Complex Analysis and Dynamics Seminar

Fall 2023 Schedule

September 8: Karl Winsor (Stony Brook University)
Toward Ratner's orbit closure theorem for absolute period foliations of strata

The absolute period foliation is a holomorphic foliation of a stratum of abelian differentials, whose leaves are navigated by moving the zeros of a differential relative to each other. In genus 2, the dynamics of this foliation are well-understood due to a close connection with homogeneous dynamics. However, for most strata in higher genus, the situation is more mysterious. In this talk, we will present a method for completely classifying the closures of leaves of the absolute period foliation of most strata, in the spirit of Ratner's orbit closure theorem.

September 22: Roland Roeder (IUPUI)
Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6

We study the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces $$S_{A,B,C,D} = {(x,y,z) \in C^3 : x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D},$$ where A,B,C, and D are complex parameters. It arises naturally in the dynamics on character varieties and it also describes the monodromy of the famous Painlevé 6 differential equation. We explore the Fatou/Julia dichotomy for this group action (defined analogously to the usual definition for iteration of a single rational map) and also the Locally Discrete / Non-Discrete dichotomy (a non-linear version from the classical discrete/non-discrete dichotomy for Lie groups). The interplay between these two dichotomies allow us to prove several results about the topological dynamics of this group. Moreover, we show the coexistence of non-empty Fatou sets and Julia sets with non-trivial interior for a large set of parameters. Several open questions related to our work will be described. This is joint work with Julio Rebelo.

September 29: Hongming Nie (Institute for Mathematical Sciences -- Stony Brook University)
(Un)bounded hyperbolic components of Newton maps

Hyperbolic maps form an open set in the space of rational maps. This descends to an open set in the moduli space of rational maps. Each component of this open set is a hyperbolic components. It is interesting to study the following two questions:

(1) Which hyperbolic components are bounded?

(2) What is the boundary of an unbounded hyperbolic components at infinity?

In this talk, I will explore these questions and present a complete classification of bounded hyperbolic components in the slice of quartic Newton maps. The proof is based on a convergence result of perturbations of graphs for Newton maps. This a joint work with Yan Gao.

October 6: Alejandro Passeggi (Universidad de La República Oriental del Uruguay)
Weak topological conditions implying annular Chaos

Annular dynamics has birth together with the theory of chaos: in the Hamiltonian setting by means of the work of Poincaré on the three body problem and in the dissipative setting by the study of the Van der Pol equation arising in problems involving electric circuits. Since then, several important examples coming from different fields have been reduced to discrete annular dynamics, and the usual question is about whether these systems are integrable or chaotic.

Although the mathematical understanding of topological chaos is nowadays quite strong being supported in some renowned models, an important problem still finds a weak answer: Given a particular system, decide whether it is chaotic or not. Those systems arising in applications which find a mathematical proof for the existence of chaos are incredibly few, and usually one needs to restrict the parameters so that the system is close to an "integrable" case. In contrast, there is an uncountable list of numerical simulations of systems which are taken as evidence of chaos.

In this talk we will comment about the evolution of this problem in view of topological theory of surface dynamics and introduce a result which intends to bring this theory to applicable versions in order to determine the existence of chaos for prescribed systems. In particular, the result creates a bridge that can turn numerical simulations of dynamical systems into rigorous tests for this aim.

(Joint work with Fabio Tal, USP)

October 13: Yongquan Zhang (Institute for Mathematical Sciences -- Stony Brook University)
Quasiconformal nonequivalence of gasket Julia sets and limit sets

The Julia set of a rational map and the limit set of a Kleinian group share many common features. In many cases, the two fractal sets can be homeomorphic. However, we do not know any "non-trivial" example where the homeomorphism is quasiconformal. In this talk, I will discuss a quasiconformal nonequivalence result for a class of Julia sets and limit sets called "gaskets". The proof is based on an analysis of the combinatorial arrangement of the connected components of Fatou sets / domain of discontinuity. In particular, our result implies that a Julia set can never be quasiconformally homeomorphic to the Apollonian gasket.

The talk is based on a joint work with Yusheng Luo.

