# Complex Analysis and Dynamics Seminar

## Department of MathematicsGraduate Center of CUNY

### Fall 2016:

#### Sep 16: Maciej Capinski (AGH University of Science and Technology, Krakow, Poland) Beyond the Melnikov Method: A Computer Assisted Approach

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds. In our approach, we do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are ‘small enough,’ as is the case in the classical Melnikov approach.

#### Sep 23: David Aulicino (Brooklyn College of CUNY) Weak Mixing for Translation Surfaces with Intermediate Orbit Closures

Work of Avila and Forni established weak mixing for the generic straight line flow on generic translation surfaces, and the work of Avila and Delecroix determined when weak mixing occurs for the straight line flow on a Veech surface. Following the work of Eskin, Mirzakhani, Mohammadi, which proved that the orbit closure of every translation surface has a very nice structure, one can ask how the orbit closure affects the weak mixing of the straight line flow. In this talk all of the necessary background on translation surfaces and weak mixing will be presented followed by the answer to this question. This is a joint work in progress with Artur Avila and Vincent Delecroix.

#### Oct 21: Mark Bell (University of Illinois at Urbana-Champaign) Slowly Converging Pseudo-Anosovs

A classical property of pseudo-Anosov mapping classes is that they act on the space of projective measured laminations with north-south dynamics. This means that under iteration of such a mapping class, laminations converge exponentially quickly towards its stable lamination. We will discuss a new construction (joint with Saul Schleimer) of pseudo-Anosovs where this exponential convergence has base arbitrarily close to one and so is arbitrarily slow.

#### Oct 28: Enrique Pujals (IMPA and the Graduate Center of CUNY) Two-dimensional Blaschke Products: Degree Growth and Ergodic Consequences

For dominant rational maps of compact, complex, Kahler manifolds there is a conjecture specifying the expected ergodic properties of the map depending on the relationship between the rates of growth for certain degrees under iteration of the map. In the present talk, we will discuss the case of two-dimensional Blaschke products, observing that they fit naturally within this conjecture, having examples from each of the three cases that the conjecture gives for maps of a surface.
The results to be discussed are included in different works with Mike Shub and Roland Roeder.

#### Nov 11: Jonah Gaster (Boston College) New Bounds for `Homotopical Ramsey Theory' on Surfaces

Farb and Leininger asked: How many simple closed curves on a finite-type surface $S$ may pairwise intersect at most $k$ times? Przytycki has shown that this number grows at most as a polynomial in $|\chi(S)|$ of degree $k^{2}+k+1$. We present narrowed bounds by showing that the above quantity grows slower than $|\chi(S)|^{3k}$. In particular, the size of a maximal 1-system grows sub-cubically in $|\chi(S)|$. The proof uses a bound for the maximum size of a collection of curves of length at most $L$ on a hyperbolic surface homeomorphic to $S$. Specializing to the case that $S$ is an $n$-holed sphere and $k=2$, we use the coloring computations of Gaster-Greene-Vlamis to show that this bound can be improved to $O(n^5 \log n)$. This is joint work with Tarik Aougab and Ian Biringer.

#### Dec 9: Lien-Yung Kao (University of Notre Dame) Entropy, Critical Exponent, and Immersed Surfaces in Hyperbolic $3$-Manifolds

Consider a $\pi_1$-injective immersion $f:\Sigma \to M$ from a compact surface $\Sigma$ to a hyperbolic $3$-manifold $(M,h)$. Let $\Gamma$ denote the copy of $\pi_{1}(\Sigma)$ in $\mathrm{Isom}(\mathbb{H}^{3})$ induced by the immersion. In this talk, I will discuss relations between two dynamical quantities: the critical exponent $\delta_{\Gamma}$ and the topological entropy $h_{top}(\Sigma)$ of the geodesic flow for the immersed surface $(\Sigma,f^{*}h)$.
More precisely, when $\Gamma$ is convex cocompact and $\Sigma$ is negatively curved, there exist two geometric constants $C_{1}(\Sigma,M)$, $C_{2}(\Sigma,M)\leq 1$ such that $C_{1}(\Sigma,M)\cdot\delta_{\Gamma}\leq h_{top}(\Sigma)\leq C_{2}(\Sigma,M)\cdot\delta_{\Gamma}$. When $f$ is an embedding, $C_{1}(\Sigma,M)$ and $C_{2}(\Sigma,M)$ are exactly the geodesic stretches (aka Thurston's intersection numbers) with respect to certain Gibbs measures. Moreover, there are rigidity phenomena arising from these inequalities. Lastly, if time permits, I will also discuss applications of these inequalities to immersed minimal surfaces in hyperbolic $3$-manifolds and derive results similar to A. Sanders' work on the moduli space of $\Sigma$ introduced by C. Taubes.