Mini Course / Dynamics Learning Seminar

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

Wednesday
February 07, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dyi-Shing Ou, Stony Brook University
Nonexistence of wandering domains for infinitely renormalizable : I Hénon maps

The plan of the lectures is to prove the theorem: A strongly dissipative infinitely renormalizable Hénon-like map with stationary combinatorics does not have a wandering domain.

I will focus on the case of the period-doubling combinatorics. After the proof, I will say a few words about extending the proof to other stationary combinatorics.

The plan of the first talk is to cover the topics:
1. unimodal renormalization,
2. Hénon renormalization,
3. dynamics of an infinite period-doubling renormalizable Hénon map.

We will begin to prove the theorem in the following talk.


Wednesday
February 14, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dyi-Shing Ou, Stony Brook University
Nonexistence of wandering domains for infinitely renormalizable Henon maps: part II

In this talk, we will prove the nonexistence of wandering domains for a strongly dissipative infinitely (period-doubling) renormalizable Henon-like map. I will classify the domain into two regions: the good region and the bad region. In the good region, the classical results from unimodal maps can be applied to Henon-like maps. In particular, if a wandering domain exists, the horizontal size of the elements in a rescaled orbit of the wandering domain (called the closest approach) expands at a definite rate. However, in the bad region, the Henon-like map behaves differently from a unimodal map and the property break down. I will show that the bad behavior can occur at most finitely many times in the rescaled orbit to conclude the theorem.

After proving the theorem, I will give some remarks on my recent work of extending the proof to other stationary combinatorics.


Wednesday
February 21, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Babak Modami, Stony Brook University
Minimal nonuniquely ergodic foliations on surfaces: I

Measured foliations (laminations) on surfaces are well-known examples of dynamical systems in low dimension. The first return maps of measured foliations are interval exchange transformations which have been studied extensively. Measured foliations also determine the trajectories of Teichmüller and Weil-Petersson geodesics in the Teichmüller space.

In this mini-course, I outline my joint work with Brock, Leininger and Rafi about construction of minimal nonuniquely ergodic laminations.
This work was inspired by a construction of Gabai and the earlier work of Lenzhen-Leininger-Rafi where laminations are realized as the limits of sequences of curves on surfaces. An advantage of our method is explicit estimates for intersection numbers of the curves in sequences and the associated subsurface coefficients. These estimates are crucial to control the behavior of geodesics and determine their limit sets in the Thurston compactification of Teichmüller space (which won't be discussed in the minicourse).


Wednesday
February 28, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Babak Modami, Stony Brook
Minimal nonuniquely ergodic foliations on surfaces: II

Measured foliations (laminations) on surfaces are well-known examples of dynamical systems in low dimension. The first return maps of measured foliations are interval exchange transformations which have been studied extensively. Measured foliations also determine the trajectories of Teichmüller and Weil-Petersson geodesics in the Teichmüller space. In this mini-course, I outline my joint work with Brock, Leininger and Rafi about construction of minimal nonuniquely ergodic laminations. This work was inspired by a construction of Gabai and the earlier work of Lenzhen-Leininger-Rafi where laminations are realized as the limits of sequences of curves on surfaces. An advantage of our method is explicit estimates for intersection numbers of the curves in sequences and the associated subsurface coefficients. These estimates are crucial to control the behavior of geodesics and determine their limit sets in the Thurston compactification of Teichmüller space (which won't be discussed in the minicourse).


Wednesday
March 07, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Michael Benedicks, KTH Royal Institute of Technology
Attractors and sinks for Hénon maps

Since M. Hénon's fundamental computer experiment in 1976, it was a natural question whether the Hénon family of quadratic maps of the plane has parameters for which the corresponding maps has strange attractors (at least attractors that are not sinks). Hénon found numerically maps with sinks but also maps which seemed so have strange attractors. In the beginning of the 1990s Carleson and the speaker proved the existence of a positive Lebesgue measure of parameters with a strange attractor. We will review the construction in this proof and also indicate how one can find sinks, close to these parameters, even multiple coexisting sinks. The later is work in progress with Liviana Palmisano.


Wednesday
March 14, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Rostislav Grigorchuk, Texas A&M University
Some rational multidimensional maps coming from self-similar groups

I will describe a non-standard method of getting multidimensional rational maps using self-similar groups. These maps are very different from the maps in general position and possess interesting properties. Then I will explain why study of dynamical properties of these maps is useful for solving spectral problems related to finite and infinite graphs and to groups. Also the KNS (Kesten-Von Neumann-Serre) spectral measures will be defined and discussed.


Wednesday
April 11, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dror Varolin, Stony Brook University
Solution of the Cauchy-Riemann Equations with $L^2$ estimates

In this three-lecture mini-school we will explain the technique introduced by Hormander to obtain solutions, with $L^2$ estimates, for the inhomogeneous Cauchy-Riemann equations. We will then demonstrate several applications of the theorem and of the technique of its proof, with a focus on the construction of subharmonic functions with certain properties. The background needed is relatively elementary, consisting only basic real and complex analysis, and a little bit of the formalism of differential forms on manifolds (though the latter is not absolutely necessary).


Wednesday
April 25, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dror Varolin, Stony Brook University
Solution of the Cauchy-Riemann Equations with $L^2$ estimates

In this three-lecture mini-school we will explain the technique introduced by Hormander to obtain solutions, with $L^2$ estimates, for the inhomogeneous Cauchy-Riemann equations. We will then demonstrate several applications of the theorem and of the technique of its proof, with a focus on the construction of subharmonic functions with certain properties. The background needed is relatively elementary, consisting only basic real and complex analysis, and a little bit of the formalism of differential forms on manifolds (though the latter is not absolutely necessary).


Wednesday
May 02, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dror Varolin, Stony Brook University
Solution of the Cauchy-Riemann Equations with $L^2$ estimates

In this three-lecture mini-school we will explain the technique introduced by Hormander to obtain solutions, with $L^2$ estimates, for the inhomogeneous Cauchy-Riemann equations. We will then demonstrate several applications of the theorem and of the technique of its proof, with a focus on the construction of subharmonic functions with certain properties. The background needed is relatively elementary, consisting only basic real and complex analysis, and a little bit of the formalism of differential forms on manifolds (though the latter is not absolutely necessary).


Wednesday
May 09, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Hongming Nie, Indiana University
Bounded hyperbolic components of Newton maps

The moduli space $mn_4$ of quartic Newton maps has complex dimension $2$. We study the hyperbolic components in $nm_4$ and prove that those possessing two distinct non-fixed attracting cycles are bounded. Under a natural assumption, we can deal with other types of hyperbolic components with an analogical argument. This is a joint work in progress with Kevin Pilgrim.


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