# Complex Analysis and Dynamics Seminar

## Department of MathematicsGraduate Center of CUNY

### Spring 2019:

#### Feb 1: Christian Wolf (City College and Graduate Center of CUNY) A Topological Classification of Locally Constant Potentials via Zero-Temperature Measures

In this talk we discuss a topological classification of locally constant functions over subshifts of finite type via their zero-temperature measures. Our approach is to analyze the relationship between the distribution of the zero-temperature measures and the boundary of higher dimensional generalized rotation sets. We also discuss the regularity of the localized entropy function on the boundary of the generalized rotation sets. The results in this talk are joint work with Yun Yang.

#### Feb 8: Mike Todd (University of St. Andrews) Stability of Measures in Interval Dynamics

Given a family of interval maps, each map possessing a canonical measure (an invariant measure absolutely continuous w.r.t. Lebesgue - an acip), we have a weak form of stability if these measures change continuously through the family.  Even for uniformly hyperbolic dynamical systems this stability can fail.  I’ll give minimal conditions for a class of non-uniformly hyperbolic interval maps to satisfy this stability property. This work forms part of a paper with Neil Dobbs, where more general thermodynamic properties are proved to be stable (entropy, pressure, equilibrium states), and I’ll give some indication of the general approach there.

#### Feb 15: Han Li (Wesleyan University) Masser’s Conjecture on Equivalence of Integral Quadratic Forms

A classical problem in the theory of quadratic forms is to decide whether two given integral quadratic forms are equivalent. Formulated in terms of matrices the problem asks, for given symmetric $n$-by-$n$ integral matrices $A$ and $B$, whether there is a unimodular integral matrix $X$ satisfying $A=X^tBX$, where $X^t$ is the transpose of $X$. For definite forms one can construct a simple decision procedure. Somewhat surprisingly, no such procedure was known for indefinite forms until the work of C. L. Siegel in the early 1970s. In the late 1990s D. W. Masser conjectured for $n \geq 3$, there exists a polynomial search bound for $X$ in terms of the heights of $A$ and $B$. In this talk we shall discuss our recent resolution of this problem based on a joint work with Professor Gregory A. Margulis, and also explain how ergodic theory is used to understand integral quadratic forms.

#### Feb 22: Fabio Tal (Universidade de São Paulo) Homeomorphisms of Surfaces with Zero Entropy

In this work we derive a new criterion to detect the existence of positive entropy (and of topological horseshoes) for surface homeomorphisms in the isotopy class of the identity. This is done using the machinery of Brouwer-Le Calvez foliations and a related dynamical forcing theory. We will describe what this new criterion is, and apply it to obtain a description of the possible behaviors of dynamical systems on surfaces with null genus with zero topological entropy. We will show that if such a system is conservative, then the dynamics is in many ways similar to that of a fully integrable system, extending a result of Franks and Handel previously known for diffeomorphisms. We will also describe transitive sets in the non-conservative case, showing that such a set must belong to one of the following possibilities:
1- It is a periodic orbit
2- It is a irrationally rotating set
3- The dynamics over this set is infinitely renormalizable, and semi-conjugate to that of an odometer map.
This is joint work with P. Le Calvez.

#### Mar 1: Rich Schwartz (Brown University) Inscribing Triangles and Rectangles in Jordan Curves

I'll describe some computer experiments I've done, and also some results I have, about inscribing triangles and rectangles in Jordan loops. One result I have is this: Call a point on a Jordan curve GOOD if it is the vertex of an inscribed rectangle. Then all but at most 4 points of any Jordan curve are good. All this effort is supposed to be in the service of trying to solve the notorious Square Peg Conjecture, which I have not done.

#### Mar 8: Enrique Pujals (Graduate Center of CUNY) Surface Dissipative Difffeomorphisms with Zero Entropy and Renormalization

We will describe the dynamics of surface diffeomorphisms with zero entropy and show that in the specific case of the disk they are renormalizable (those that are at the zero entropy boundary are infinitely renormalizable).

#### March 15: Tao Chen (Laguardia Community College of CUNY) Dynamics of General Tangent Family

This talk will be on the joint work with Linda Keen on the parameter space of the family $\lambda \tan^p z^q$, where $p$, $q$ are positive integers. Any map in this family has only one or two symmetric asymptotic values and the dynamics depends on its orbits. Therefore, the parameter space can be decomposed into shell components and capture components. Moreover, a topological and combinatoric description of these components will be given.

#### March 22: Yunping Jiang (Queens College and Graduate Center of CUNY) Shape of the Metric Entropy

The topological entropy measures the complexity of a dynamical system. However, to measure the level of the complexity of a dynamical system, one needs the metric entropy. Given a smooth dynamical system with hyperbolic attractor (e.g., an Anosov diffeomorphism or a smooth circle expanding endomorphism), the metric entropy is the measure-theoretic entropy with respect to its SRB measure. Consider the space of all smooth dynamical systems with hyperbolic attractors conjugating this given one. The metric entropy is a positive functional defined on this space. It is actually a positive functional defined on the Teichmueller space of this space (or the space of all smooth conjugacy classes). In joint work with Huyi Hu and Miaohua Jiang, we used to prove that there is a smooth path starting from any given dynamical system with hyperbolic attractor such that the metric entropy on this path can be as small as possible. Thus, the infimum of the metric entropy on the space is zero. We also proved a version for area-preserving Anosov dynamical systems. A challenging question is to see a global picture of the metric entropy on the Teichmueller space. In this talk, I will show a global picture of the metric entropy on the Teichmueller space of the space of all degree two expanding Blaschke products.

#### March 29: Saeed Zakeri (Queens College and Graduate Center of CUNY) Cyclic Permutations, Periodic Orbits and Complex Polynomials

There is a well-known connection between the dynamics of complex polynomial maps of degree $k \geq 2$ and the multiplication-by-$k$ map $m_k(x) = kx$ (mod 1) acting on the circle at infinity. Motivated by this link, we study the combinatorial types of periodic orbits of $m_k$ and the frequency with which they occur. In fact, for every $q$-cycle $\sigma$ in the permutation group $S_q$ we give a full description of the set of period $q$ orbits of $m_k$ that realize $\sigma$ and count precisely how many such orbits there are. The description is based on an invariant called the fixed point distribution'' vector and is achieved by reducing the realization problem to finding the stationary state of an associated Markov chain. This is joint work with C. L. Petersen.

#### April 5: Yun Yang (Graduate Center of CUNY) Random Perturbations of Predominantly Expanding Maps

In this talk, we will consider the random perturbations of deterministic dynamical systems and seek to understand the corresponding asymptotic behavior, i.e., the phenomena that can be observed under long-term iteration. In particular, we will study the random perturbations of a family of circle maps $f_a$. We will obtain a checkable, finite-time criterion on the parameter $a$ for small random perturbation of $f_a$ to exhibit a unique and thus ergodic stationary measure with positive Lyapunov exponents. This is joint work with Alex Blumenthal.

#### April 12: Claire Burrin (Rutgers University) Effective Counting via Eisenstein Series

Various questions concerning translation surfaces depend on counting saddle connections. For a certain class of translation surfaces, this reduces to the more general, yet more tractable problem of counting points in discrete orbits for the linear action of a lattice of $SL(2,{\mathbb R})$ on the Euclidean plane. This can be done effectively, using either methods from ergodic theory or from number theory. We will illustrate both methods and discuss certain of their aspects. This is joint work with Amos Nevo, René Rühr, and Barak Weiss.

#### May 3: Linda Keen (Lehman College and Graduate Center of CUNY) Parameter Spaces for Families of Meromorphic Functions with Two Asymptotic Values

Just as in the study of dynamics of rational maps, those of degree 2 are a natural next step after quadratic polynomials, in the study of the dynamics of meromorphic functions, those with two asymptotic values and no critical points are the natural next step after tangent functions. Today we will describe the parameter spaces of families with one attracting fixed point. We will use techniques adapted from studies of rational functions of degree two as well as techniques that depend on the transcendental properties of the functions. This is work in progress and joint with Tao Chen and Yunping Jiang.

#### May 10: Bryce Gollobit (Graduate Center of CUNY) Generalized Hyperbolic Linear Operators

An open class of linear automorphisms, containing the hyperbolic ones, is introduced for infinite dimensional Banach spaces. This class shares many classic properties of hyperbolic operators, including the shadowing property and $C^0$ structural stability. While robust transitivity cannot exist for linear maps, this class acts transitively on its non-wandering set, which is always infinite dimensional unless it is hyperbolic. This is joint work with Patricia Cirilo (Federal University of Sao Paulo) and Enrique Pujals.