Graduate Center, New York, NY
Title: Divergent Fourier series with respect to biorthonormal systems in function spaces near \(L^1\)
Abstract: In this paper, we generalize Bochkarev's theorem, which states that for any uniformly bounded biorthonormal system \(\Phi\), there exists a Lebesgue integrable function whose Fourier series with respect to the system \(\Phi\) diverges on a set of positive measure. We find the class of variable exponent Lebesgue spaces \(L^{p(\cdot)}([0,1]^n)\), where \(p(x)\in (1,\infty)\) almost everywhere on \([0,1]^n\), such that the aforementioned Bochkarev's theorem holds.
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
Yang Fan , Wake Forest University, Winston-Salem, NC
Title: Equilibrium states for star flows and the spectral decomposition conjecture
Abstract: In this talk, we will discuss recent progress in the theory of smooth star flows that contain singularities and consider their expansiveness, continuity of the topological pressure, and the existence and uniqueness of equilibrium states. We will prove an ergodic version of the Spectral Decomposition Conjecture: \(C^1\) open and densely, every singular star flow has only finitely many ergodic measures of maximal entropy, and only finitely many ergodic equilibrium states for Holder continuous potentials satisfying a mild yet optimal condition. Joint with M.J. Pacifico and J. Yang.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Xing Hao , CUNY - Graduate Center, New York, NY
Title: Homogeneous dynamical problems arise from Euclidean sublattices
Abstract: In this talk, I will talk about some interesting problems and challenges in homogeneous dynamics that arise from the study of the space of Euclidean sublattices, such as the density of orbits, non-escape of mass and the equidistribution of closed orbits, as well as the available techniques within my knowledge. This talk will be made accessible to audience without a background in homogeneous dynamics.
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
Emma Dinowitz , CUNY - Graduate Center, New York, NY
Title: A point to set principle for topological entropy with applications to formulas relating dimension, entropy, and Lyapunov exponents
Abstract: We prove a point to set principle for topological entropy by extending the orbit complexity framework established by Galatolo, Hoyrup, and Rojas. We use this to establish a number of classical results in dynamical systems relating dimension, entropy, and Lyapunov exponents, and prove several new dimension formulas in the setting of nonuniformly hyperbolic dynamical systems.
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
David Aulicino , CUNY - Graduate Center, New York, NY
Title: Weak Mixing of Translation Flows in Rank One Loci
Abstract: Translation surfaces admit natural straight-line flows in the full circle of directions. These flows exhibit a range of dynamic properties. Generically they are known to be uniquely ergodic but not mixing. Recently work of Arana-Herrera, Chaika, and Forni classified when the flow exhibits the intermediate property of weak mixing. In the present work, we restrict to a special class of translation surfaces and give an alternate proof of the result of Arana-Herrera, Chaika, and Forni, which is accomplished by proving a stronger result for this special class by extending previous work of A. Avila and V. Delecroix. All necessary background will be given. This is joint with Artur Avila and Vincent Delecroix.
Learning Seminar 11:00 AM-12:00 PM, GC 5417
Bruno Nussenzveig, CUNY - Queens College, New York, NY
Title: Destroying infinite renormalizability of unimodal maps of the interval
Abstract: We will discuss an ongoing project by the student and his advisor to prove, using relatively "cheap" techniques, that the
set \(\mathbb{U}_\infty^r\) of \(C^r\) unimodal maps, \(r \geq 3\), which are infinitely period doubling renormalizable has no interior in the \(C^r\) topology on the space of all unimodal maps; that is, that one can destroy infinite renormalizability (in the period doubling combinatorics) by an arbitrarily \(C^r\)-small perturbation. The main source of inspiration is the paper [JMS] by Jiang, Morita and Sullivan, where an expanding direction for the period doubling operator \(\mathcal{R}\) is constructed at the fixed point while avoiding the main complications related to the non-differentiability (in the Fréchet sense) of this operator. The primary focus of the talk will be twofold: first, we will discuss a more flexible approach to the period doubling operator which allows one to make use of most of the ideas from [JMS] at an arbitrary infinitely renormalizable map; and secondly, we will construct a tangent cone field on \(\mathbb{U}_\infty^r\), invariant under the Gateaux derivative of \(\mathcal{R}\), which is a prime candidate for expansion. Hopefully, by the time of the presentation, we will have been able to show that tangent vectors in this cone field are indeed expanded, which will close out the argument.
Though the result described above has been well-known for decades, it has been previously attained as a byproduct of the complicated theory of hyperbolicity for the renormalization operator, which (in its current form) is inherently one-dimensional. Our goal is to find a sufficiently simple and flexible approach that could be lifted to Hénon-like and mild dissipative diffeomorphisms of the disk, where the corresponding result is as of yet unproven.
[JMS] Jiang, Y.; Morita, T.; Sullivan, D. Expanding direction of the period doubling operator (1992). Comm. Math. Phys. 144, No. 3, 509-520.
Ergodic Theory and Dynamical Systems Seminar, 2:30 PM - 3:30 PM, GC 5383
Simon Locke , CUNY - Graduate Center, New York, NY
Title: Dynamics of Learning in Physical Networks
Abstract: Recent work has shown that physical systems—such as electrical circuits or mass–spring networks—can be trained to perform computational tasks by adapting internal parameters in response to external signals. These physical learning networks can be viewed as dynamical systems with two coupled timescales: fast variables describing the system’s physical state and slow variables encoding changes in learning parameters. In this talk, I will introduce a mathematical framework for these networks and derive a local learning rule, known as equilibrium propagation, that can be used to train them. I will highlight connections to classical ideas in dynamical systems theory—fixed points, stability, and energy-based formulations—and discuss open questions at the interface of dynamics, computation, and physics.