October 20: Mark Demers (Fairfield University)
Thermodynamic Formalism for Sinai Billiards

For a finite horizon Sinai billiard, we study the family of geometric potentials $– t \log (J^uT)$, $t>0$, where $J^uT$ is the Jacobian of T in the unstable direction. For a dispersing billiard, this potential is not Hölder continuous on any open set. We find $t_* >1$ such that for each $t \in (0, t_*)$, there exists a unique equilibrium state $\mu_t$ for the potential. We show that $\mu_t$ is exponentially mixing for Hölder observables, and that the pressure function $P(t)$ is analytic on $(0,t_*)$ and satisfies a variational principle. In the limit $t=0$, we lose the spectral gap for the associated transfer operator, but are still able to prove the existence and uniqueness of a measure of maximal entropy. This is joint work with Viviane Baladi.

October 27: Sven Sandfeldt (KTH Royal Institute of Technology)
Centralizer rigidity of partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds

The smooth centralizer of a diffeomorphism $f$ is the set of smooth maps $g$ that commute with $f$. A conjecture due to Smale asserts that a generic $f$ has a small centralizer, consisting entirely of iterates of $f$. In this talk I will discuss the centralizers of a class of partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds. In particular, I will classify all possible groups that can arise as the centralizer of a diffeomorphism within this class. Moreover, I will discuss a rigidity result for diffeomorphisms with large centralizer, showing that, within this class of partially hyperbolic diffeomorphisms, the complement of the generic set from Smales conjecture consists of diffeomorphisms that are smoothly conjugated to some algebraic model system.

November 3: Tao Chen (Laguardia Community College and the Graduate Center)
A family of meromorphic functions: ergodic or non-ergodic

McMullen's dichotomy states that a rational map is ergodic (the Julia set is the whole sphere, and the map is ergodic) or attracting (almost every point on the sphere is attracted to the post-critical set). However, it may be challenging to determine which case applied to a given map, the Julia set of which is the sphere.

In this talk, for a family of meromorphic maps with finite many asymptotic values and no critical points, called Nevanlinna functions in the literature, we give a criterion to determine whether it is ergodic or not.

This is a joint work with Yunping Jiang and Linda Keen.

November 10: Michael Yampolsky (University of Toronto)
Renormalization of analytic circle maps, KAM theory, and smoothness of Arnold tongues

I will describe a renormalization theory which includes critical circle maps and analytic diffeomorphisms, which we recently constructed with Natasha Goncharuk. Among applications are KAM-type results such as Risler's Theorem, and statements on smoothness of irrational Arnold tongues.

November 17: Huiping Pan (South China University of Technology)
Envelopes of the Thurston metric on Teichmuller space

Aiming for a geometric understanding of the Teichmuller space from the point of view of hyperbolic geometry, Thurston introduced an asymmetric Finsler metric on the Teichmuller space, the Thurston metric. The deficiency of this metric from being uniquely geodesic is described by the envelope, defined as the union of geodesics from the initial point to the terminal point. In this talk, I will talk about the shapes and continuity of envelopes. This is a joint work with Michael Wolf.

December 1: Lasse Rempe (University of Liverpool)
Building surfaces from equilateral triangles

In this talk, we consider the following question. Suppose that we glue a (finite or infinite) collection of closed equilateral triangles together in such a way that we obtain an orientable surface. The resulting surface is a Riemann surface; that is, it has a natural conformal structure (a way of measuring angles in tangent space). We ask which Riemann surfaces are *equilaterally triangulable*; i.e., can arise in this fashion.

The answer in the compact case is given by a famous classical theorem of Belyi, which states that a compact surface is equilaterally triangulable if and only if it is defined over a number field. These *Belyi surfaces* - and their associated "dessins d’enfants" - have found applications across many fields of mathematics, including mathematical physics.

In joint work with Chris Bishop, we give a complete answer of the same question for the case of infinitely many triangles (i.e., for non-compact Riemann surfaces). The talk should be accessible to a general mathematical audience, including postgraduate students.

December 8: Axel Kodat (The Graduate Center)
An average intersection estimate for families of diffeomorphisms

Poincaré’s formula in integral geometry states that for any submanifolds of V, W of a compact homogeneous space M = G/H with G acting transitively on tangent planes, the average volume of the intersection between g(V) and W is equal to Cvol(V)vol(W) for C a universal constant depending only on the dimensions of V and W. We discuss an adaptation of this result to general (non-homogeneous) closed manifolds, where the transformation group G is replaced by a compact family of diffeomorphisms, and the formula now holds up to uniform multiplicative error. We also sketch some possible applications of this result to the dynamics of diffeomorphisms with exponential volume growth. This is joint work with Michael Shub.

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