Dynamical Systems Seminar

from Friday
June 01, 2018 to Monday
December 31, 2018
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Tuesday
June 05, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Yair Minsky, Yale University
Weil-Petersson geometry, Dehn filling and branched surfaces

We study pseudo-Anosov mapping classes with bounded normalized Weil-Petersson translation distance (and unbounded genus). In analogy with a result of Farb-Leininger-Margalit for Teichmuller translation distances, we show all such mapping classes fit together into a finite collection of cusped hyperbolic 3-manifolds, where the cusps are filled to become either vertical (transverse to fibers) or horizontal (parallel to fibers). After a reduction using work of Schlenker, Kojima-McShane and Brock-Bromberg, the argument uses the theory of branched surfaces in 3-manifolds. This is joint work with Chris Leininger, Juan Souto and Sam Taylor.


Tuesday
June 12, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Yusheng Luo, Harvard University
On the inhomogeneity of the Mandelbrot set

We will show that the Mandelbrot set is totally locally conformally inhomogeneous: the only orientation preserving conformal map $f:U → V$ with $U\cap ∂ M \neq \emptyset$ and satisfying $f(U\cap∂ M) ⊂ ∂ M$ is the identity map. The proof uses the study of the local conformal symmetries of the Julia sets: we will show in many cases, the dynamics can be recovered from local conformal structures of the Julia sets.


Thursday
June 21, 2018

1:30 PM - 2:30 PM
Math Tower P-131
Roland Roeder, IUPUI
Limiting Measure of Lee-Yang Zeros for the Cayley Tree

I will explain how to use detailed properties of expanding maps of the circle (Shub-Sullivan rigidity, Ledrappier-Young formula, large deviations principle, ...) to study the limiting distribution of Lee-Yang zeros for the Ising Model on the Cayley Tree. No background in mathematical physics is expected of the audience. This is joint work with Ivan Chio, Caleb He, and Anthony Ji.


Friday
August 31, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Mahan Mj, TIFR Mumbai
A survey of Cannon-Thurston Maps

We shall survey the theory of Cannon-Thurston Maps. These form a connecting link between the hyperbolic geometry and the complex dynamics of Kleinian groups. We shall also discuss a generalization to geometric group theory and end with some open problems.


Friday
September 07, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Mark Pollicott, University of Warwick
Dynamical Zeta functions

The famous Selberg zeta function can be interpreted as a complex function defined in terms of closed orbits on a compact
surface with constant negative curvature. We want to discuss generalizations of this: firstly to surfaces of variable negative curvature; and secondly to higher Teichmuller theory.


Friday
September 14, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Peter Veerman, Portland State University
Strange Convex Sets

Given a closed convex set $Ω∈\mathbb{R}^n$, the metric projection of a given point $x∈\mathbb{R}^n$ is given by the unique point $Π(x)∈Ω$ that minimizes the (Euclidean) distance $\lbrace\vert y-x\vert\ \vert\ y∈Ω\rbrace$ between $Ω$ and $x$. Most mathematicians tend to think of convex sets in $\mathbb{R}^n$ as very tame objects. It is therefore surprising that it is easy to construct a compact convex set $Ω$ in $\mathbb{R}^2$ with the following strange property [Shapiro, 1994]: There is a point $x\notinΩ$ and a vector $v$ such that the directional derivative $$\lim_{t→ 0}\frac{Π(x+vt)-Π(x)}{t}$$ fails to exist. Note that for example convex polygons are not strange in this sense.

We revisit and modify that construction to obtain a convex curve in $\mathbb{R}^2$ that is $C^{1,1}$ or differentiable with Lipschitz derivative. We show that the convex set bounded by this curve has the property that the directional derivative of the projection is not defined. This construction can be made $C^n$ for $n≥ 2$ except at a single point, and such that directional differentiability still fails.


Friday
September 21, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Nguyen-Bac Dang, Stony Brook University
Spectral gap in the dynamical degrees of tame automorphisms preserving an affine quadric threefold

In this talk, I will present the tame automorphisms group preserving an affine quadric threefold. The main focus of my talk is the understanding of the degree sequences induced by the elements of this group. Precisely, I will explain how one can apply some ideas from geometric group theory in combination with valuative techniques to show that the values of the dynamical degrees of these tame automorphisms admit a spectral gap.Finally I will apply these techniques to characterize when the Lyapounov exponents of a random walk on this particular group are strictly positive.


Friday
September 28, 2018

2:00 PM - 3:00 PM
Math Tower P-131
Dragomir Saric, Queens College
Asymptotics of moduli of curves and applications

In a joint work with H. Hakobyan, we prove that each Teichmuller geodesic in the universal Teichmuller space has a unique limit point on Thurston boundary. The main result depends on asymptotic estimates of moduli of curves. Another application of the asymptotics, in a joint work with A. Basmajian and H. Hakobyan, is to give a sufficient condition on Fenchel-Nielsen coordinates for infinite surfaces to guarantee that the surfaces have ergodic geodesic flows, i.e. of type $O_G$. In a joint work with H. Miyachi, we show that the Teichmuller disk in the universal Teichmuller space extends by continuity to a closed disk in Thurston bordification. Thurston boundary to arbitrary Teichmuller spaces are recently introduced in a joint work with F. Bonahon.


