- Ergodic Theory and Dynamical Systems Seminar
- CUNY Graduate Center
- Fridays 11:45 AM -12:45 PM, Room 5417 (Learning Seminar) and 3:00 PM - 4:00 PM, Room 6417 (ETDS Seminar)
- Organizers: Yunping Jiang, Tamara Kucherenko, and Enrique Pujals
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Mattew Grote, CUNY - Graduate Center, New York, NY
Title: Differentiable stacks in dynamical systems
Abstract: We will briefly review differentiable stacks and their natural dynamical flavor. I will demonstrate some of my original construction for a smooth action on a manifold ascending to the stack of an action groupoid. Then we will exploit general structure theorems offered by stacks to describe conditions on dynamical properties, e.g. conjugacy, as well as some examples demonstrating the thrills and pitfalls that stacks offer the dynamicist.
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Linda Keen, CUNY - Graduate Center, New York, NY
Title: Parameter Spaces for the Dynamics of Meromorphic Functions With 2 Asymptotic Values
Abstract: Just as the study of the parameter spaces of degree 2 polynomials was the basis for understanding the general theory for rational maps, studying parameter spaces of meromorphic functions with 2 asymptotic values is the basis for understanding the dynamics of meromorphic maps in general. Because these spaces are in ℂ2, one studies them by looking at one (complex) dimensional "slices". In this talk I will summarize the descriptions of several of these slices that have been previously studied and present some new results about a new slice.
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Simon Locke, CUNY - Graduate Center, New York, NY
Title: In-context learning across the logistic family
Abstract: Prior theoretical studies of in-context learning (ICL) in transformers have formulated task inference in a regression setting, where tasks are weight vectors in parameter space, and the data are independently sampled from a known distribution. In contrast, time series prediction can involve observations generated from an underlying dynamical system that can exhibit temporal dependencies while the available representation may only preserve part of the underlying state. We study ICL in transformers trained to perform next token prediction on trajectories generated by the logistic family. Using the logistic family as a controlled setting, we investigate how training-task diversity, context length, and observation resolution affect model performance. At a fixed trajectory budget, generalization to new maps emerges sharply beyond a critical level of training-task diversity. The resulting generalization structure depends on the observation representation. Finer partitions require greater task diversity to generalize but attain lower limiting error. Under restricted training support, transfer is strongly asymmetric. Models trained on intervals containing chaotic dynamics often predict lower-complexity systems accurately, whereas reverse transfer is poor. Our results show that in-context generalization depends jointly on task diversity, observation representation, and the dynamical regimes represented during training.
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Bryce Gollobit, CUNY - Graduate Center, New York, NY
Title: TBA
Abstract: TBA
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Arshiya Farhath Gulam Dasthagir, CUNY - Graduate Center, New York, NY
Title: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Melkana Brakalova, Fordham University, New York, NY
Title: TBA
Abstract: TBA
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Nikoloz Devdariani, CUNY - Graduate Center, New York, NY
Title: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Hao Xing, CUNY - Graduate Center, New York, NY
Title: TBA
Abstract: TBA
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Enrique Pujals , CUNY - Graduate Center, New York.
Title: Wild dynamics
Abstract: we will recall the result by Newhouse about the unfolding of tangencies and the emergences of what is usually called "wild phenomena" and its relation to a conjecture by Palis about the finiteness of attractors.
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Enrique Pujals , CUNY - Graduate Center, New York.
Title: Taming wild dynamics
Abstract: I will ty to explain a joint work with Sylvain Crovisier about how wild dynamics (the ones introduced in the learning seminar) can be organized using (external) homoclinic classes and how that may be related to a weak version of Palis conjecture.
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Chenxi Wu, University of Wisconsin at Madison
Title: Thurston’s algorithm for core entropy
Abstract: I will introduce William Thurston’s algorithm for core entropy of complex quadratic maps, its connection with Handel-Miller kneading theory, and how to use it to show the continuity of core entropy with respect to the turning angle. If time permits I may cover prior works by Giulio Tiozzo, Kathryn Lindsey and me.
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Chenxi Wu, University of Wisconsin at Madison
Title: Entropy and Zeta function on Abstract Hubbard Trees
Abstract: For complex quadratic maps which are non renormalizable but not necessarily postcritically finite, one can define the concept of abstract Hubbard trees which generalizes the concept of Hubbard trees for postcritically finite maps. With Yunping Jiang, we developed an algorithm for calculating the topological entropy and Artin-Mazur zeta function for the induced map, and showed that the latter is meromorphic on the unit disc, generalizing a consequence of the Milnor-Thurston kneading theory. I will also discuss possible generalizations of the algorithm to the non Archimedean setting.
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Emma Dinowitz , CUNY Graduate Center, New York
Title: Entropy of generic points under vague specification
Abstract: By the ergodic theorem, every ergodic measure has generic points, and by a theorem of Bowen the set of generic points of an ergodic measure has Bowen topological entropy equal to the measure-theoretic entropy; the analogous statement for packing entropy is also known. A general invariant measure need not have any generic points, but specification-like properties guarantee that it does. Vague specification, equivalent to the asymptotic average shadowing property, is among the weakest such properties known. We show that under vague specification the set of generic points of every invariant measure has both Bowen and packing entropy equal to the entropy of the measure, extending results of Pfister–Sullivan and others obtained under stronger hypotheses. The proof uses the tools of orbit complexity, algorithmic randomness, and the point-to-set principle for entropy.
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Tao Chen, CUNY - LaGuardia Community College, New York
Title: Accessibility of some points on bifurcation locus.
Abstract: In the exponential family, escaping parameters are those for which the orbit of the asymptotic value tends to infinity. These points lie on the bifurcation locus. It has been shown that these points can be approached by hyperbolic exponential maps; however, they do not lie on the boundary of any hyperbolic component. In this talk, we show that these escaping parameters do lie on the boundary of a hyperbolic component in the space of meromorphic maps. We also discuss extensions of this result to some meromorphic functions. This is joint work with Yunping Jiang and Linda Keen.
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Tamara Kucherenko, CUNY - City College, New York
Title: Maximizing Measures and the Aubry Set.
Abstract: Given a potential on a dynamical system, which invariant measures maximize its integral? In this introductory talk, we study the two sets that describe where these maximizing measures live: the Mather set, the union of the supports of all maximizing measures, and the Aubry set, which also contains orbits that are only "almost maximizing" in a precise sense. We work in the symbolic setting of subshifts of finite type, where both sets can be described concretely, and use simple examples to show how they can differ. Along the way, we meet the Mañé classes, the building blocks of the Aubry set, and see why their structure matters for what happens in the zero-temperature limit, where equilibrium states converge to maximizing measures.
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Tamara Kucherenko, CUNY - City College, New York
Title: Exponential convergence of the pressure at zero temperature: a characterization via the Aubry set
Abstract: As the temperature goes to zero, the pressure of a potential approaches a slant asymptote. How fast does it get there? For a Walters potential on a mixing subshift of finite type, we show that the rate of convergence, measured on an exponential scale, always exists. Moreover, this rate is strictly positive if and only if the union of the maximal entropy components of the Aubry set is a subshift of finite type. We also establish a dichotomy for Hölder potentials: if this union is a subshift of finite type, the pressure converges to its asymptote exponentially fast, while if it is sofic (but not of finite type), the convergence is polynomial. This is joint work with J.-R. Chazottes, A. Quas, and R. Leplaideur.
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Yunping Jiang, CUNY - Queens College, New York, NY
Title: Gibbs measures in thermodynamical formalism
Abstract: We will introduce Gibbs measures in thermodynamic formalism and give a proof of the existence and uniqueness of the Gibbs measure associated with a Hölder continuous potential. If time permits, we will extend the argument to Dini continuous potentials and discuss the similarities and differences between the Hölder and Dini settings.
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Yunping Jiang, CUNY - Queens College, New York, NY
Title: The space of geometric Gibbs measures: Uniqueness, ergodicity, and Teichmüller complex manifold structure
Abstract: From the morning seminar, we know that the space of Gibbs measures is in one-to-one correspondence with the space of cohomology classes of Hölder continuous potentials, and that each cohomology class contains a unique normalized g-function. When a g-function satisfies a certain compatibility condition, it determines a circle endomorphism for which the g-function becomes the dual derivative. We call such a g-function a smooth geometric g-function.
Every expanding circle endomorphism is quasisymmetrically conjugate to the standard model. This allows us to introduce a Teichmüller metric on the space of smooth geometric g-functions using boundary quasiconformal dilatation. However, this Teichmüller space is not complete.
We have studied its completion and identified it with the Teichmüller space of continuous geometric g-functions. In our earlier work, we proved the existence of a geometric Gibbs measure associated with each continuous geometric g-function, while uniqueness remained an open problem for a long time. Conceptually, a geometric Gibbs measure need not be a classical Gibbs measure, although the two notions share several properties. We recently proved uniqueness, thereby establishing a one-to-one correspondence between continuous geometric g-functions and geometric Gibbs measures. Consequently, the completion of the space of smooth geometric Gibbs measures can be identified with the Teichmüller space of continuous geometric Gibbs measures.
