Dynamical Systems Seminar

from Thursday
June 01, 2017 to Sunday
December 31, 2017
Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars

Friday
September 08, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Tetsuo Ueda, Kyoto University
Some problems on the local structure of holomorphic maps

We will discuss the relations between local holomorphic dynamics and structure of complex manifolds. Examples will be taken from Schroeder equation for of attracting fixed point, Ecalle-Voronin theory for parabolic fixed point in one variable and saddle or semi-attracting fixed point in two variables.


Friday
September 15, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Tali Pinsky, TIFR
On the Lorenz flow and the modular surface

The talk addresses three famous, albeit fundamentally different, three dimensional flows: The geodesic flow on the modular surface, the chaotic Lorenz equations and the geometric Lorenz model. I will describe a new approach, potentially showing that these flows are orbit equivalent for the correct choice of parameters.
This will be an introductory talk.


Friday
September 22, 2017

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

Suppose that within the space of certain one-dimensional dynamical systems, a suitably defined renormalization operator has a hyperbolic fixed point. By incorporating a non-linear change of coordinates to the rescaling map, I will show that this renormalization operator can be extended to nearby two-dimensional dynamical systems without losing its hyperbolicity.


Friday
September 29, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Nataliya Goncharuk, Cornell University
Complex rotation numbers

Given an analytic circle diffeomorphism f ∶ ℝ/ℤ → ℝ/ℤ and a complex number w, ℑw > 0, consider the quotient space of the annulus 0 < ℑz < ℑw, z ∈ ℂ/ℤ, by the action of f + w. This quotient space is a torus, and we can ask about its modulus. This modulus is called the *complex rotation number* of f + w.

The limit of the complex rotation number as w tends to w₀ ∈ ℝ/ℤ coincides with the rotation number of f + w₀, if this rotation number is irrational. In general, limit values of the complex rotation number on ℝ/ℤ form a bubbly picture in the upper half-plane: infinitely many bubbles (analytic curves) grow from rational points of the real axis. Bubbles are complex analogue to Arnold tongues.

In the talk, I'll give a survey of results on shapes of bubbles, with some proofs, and list open questions.

Mostly based on the joint work with X. Buff.


Friday
October 06, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Dmitry Jakobson, McGill University
Distances between closed geodesics

We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrary small gaps is topologically generic: this is established both for surfaces of constant negative curvature, and for the space of negatively curved metrics. While arbitrary small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric; we discuss a result in this direction. Joint work with Dima Dolgopyat.


Friday
October 13, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Peter Makienko, Instituto de Matematicas UNAM
On the summability and instability for rational maps

In this talk we discuss an approach to the Fatou conjecture
based on different types of summability of the spectrum of critical values.


Friday
October 20, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Yury Kudryashov, Cornell University
Bifurcations of planar vector fields

In 1985 V. Arnold stated six problems aimed to outline the future development of the global bifurcation theory in the plane. In particular, he conjectured that a generic finite-parametric family of vector fields on the two-sphere is structurally stable. Recently Yulij
Ilyashenko, Ilya Schurov and I disproved this conjecture [1]. It turns out that 3-parametric families of vector fields may have numerical invariants of (moderate) topological classification, and 6-parametric families may have functional invariants.

I will discuss the main construction of [1], then I'll give a survey of the results obtained by Yu. Ilyashenko and his students N. Goncharuk, me, I. Schurov, N. Solodovnikov, V. Starichkova.

I will also discuss some ongoing research in this field that may lead to many independent numerical invariants of the 3-parametric family of vector fields constructed in [1].

[1]: http://arxiv.org/abs/1506.06797


Friday
October 27, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Babak Modami, Stony Brook University
Renormalization in Teichmuller dynamics

After a brief review of Masur's criterion which is an example of the renormalization technique in Teichmuller dynamics, we show that an analogue of the criterion does not hold for the Weil-Petersson geodesic flow. The talk is based on a joint work with Jeff Brock from 2015.


Friday
November 03, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Pierre Berger, Université Paris 13
Zoology in the Hénon family: twin babies and Milnor's swallows

We study $C^{d,r}$-Hénon-like families $(f_{a, b})_{a, b}$ with two parameters $(a,b)∈ \mathbb{R}^2$. We show the existence of an open set of parameters $(a,b)∈ D$, so that a renormalization chart conjugates an iterate of $f_{a, b}$ to a perturbation of $(x,y)→ ((x^2+c_1)^2+c_2,0)$. We prove that the map $(a,b)∈ D→ (c_1,c_2)$ is a $C^d$-diffeomorphism ; as first numerically conjectured by Milnor in 1992.

Furthermore, we show the existence of an open set of parameters $(a,b)$ so that $f_{a, b}$ displays exactly two different renormalized Hénon-like maps which attract Lebesgue a.e. point with bounded forward orbit. A great freedom in the choice of the renormalized parameters enables us to deduce in particular the existence of a Hénon map with exactly $2$ sinks (an answer to a Question by Lyubich).

The proof is based on a generalization of puzzle pieces for Hénon-like maps, and on a generalization of both the affine-like formalism of Palis-Yoccoz and the cross map of Shilnikov. The distortion bounds enable $C^{d,r}$-renormalizations without loss of regularity.


Friday
November 10, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Michael Yampolsky, University of Toronto
New results on renormalization and rigidity of analytic critical circle maps

I will review the renormalization theory of maps of the circle with critical points, and will present some recent results and applications, and discuss the problems which remain open.


Friday
November 17, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Enrique Pujals, IMPA
Dissipative diffeomorphisms of the disk with zero entropy: structure of periodic points and infinite renormalization

We will discuss a class of volume-contracting surface diffeomorphisms (named strong dissipative diffeomorphisms) whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics.

For the particular case of the disk, we will consider the ones that have zero entropy, explaining their structure of periodic orbits and showing that they are infinitely renormalizable if they are in the boundary of zero entropy.

The talk is a joint works with Sylvain Crovisier and Charles Tresser.


Friday
December 01, 2017

2:30 PM - 3:30 PM
Math Tower P-131
Sabya Mukherjee, Stony Brook University
TBA

TBA


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