Friday
October 05, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Lasse Rempe-Gillen, University of Liverpool
Taming wild entire functions

The study of the dynamics of transcendental entire functions has a long history, going back to Fatou. It has recently garnered much interest, partly due to intriguing connections with other areas of holomorphic dynamics. While some examples and families have been well-understood since the 1980s, only recently tools have become available to advance a detailed understanding of large and very general classes of transcendental entire functions.

In this talk, I will discuss some of these developments. In particular, I will discuss analogues of the local connectivity of Julia sets, which play a crucial role in polynomial dynamics. In joint work with Sixsmith (and partly with Alhamd), we introduce a notion of "docile" entire functions, and show in particular that a large class of functions, known as "strongly geometrically finite functions", are docile. I will also discuss work in progress with Sixsmith, with the goal of bringing the "puzzle" techniques of Yoccoz to bear on transcendental dynamics.

In the course of the discussion, I intend to also touch upon work of Mihaljevic, and joint work with Albrecht and Benini (concerning the dynamics of entire functions of finite order), with Benini (concerning an analogue of the Douady-Hubbard landing theorem), and with Pfrang (on Hubbard trees).


Friday
October 12, 2018

2:30 PM - 3:30 PM
SCGP 102
Alena Erchenko, Ohio State University
Flexibility of Lyapunov exponents on the circle and the torus

There are several interesting classes of measures. For two special classes of dynamical systems, we will concentrate on the invariant measure that is absolutely continuous with respect to the Lebesgue measure and the measure of maximal entropy. First, we show that Lyapunov exponents with respect to these two probability measures for smooth expanding circle maps of a fixed degree $≥ 2$ take on all values that satisfy some well-known inequalities. Then, we demonstrate a similar result for positive Lyapunov exponents with respect to these two measures for Anosov area-preserving diffeomorphisms on a two-torus that are homotopic to a fixed area-preserving Anosov automorphism (work in progress).


Friday
October 26, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Vasiliki Evdoridou, The Open University
Singularities of inner functions associated to entire maps in the class $\mathcal{B}$

Let $f$ be a transcendental entire function and $U$ be an unbounded, invariant Fatou component of $f$. We can associate an inner function, $g$ say, to the restriction of $f$ to $U$. We consider two classes of functions in $\mathcal{B}$ having finitely many tracts. We show that if $f$ belongs to either of these two classes the number of singularities of $g$ on the unit circle is equal to the number of tracts of $f$. This is joint work with N. Fagella, X. Jarque and D. Sixsmith.


Friday
November 02, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Yotam Smilansky, The Hebrew University of Jerusalem
Multiscale substitution schemes and Kakutani sequences of partitions

Substitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. In this talk we will use new path counting results for directed weighted graphs to show that such sequences of partitions are uniformly distributed, thus extending Kakutani's original result. Furthermore, we will describe certain limiting frequencies associated with sequences of partitions, which relate to the distribution of tiles of a given type and the volume they occupy.


Friday
November 09, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dimitrios Ntalampekos, Stony Brook University
Removability of planar sets: old and new results

Removability of sets for quasiconformal maps and Sobolev functions has applications in Complex Dynamics, in Conformal Welding, and in other problems that require "gluing" of functions to obtain a new function of the same class. We, therefore, seek geometric conditions on sets that guarantee their removability. In this talk, I will give a survey of old results and discuss some very recent results on the (non)-removability of the Sierpiński gasket and of Sierpiński carpets.

A first result is that the Sierpiński gasket is removable for continuous functions of the class $W^{1,p}$ for $p>2$. The method used applies to more general fractals that resemble the Sierpiński gasket, such as the Apollonian gasket and generalized Sierpiński gasket Julia sets.

Then, I will sketch a proof that the Sierpiński gasket is non-removable for quasiconformal maps and thus for $W^{1,p}$ functions, for $1≤ p≤ 2$. The argument involves the construction of a non-Euclidean sphere, and then the use of the Bonk-Kleiner theorem to embed it quasisymmetrically to the plane.


Friday
November 16, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dimitry Turaev, Imperial College
On wandering domains near homoclinic tangtencies

Given a map, we define a wandering domain as an open region such that the diameter of its images by the iterations of the map shrinks to zero but the corresponding limit set is not a periodic point. It is known that many finitely smooth two-dimensional diffeomorphisms have wandering domains while it is not known if a polynomial diffeomorphism of a plane can have one. We discuss wandering domains whose limit sets are homoclinic tangencies. We show the existence of real analytic planar diffeomorphisms with wandering domains and discuss how to find wandering domains for polynomial diffeomorphisms of the three-dimensional space.


Friday
November 30, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Christopher J. Leininger, University of Illinois at Urbana-Champaign
Polygonal billiards, Liouville currents, and rigidity

A particle bouncing around inside a Euclidean polygon gives rise to a biinfinite "bounce sequence" (or "cutting sequence") recording the (labeled) sides encountered by the particle. In this talk, I will describe recent work with Duchin, Erlandsson, and Sadanand, in which we prove that the set of all bounce sequences---the "bounce spectrum"---essentially determines the shape of the polygon. This is consequence of our main result about Liouville currents on surfaces associated to nonpositively curved Euclidean cone metrics. In the talk I will explain the objects mentioned above, how they relate to each other, and give some idea of the proof of the main theorem.


Friday
December 07, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Zoran Sunic, Hofstra University
TBA

TBA


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