Dynamical Systems Seminar

from Monday
January 01, 2018 to Thursday
May 31, 2018
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Instructions for subscribing to Stony Brook Math Department Calendars

Friday
February 02, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Scott Sutherland, Stony Brook University
On the Lebesgue Measure of the Feigenbaum Julia set

In joint work with Artem Dudko (IMPAN), we show that the Julia set of the quadratic Feigenbaum map has Hausdorff dimension less than two and consequently zero Lebesgue measure, answering a long-standing open question. This is established by a combination of new estimation techniques and a rigorous computer-assisted computation.


Friday
February 09, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dong Chen, Ohio State University
KAM-nondegenerate nearly integrable systems with positive metric entropy on arbitrarily small invariant sets

In 1950s Kolmogorov asked the following question, which is closely related to the celebrated KAM theory: Can a non-degenerate nearly integrable Hamiltonian system have a positive Kolmogorov-Sinai entropy (a.k.a. metric entropy)? In this talk we give a positive answer to this question.

In fact, examples with positive metric entropy can be constructed by a C^∞ small Lagrangian perturbation of the geodesic flow on any flat Finsler torus. Moreover positive metric entropy is generated in an arbitrarily small tubular neighborhood of any single trajectory. Similar construction applies to general completely integrable Hamiltonian systems as well. This is a joint work with D. Burago and S. Ivanov.


Friday
February 16, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Kei Irie, Kyoto University
$C^∞$-closing lemma for three-dimensional Reeb flows via embedded contact homology

I will explain a proof of $C^∞$-closing lemma for three-dimensional Reeb flows and Hamiltonian surface maps, using embedded contact homology (ECH). In particular, the key ingredient of the proof is an asymptotic formula for spectral invariants in ECH, which was proved by Cristofaro-Gardiner, Hutchings, and Ramos. If time permits, I will discuss a conjecture which gives a quantitative refinement of this result.


Friday
February 23, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Zhiqiang Li, Stony Brook University
Prime orbit theorems for expanding Thurston maps

Analogues of the Riemann zeta function were first introduced into geometry by A. Selberg and into dynamics by M. Artin, B. Mazur, and S. Smale. Analytic studies of such dynamical zeta functions yield quantitative information on the distribution of closed geodesics and periodic orbits.

We obtain the first Prime Orbit Theorem, as an analogue of the Prime Number Theorem, in complex dynamics outside of hyperbolic maps, for a class of branched covering maps on the $2$-sphere called expanding Thurston maps $f$. More precisely, we show that the number of primitive periodic orbits of $f$, ordered by a weight on each point induced by a non-constant real-valued Hölder continuous function on $S^2$ satisfying some additional regularity conditions, is asymptotically the same as the well-known logarithmic integral, with an exponentially small error term. Such a result follows from our quantitative study of the holomorphic extension properties of the associated dynamical zeta functions and dynamical Dirichlet series.

In particular, the above result applies to postcritically-finite rational maps whose Julia set is the whole Riemann sphere. Moreover, we prove that the regularity conditions needed here are generic; and for a Lattès map $f$, a continuously differentiable function satisfies such a condition if and only if it is not cohomologous to a constant. This is a joint work with T. Zheng.


Friday
March 02, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Serge Troubetzkoy, Institut de Mathématiques de Marseille
On interval maps preserving the Lebesgue measure

This is joint work with Jozef Bobok. We consider the space $C([0,1]),λ)$ of continuous maps of the interval which preserve the Lebesgue measure $λ$ endowed with the uniform metric. I will present Baire typical dynamical and differentiability properties of such maps.


Friday
March 09, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Vadim Kaimanovich, University of Ottawa
Stochastic homogenization and dynamics

The key point of Shannon's information theory consists in passing from finite strings of symbols to infinite ones with a subsequent study of shift-invariant measures on infinite words. One can extend this idea from strings of symbols (i.e., linear graphs) to general finite graphs. In this case the role of the space of infinite words is played by the space of locally finite infinite rooted graphs. This space is endowed with a natural root moving equivalence relation, so that one can talk about the measures invariant with respect to this relation. Random graphs sampled from an invariant measure are called stochastically homogeneous. Similar notions of unimodular random graphs and invariant random subgroup are currently quite popular in probability and group theory. In this talk (partially based on joint work with Paul-Henry Leemann and Tatiana Nagnibeda) I will discuss a new example of stochastic homogenization arising from the homoclinic equivalence relations of symbolic dynamical systems.


Friday
March 23, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Juan Rivera-Letelier, University of Rochester
Dynamics on the moduli space of $p$-adic elliptic curves

We describe the dynamics of Hecke correspondences and the asymptotic distribution of CM points in the moduli space of $p$-adic elliptic curves. The main ingredient is a $p$-adic version of Linnik's theorem on the equidistribution of integer points in spheres.


Friday
April 06, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Jonguk Yang, Stony Brook University
Renormalization of Semi-Siegel Hénon Maps

Renormalization is the technique of analyzing a dynamical system by understanding its small-scale behavior. This approach has proved to be very powerful, and has produced many deep results that would have been inaccessible through more conventional methods. In my talk, I will motivate the use of renormalization in the study of holomorphic dynamical systems with a rotation domain (called a Siegel disk). The focus will be in the two-dimensional setting, where many of the essential tools of one-dimensional holomorphic dynamics are not available. As the main result, I will prove that the boundary of a golden-mean Siegel disks in two-dimensions is typically not a quasi-circle.


Friday
April 13, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Anand P. Singh, Central University of Rajasthan
Escaping sets of composition of transcendental entire functions

If $f$ and $g$ are transcendental entire functions then so are $f ○ g$ and $ g ○ f $. Thus one would be interested in looking for the relation between the dynamics of $f ○ g$ and $ g ○ f $ and also with respect to its factors $f $ and $g$. Here we shall discuss these relations with respect to escaping sets, fast escaping sets, and also some relations involving permutable transcendental entire functions which throw some light on the conjecture of equality of Julia sets of permutable transcendental entire functions.


Friday
April 20, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Joshua Bowman, Seaver College
Algebra and dynamics of Chebyshev-like maps

Chebyshev-like maps are polynomial endomorphisms of $\mathbb{C}^n$ that are semiconjugate to power maps. They were introduced in the 1980s by Veselov (in terms of Lie algebras) and Hoffman--Withers (in terms of folding dynamics). We establish a general framework for studying these maps that allows us to characterize families of Chebyshev-like maps, compute their coordinate components, and determine dynamical properties (such as Julia sets and Fatou components). We also introduce some infinite-dimensional analogues of these maps.


Friday
April 27, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Romain Dujardin, Université Pierre et Marie Curie
A closing lemma for polynomial automorphisms of $\mathbb{C}^2$

A basic and still open question in the dynamics of polynomial automorphisms of $\mathbb{C}^2$ is whether periodic saddle points are dense in the Julia set. In this talk I will explain the following
“ergodic closing lemma”: in the dissipative setting, the support of any invariant measure is (apart from a few obvious cases) contained in the closure of the set of saddle points.


Friday
May 04, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Leonid Bunimovich, Georgia Tech
Finite Time Dynamics

Traditionally dynamical systems theory deals with asymptotic in time properties like ergodic theorems, mixing, etc, unless solutions are known and thus could be computed for any moment of time. Analogously probability theory deals with limit, i.e. again asymptotic in time, theorems. Moreover, all basic notions we use like Lyapunov exponents, entropies, various types of mixing involve taking a limit when time tends to infinity. I will demonstrate that some interesting finite time properties of "the most chaotic" dynamical systems can be rigorously studied. This direction of research appeared "by chance" in attempt to answer a traditional kind of question on dynamics which involves infinite time limit.


Friday
May 25, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Alena Erchenko, Penn State University
Flexibility, negative curvature, and conformal classes

Consider a closed orientable surface of negative Euler characteristic. In joint work with A. Katok, we showed the flexibility of metric and topological entropies of geodesic flow in the class of negatively curved metrics of fixed total area. In this talk, we will discuss the flexibility of entropies under the additional restriction that the metrics we consider are conformally equivalent to a fixed hyperbolic metric. It turns out that some restrictions arise. Also, we will point out connections with flexibility of some geometrical data (joint with T. Barthelmé).


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