In this talk, I will explain these results, discuss future the ergodic properties of geometric Gibbs measures, and describe the Teichmüller complex manifold structure on the space of continuous geometric Gibbs measures.
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Christian Wolf, Mississippi State University, Starkville, MS
Title: Computability in Dynamical Systems
Abstract: In this talk, we present a survey of recent advances in the study of computability within dynamical systems, a field that has experienced significant growth over the past two decades, yielding many surprising and insightful results. We will discuss developments related to the computability of entropy and pressure, the effective description of dynamically significant sets such as Julia sets and rotation sets, and the computability of certain natural invariant measures.
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Christian Wolf, Mississippi State University, Starkville, MS
Title: Computability Properties of Hyperbolic Henon Maps
Abstract: In this talk we address the question if the Julia set of a complex H\'enon map can be computed rigorously by an algorithm. More broadly, we study computability properties of polynomial diffeomorphisms of \(\mathbb{C}^2\). We show that, whenever such a map is hyperbolic, its Julia set is computable to arbitrary prescribed accuracy. The approach uses finite box models of the chain recurrent set together with the density of saddle periodic points in the Julia set. We also describe an algorithm that semi-decides hyperbolicity by constructing rigorously verified stable and unstable cone fields. As a consequence, the hyperbolicity locus of generalized Henon families is lower semi-computable. I will explain the main ideas behind these algorithms and discuss further consequences, including the semi-decidability of disconnected Julia sets and extensions to higher-dimensional polynomial automorphisms. If time allows, we will also discuss computability results for certain non-hyperbolic H\'enon maps. The material discussed in this talk is joint work with Suzanne Boyd, see https://arxiv.org/abs/2605.26306 and https://arxiv.org/abs/2608.01517.
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Marian Gidea, Yeshiva University, New York, NY
Title: Arnold Diffusion in Some Models from Celestial Mechanics
Abstract: In a landmark paper, V.I. Arnold conjectured that
integrable Hamiltonian systems of general type, subjected to small,
generic perturbations, exhibit diffusing orbits, along which the
action variable/energy changes by an amount that is independent of
the perturbation parameter.
He devised a mechanism based on transition chains of tori, and claimed
``I believe that [this mechanism of instability] is applicable to the
general case (for example, to the problem of three bodies)". Despite
the long history of the problem, there are only a few rigorous results
on Arnold diffusion in celestial mechanics, particularly with
realistic parameters (e.g., for the masses of the bodies, or for the
energy levels).
We will describe a geometric mechanism of diffusion and its
applications to models from celestial mechanics. Our approach is
based on identifying the geometric structures that organize the
dynamics -- fixed points, periodic orbits, invariant tori, normally
hyperbolic invariant manifolds (NHIMs) and their stable and unstable
manifolds -- and on constructing trajectories following these
structures.
We will rely on the `two-dynamics' approach, which amounts to
combining the `inner dynamics' -- along orbits restricted to the
NHIMs -- with the `outer dynamics' -- along homoclinic/heteroclinic
orbits. We encode the asymptotic behavior of homoclinic/heteroclinic
orbits via scattering maps -- geometrically defined maps acting on
the NHIMs, that relate the past asymptotic of an orbit to its future
asymptotic. The scattering maps can be computed either perturbatively
or numerically. The main application of the scattering map is that it
allows to compute the change of action/energy along a
homoclinic/heteeroclinic orbit, and to select which orbits to follow
in order to obtain a desired effect.
We will use the geometric mechanism to show Arnold diffusion in
several models: the planar elliptic restricted three-body problem, the
spatial circular restricted three-body problem, and the elliptic Hill
for body-body problem.
Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Marian Gidea, Yeshiva University, New York, NY
Title: Arnold Diffusion in the Full Three-Body Problem: A Computer
Assisted Proof
Abstract: We consider the full three-body problem describing the
motion of three positive masses interacting through their mutual
gravitational attraction.
We assume that one of the three masses is small and reformulate the
system as a perturbation of two uncoupled subsystems: a Kepler problem
and a restricted three-body problem, with the small mass serving as
the perturbation parameter. We show that the perturbed system exhibits
Arnold diffusion, in the sense that there is a transfer of energy
-- by an amount independent of the perturbation parameter -- between
the Kepler problem and the restricted three-body problem.
Our argument is based on the topological method of correctly aligned
windows, which is implemented into a computer assisted proof.
We show that the approach applies to physically relevant masses of the
bodies, choosing a Neptune-Triton-asteroid system as an example.
In this case, we obtain explicit estimates for the range of the
perturbation parameter and for the time required for the energy
transfer.
This is based on joint work with Maciej Capinski.