Dynamical Systems Seminar

from Monday
January 01, 2018 to Thursday
May 31, 2018
Show events for:
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
TBA

TBA


Friday
March 09, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Vadim Kaimanovich, University of Ottawa
TBA

TBA


Friday
March 23, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Juan Rivera-Letelier, University of Rochester
TBA

TBA


Friday
April 13, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Anand P. Singh, Central University of Rajasthan
TBA

TBA


Friday
April 20, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Joshua Bowman, Seaver College
TBA

TBA


Friday
April 27, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Romain Dujardin, Université Pierre et Marie Curie
TBA

TBA


Friday
May 04, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Leonid Bunimovich, Georgia Tech
TBA

TBA


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars