# Complex Analysis and Dynamics Seminar

## Spring 2022 Schedule

#### Feb 25: Fred Gardiner (Brooklyn College and Graduate Center of CUNY) Smale's Mean Value Conjecture

Smale's mean value conjecture says for every polynomial $p$ of degree $n \geq 2$ with $n-1$ critical points $c_j$ and with $p(0)=0,$ $$\min_{1 \leq j \leq n-1} \left|\frac{p(c_j)}{c_j} \cdot \frac{1}{p'(0)}\right| \leq \frac{n-1}{n}.$$ To prove this one follows a strategy proposed by Dubinin. To prove the statement for the full class ${\mathcal P}$ one should first prove a stronger form of it for a subclass that has a strong form of symmetry. He calls this part symmetrization. To complete the proof one then proves a weaker result that applies to the larger class. He calls this second step disymmetrization. The first part of the first step is achieved by finding the appropriate subclass $\widehat{\mathcal P}$ of ${\mathcal P}.$ We find this class by using a form of symmetry shared by Lebesgue measure. For this reason, we call it the Lebesgue class and it is defined by declaring that a certain projection of general measures projects to measures in the smaller class: those with measure densities that coincide with the measure density of Lebesgue measure.

Theorem (Symmetrization). For any polynomial $\widehat{p}$ in the symmetric subclass $\widehat{\mathcal P}$ the geometric mean value (GMV) of the Smale quotients satisfies $$|GMV(S_j(\widehat{p})| = \left|\prod_{j=1}^{n-1} S_j(\widehat{p})\right|^{1/(n-1)} = \frac{(n-1)}{n}.$$

The disymmetrization part is accomplished by using a double pole quadratic differential $q$ realized from forming the maximal reduced modulus punctured, simply connected domain ${\mathcal D}$ centered at $z=0$ contained in the extended complex plane $\overline{\mathbb C}= {\mathbb C}\cup\{\infty\}$ that excludes the $n-1$ critical points of $p.$ It is maximal in the sense that among all other such domains that exclude the critical points it has the maximal reduced modulus centered at $z=0.$ By Strebel's double pole theorem, this extremal domain realizes a holomorphic quadratic differential $q$ with double pole at $z=0.$

The same is true for the double pole holomorphic quadratic differential $\widehat{q}$ corresponding to the polynomial $\widehat{p}.$ In this way one obtains two bounded flows of quasiconformal self mappings of the extended complex plane corresponding to the Teichmuller discs $[t|q|/q\ ]$ and $[t|\widehat{q}|/\widehat{q}\ ].$ One obtains two holomorphically moving families of polynomials ${p}_t$ and $\widehat{p}_t$ with the following properties:

1. the geometric mean value $GMV(S_j(p_t))$ is a holomorphic function of $t$ for $|t|\leq 1$ and $|\operatorname{Im} \log GMV(S_j(p_t))|$ is subharmonic,
2. the geometric mean value $GMV(S_j(\widehat{p}_t))$ is holomorphic for $|t|\leq 1$ and $\left|\operatorname{Im} \log GMV(S_j(\widehat{p}_t))\right|$ is harmonic,
3. on the boundary $|t|=1$ of the unit circle the polynomials $p_t$ and $\widehat{p}_t$ are identical and of Lebesgue type.

Items (1), (2) and (3) lead to $\int_{|t|=1} | \operatorname{Im} \log GMV(S_j(p_t))| d\eta \leq \int_{|t|=1}| \operatorname{Im} \log GMV(S_j(\widehat{p}_t))| d\eta.$ The boundedness of the flows along the circle $|t|=1$ is a consequence of Minsky's intersection inequality, which is a two dimensional version of Heisenberg's uncertainty principle. That the two flows coincide along the circle $|t|=1$ follows from the Koebe distortion argument for univalent functions in the unit disc.

#### March 4: Tengren Zheng (National University of Singapore) Entropy Rigidity for Cusped Hitchin Representations

Let $G$ be a geometrically finite subgroup of $PSL(2,\mathbb{R})$. We say that a representation $r$ from $G$ to $PGL(d,\mathbb{R})$ is a Hitchin representation if there is an $r$-equivariant positive map from the real projective line to the space of complete flags in $\mathbb{R}^d$. We define the notion of an Anosov representation from $G$ to $PGL(d,\mathbb{R})$, and show that Hitchin representations from $G$ to $PGL(d,\mathbb{R})$ are Borel-Anosov. We then prove a rigidity result for the entropy of Hitchin representations, generalizing previous work of Potrie-Sambarino. This is joint work with Richard Canary and Andrew Zimmer.

#### March 11: Richard Schwartz (Brown University) The Farthest Point Map on the Regular Dodecahedron

On any compact metric space $X$, you can look at the map which sends the point $p$ to the set of points $F(p)$ which are metrically farthest from $p$. In good cases $F(p)$ is generically a single point, and then you can iterate the map $F$ and study its dynamics. I'll explain what happens when $X$ is a platonic solid equipped with its intrinsic path metric. The prettiest case is that of the regular dodecahedron. I'll focus on this case and show computer demos which illustrate the structure and the results.

#### March 18: Tamara Kucherenko (City College of New York) Katok's Flexibility Paradigm and its Realization in Symbolic Dynamics

Katok launched the flexibility program which has been described in a nutshell as follows: "there should be no restrictions on the dynamical characteristics apart from a few obvious ones." This is a novel direction in dynamics, yet the core problems are clear and accessible to a rather broad community of mathematicians and this has made the program develop at a rapid pace. I will outline the flexibility program and showcase related results obtained within the last few years. Then I will present a striking application of the flexibility paradigm to the pressure function on compact symbolic systems. This is based on joint work with Anthony Quas.

#### March 25: Howard Masur (University of Chicago) Counting Pairs of Saddle Connections on a Translation Surface

A translation surface can be thought of in several ways. A classical way is as an Abelian differential on a closed Riemann surface. A second way is as a polygon in the plane with pairs of sides identified by parallel translation. Associated to a translation surface are saddle connections which are straight lines joining the vertices of the polygon. A saddle connection determines a vector in the plane. The problem of the asymptotics of the number of saddle connections less than a given length was initiated by W. Veech. I will recount some of the known results in this subject. I will then discuss the problem of counting pairs of saddle connections. The motivation is in part a result of J. Smillie and B. Weiss who showed that for a Veech or lattice surface there are no small area triangles, so that any pair of saddle connections with small cross product are in fact parallel. In this talk I will discuss for a generic surface the asymptotics of the number of pairs of saddle connections which have a bound on their cross product. This is joint work with Jayadev Athreya and Samantha Fairchild.

#### April 8: Bryce Gollobit (Graduate Center of CUNY) Chain Recurrence for Linear Systems

We discuss the chain recurrent set of a linear homeomorphism of a Banach space, and present an analog of Conley's theorem. We will give concrete examples for finite dimensional systems, basis shifts, and operators induced from dynamics on compact manifolds.

#### April 29: Dragomir Saric (Queens College and Graduate Center of CUNY) Quadratic Differentials and Foliations on Infinite Riemann Surfaces

For a compact Riemann surface $S$ of genus at least two, Hubbard and Masur proved that the space of holomorphic quadratic differentials $Q(S)$ on $S$ is in a one to one correspondence with the space of measured foliations on $S$ by associating with each holomorphic quadratic differential its corresponding vertical foliation. Thurston proved that the space of measured foliation on $S$ is in a one to one correspondence with the space of measured geodesic laminations $ML(S)$. Therefore, there is a bijection between $Q(S)$ and $ML(S)$ obtained by straightening the horizontal leaves of the quadratic differentials into hyperbolic geodesics on $S$ and pushing forward the transverse measure.

Let $X$ be an infinite Riemann surface whose covering group is of the first kind. We give an analogue of the Hubbard and Masur theorem for integrable holomorphic quadratic differentials on $X$. Namely, we give a complete characterization of the class of measured geodesic laminations on $X$ that correspond to integrable holomorphic quadratic differentials thus establishing a bijection between the two spaces. Some applications of this result will be considered.

#### May 6: Alena Erchenko (Stony Brook University) Flexibility and Rigidity Questions in Dynamics

The interplay between rigidity and flexibility is a general concept in various fields of mathematics. We will demonstrate the principles involved and some of the approaches to such questions in dynamical systems. We will largely proceed by analyzing one class of examples: smooth Anosov volume preserving diffeomorphisms on a torus.

#### May 13: Federico Rodriguez Hertz (Penn State) Cohomology and Dynamics

In this talk I plan to discuss some problems and ongoing projects related to cohomology and dynamics. Most of these problems are for hyperbolic dynamics, but some are in a more general setting.