Ergodic Theory and Dynamical Systems Seminar 12:30-1:30 PM, GC 6417
Alex Stokolos, Georgia Southern University, Statesboro, GA
Title: Geometric Complex Analysis and detection of cycles in nonlinear dynamical systems
Abstract: In the talk, I will discuss a remarkable connection between the problem of long cycles detection in nonlinear autonomous dynamical systems and geometric complex analysis. The presentation will be accessible to graduate students and non-experts.
Ergodic Theory and Dynamical Systems Seminar 12:30-1:30 PM, GC 6417
Christian Wolf, CUNY - Graduate Center, New York, NY
Title: Ergodic theory on coded shifts spaces
Abstract: In this talk we present results about ergodic-theoretic properties of
coded shift spaces. A coded shift space is defined as a closure of all bi-infinite
concatenations of words from a fixed countable generating set. We derive
sufficient conditions for the uniqueness of measures of maximal entropy and
equilibrium states of Hoelder continuous potentials based on the partition of the coded shift into its sequential set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). We also discuss flexibility results for the entropy on the sequential and residual set. Finally, we present a local structure theorem for intrinsically ergodic coded shift spaces which shows that our results apply to a larger class of coded shift spaces compared to previous works by Climenhaga, Climenhaga and Thompson, and Pavlov. The results presented in this talk are joint work with Tamara Kucherenko and Martin Schmoll.
Learning Seminar 10:30 -11:30 AM, GC 6417
Tamara Kucherenko, CUNY - City College, New York, NY
Title: Smooth conjugacy problem for Anosov diffeomorphisms on tori
Abstract: From the results by Franks and Manning in early '70s we know that any two Anosov diffeomorphisms in the homotopy class of a fixed hyperbolic automorphism are conjugate. Generally, the conjugacy may be merely Hölder even for \(C^\infty\) diffeomorphisms. The problem of determining when the conjugating homeomorphism has the same regularity as the maps is known as the smooth conjugacy problem. In a series of papers De la Llave, Marco, and Moriyon proved that the equality of the Lyapunov exponents at corresponding periodic orbits is a sufficient condition on \(\mathbb{T}^2\). Jointly with A. Quas we show that a natural weakening of this condidion does not lead to the same conclusion, hence providing a negative answer to a question by F. Rodriguez Hertz.
Ergodic Theory and Dynamical Systems Seminar 12:30-1:30 PM, GC 6417
Fabio Armando Tal, University of São Paulo, São Paulo, Brazil
Title: Fully chaotic models for surface homeomorphisms
Abstract: We study homeomorphisms of closed surfaces homotopic to the identity. We show that, whenever the rotational behavior is sufficiently rich (in the \(\mathbb{T}^2\) case, has a rotation set with nonempty interior), then they are semiconjugated through a monotone quotient to a map with the same rotational behavior, but that also preserves area, is topologically mixing and has dense periodic points.
J. with A. De Carvalho, A. Koropecki and A. Garcia
10:30 AM -1:30 PM, GC 6417 (lunch break 11:30-12:30)
Enrique Pujals, CUNY - Graduate Center
Title: Renormalization of unicritical diffeos of the disk
Abstract: In a joint paper 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 recent 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 (at the Learningt Seminar) will be about the context of the problem. The second part will be focused on the outline of the proof's strategies.
Learning Seminar 10:30 -11:30 AM, GC 6417
Yunping Jiang, CUNY - Queens College, New York, NY
Title: Branched Coverings of the 2-Sphere and Thurston's Theorem
Abstract: This lecture introduces fundamental concepts surrounding branched coverings of the 2-sphere. A notable example is the rational map. Within the realm of complex dynamical systems, a key inquiry revolves around determining when a branched covering can be expressed as a rational map and whether this representation is unique. We establish definitions for combinatorial equivalence among branched coverings, critically finite branched coverings, geometrically finite branched coverings, Thurston's obstruction, and the orbifold associated with a critically finite branched covering. Lastly, Thurston's theorem is presented, which asserts that a critically finite branched covering with a hyperbolic orbifold can be uniquely represented by a rational map under the combinatorial equivalence if and only if it has no Thurston obstructions.
Ergodic Theory and Dynamical Systems Seminar 12:30-1:30 PM, GC 6417
Yunping Jiang, CUNY - Queens College, New York, NY
Title: Tameness Conditions in Complex Dynamics
Abstract: The extension of mathematical results from finite contexts to infinite ones presents significant challenges. An example is to extend Thurston's theorem from critically finite to critically infinite branched coverings.
This talk will provide an overview of our progress in this area. Initially, we offer a counter-example of a critically infinite branched covering, illustrating the infeasibility of a direct generalization of Thurston's theorem, even within the realm of geometrically finite branched coverings. Subsequently, we introduce a manageable "tameness" condition within the context of geometrically finite cases. Finally, we demonstrate that under this tameness condition, Thurston's theorem can be generalized to all sub-hyperbolic tame geometrically finite branched coverings under the tame combinatorial equivalence.
Learning Seminar 10:30 -11:30 AM, GC 6417
Yushan Jiang, CUNY - Graduate Center, New York, NY
Title: From the Borel Conjecture to geodesic dynamics
Abstract: Borel conjecture is a central conjecture in geometric topology (and still open for higher dimension): any homotopy equivalent between two closed aspherical manifolds (i.e. universal cover is contractable) is homotopic to a homeomorphism.
There are a lot of crucial works in this topic. I will introduce some backgrounds first. Then I will focus on Gromov’s approach in the case of negative curvature (in this case, Borel Conjecture was completely proved by Farrell-Jones later) from the point of view of geodesic dynamics:
Let \(M\) and \(N\) be two closed negatively curved manifolds, if \(M\) is homotopy equivalent to \(N\), then the geodesic foliations \(G(M)\) and \(G(N)\) are homeomorphic (i.e. there is a homeomorphism between unit tangent bundles \(S(M) \to S(N)\) sending leaves from \(G(M)\) into leaves from \(G(N)\)).
In the end, if time permits, I will try to replace the condition “negatively curved” by weaker one “with geodesic flow of Anosov type” and see what we can say about this kind of manifolds.
Ergodic Theory and Dynamical Systems Seminar 12:30-1:30 PM, GC 6417
Jan Boronski, Jagiellonian University, Krakow, Poland
Title: A classification of Hénon maps in the presence of strange attractors
Abstract: In my talk I shall present my work with Sonja Štimac on Hénon maps with strange attractors (Wang-Young parameters). First I shall explain a construction (inspired by a work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane) of conjugacy of these maps to the shift homeomorphisms on inverse limits of dendrites with dense set of branch points, and a characterization of orbits of critical points in terms of these inverse limits. Then I will explain how this leads to a classification of conjugacy classes of such maps in terms of a single sequence of 0s and 1s.
Learning Seminar 10:30 -11:30 AM, GC 6417
Bryce Gollobit, CUNY - Graduate Center, New York, NY
Title: Fundamental theorem of dynamical systems
Abstract: In this talk, we'll state and prove Conley's theorem, also known as the fundamental theorem of dynamical systems, which relates the chain recurrent set of a homemorphism to its attractor/repeller pairs.
We'll review the concepts of attractors and chains, which were introduced by Bowen in the 1970's when he was studying the statistical properties Axiom A diffeomorphisms on their basic sets. We'll provide examples in the context of diffeomorphisms.
If there is time, we'll go over some infinite dimensional linear algebra that might be useful for the afternoon talk.
Ergodic Theory and Dynamical Systems Seminar 12:30-1:30 PM, GC 6417
Bryce Gollobit, CUNY - Graduate Center, New York, NY
Title: Tangent bundle dynamics
Abstract: In this talk we'll discuss the interplay of the dynamics of \(C^1\) diffeomorphisms on compact manifolds to the infinite dimensional linear dynamics of its pushforward on continuous vector fields. The most famous example of such a correspondence is Mather's theorem, which states that a \(C^1\) diffeomorphism is Anosov if and only if its pushforward is a hyperbolic linear map.
We will present a generalization of Mather's result. The essential dynamical features of a hyperbolic linear map are the shadowing property and uniform expansivity, and each of these properties characeterize hyperbolicity for finite dimensional linear maps. This is no longer true in infinite dimensions. We'll show that:
1) A \(C^1\) diffemorphisms satisfies Axiom A and the strong transversality condition if and only if its pushforward has the shadowing property.
2) A \(C^1\) diffeomorphism is quasi-Anosov if and only if its pushforward is uniformly expansive.
We'll indicate some new results about invariant foliations for quasi-Anosov diffeomorhpisms that rely on the dynamics of its pushforward. We'll also briefly discuss applications of this approach outside of the context of Axiom A diffeomorphisms.
Learning Seminar 10:30 -11:30 AM, GC 6417
Axel Kodat, CUNY - Graduate Center, New York, NY
Title: A very short introduction to Yomdin theory
Abstract: At its core, Yomdin theory amounts to a single estimate bounding the number of cubes needed to partition a given \(C^r\) disk into pieces of uniformly \(C^r\)-small size. This technical result implies a wealth of non-obvious constraints on the geometry of \(C^r\) maps and has proven to be a powerful tool in the analysis of smooth dynamical systems. While its notoriously difficult proof relies on semi-algebraic geometry and lies well beyond the scope of this talk, I will attempt to sketch some of the main ideas, with a particular focus on Yomdin’s original application to prove the \(C^\infty\) entropy conjecture. If time permits, I will also outline a proof of the main lemma for the one-dimensional case; while substantially simpler than the general version, this result is already sufficient for several deep applications, as in e.g. the recent work of Buzzi-Crovisier-Sarig on \(C^\infty\) surface diffeomorphisms.
Ergodic Theory and Dynamical Systems Seminar 12:30-1:30 PM, GC 6417
Maurice Rojas, Texas A&M University, College Station, TX
Title: Descartes' Rule and Fewnomials over the p-adics
Abstract: Descartes stated, around 1637, a tight upper bound for the number of
real roots of a univariate polynomial with exactly t monomials: \(2t-1\). However,
the analogous result for systems of \(n\) polynomials in \(n\) variables states a much
looser bound. For instance, the maximal number of non-degenerate roots of a
\(2\) by \(2\) system with \(t\) distinct monomials has only been proved to be exponential
in \(t\) (following work of Khovanskii, Li, Rojas, Wang, Bihan, Sottile, and
others from 1980 to 2009), even though the true bound is conjectured to be
quadratic in \(t\) (via more recent work of Burgisser, Ergur, Tonelli-Cueto, and others).
Sharp bounds in Fewnomial Theory are important because sufficiently sharp
bounds would imply new complexity results like the hardness of the permanent.
Fewnomial Theory in fact extends to arbitrary local fields, e.g., the
\(p\)-adic rationals \(Q_p\): Denef and van den Dries proved around 1988 that upper
bounds that are a function solely of \(p\), \(n\), and \(t\) exist. However, non-trivial
explicit bounds remained unknown... until now: We prove that \(n\) by \(n\) systems
with exactly \(n+2\) distinct exponent vectors have at most \((n+1)p\) non-degenerate
roots over \(Q_p\), provided \(p>n+2\). The key new technique is a combinatorial
encoding of p-adic integer roots (discovered independently by Rojas and Saxena)
that may be of independent interest. We assume no background in number theory.
This is joint work with Joshua Goldstein.
Learning Seminar 10:30 -11:30 AM, GC 6417
Hao Xing, Ohio State University, Columbus, OH
Title: Equidistribution of definable curves in a polynomially bounded structure in homogeneous spaces
Abstract: We describe closures and limiting distribution of trajectories \(\{\phi(t)\mathbb{Z}^n:t\in [1,\infty]\}\) in the (finite volume) space of unimodular lattices in \(\mathbb{R}^n\), under appropriate conditions, where \(\phi(t)\) is a \(n\times n\) matrix of determinant \(1\) whose coordinate functions are `definable in a polynomially bounded o-minimal structure’. The work uses Ratner’s theorems on unipotent flows. It extends the earlier work of Shah for polynomial trajectories, and the work of Peterzil and Starchenko on trajectories on nilmanifolds that are `definable in a polynomially bounded o-minimal structure’. This reports my ongoing joint work with Michael Bersudsky and Nimish Shah. This talk will be non-technical and aimed at graduate students with no prior background including model theory (definable functions) assumed.
Ergodic Theory and Dynamical Systems Seminar 12:30-1:30 PM, GC 6417
Ethan Akin, CUNY - City College, New York, NY
Title: Simplicial Dynamical Systems
Abstract: Any continuous self-map of a polyhedron, e.g. a triangulable manifold, can be \(C^0\) approximated by a simplicial
dynamical system. Such a system is a simplicial map to a triangulation K from a suitable subdivision \(K^*\) (a so-called proper subdivision). We show that the dynamics of such a system are easy to analyze: The system is described by finite data and the set of chain recurrent points is the union of finitely many pieces on each of which the dynamics is an almost one-to-one quotient of a subshift of finite type.
Learning Seminar 10:30 -11:30 AM, GC 6417
Marian Gidea , Yeshiva University, New York, NY
Title: Geometric properties of normally hyperbolic invariant manifolds and scattering maps for conformally symplectic systems
Abstract: Conformally symplectic systems appear naturally in physics (e.g., mechanical systems with dissipative forces
proportional to the velocity), celestial mechanics (e.g., the spin-orbit models), economics (e.g., discounted systems), transport problems (e.g., thermostats), etc.
We will focus on conformally symplectic maps, which transforms the symplectic structure into a multiple of itself by a conformal factor.
First, we study geometric properties of normally hyperbolic manifolds(NHIMs) for conformally symplectic maps.
We show that conditions among rates and the conformal factor are equivalent to the NHIM being symplectic.
We also show that the hyperbolicity rates satisfy pairing rules similar to those for Lyapunov exponents of periodic orbits.
Second, we show that the scattering map -- which relates the past asymptotic trajectory of any orbit in the homoclinic manifold to the future asymptotic trajectory -- is symplectic. Joint work with R. de la Llave and T. M-Seara.
Ergodic Theory and Dynamical Systems Seminar 12:30-1:30 PM, GC 6417
Marian Gidea , Yeshiva University, New York, NY
Title: Arnold diffusion in Hamiltonian systems with small dissipation
Abstract: We consider a mechanical system consisting of a rotator and a pendulum, subject to a small, conformally symplectic perturbation. The resulting system has energy dissipation. We provide explicit conditions on the dissipation parameter, so that the resulting system exhibits Arnold diffusion. More precisely, we show that there are diffusing orbits along which the energy of the rotator grows by an amount independent of the smallness parameter. The fact that Arnold diffusion may play a role in systems with small dissipation was conjectured by Chirikov. Our system can be viewed as a simplified model for an energy harvesting device, in which context the energy growth translates into generation of electricity. Joint work with S.W. Akingbade and T-M. Seara.