# Complex Analysis and Dynamics Seminar

## Fall 2018 Schedule

#### Sep 7: Yunping Jiang (Queens College, Graduate Center of CUNY and NSF) Order of Oscillating Sequences Motived by Sarnak's Conjecture

In view of Sarnak's conjecture in number theory, we first review a result in ergodic theory and then oscillating sequences and minimal mean attractable and minimal mean-$L$-stable flows. After that we give a definition of an oscillating sequence of order $d\geq 2$ and a definition of an oscillating sequence of order $d\geq 2$ in the arithmetic sense. The sequences generated by the Mobius function and other arithmetic functions in number theory are examples for all $d\geq 2$. There is another example we recently found in view of expanding dynamical systems also for all $d\geq 2$. Furthermore, we show why the order is important in the study of Sarnak's conjecture by proving that any oscillating sequence of order $d\geq 2$ is linearly disjoint from all affine distal flows on the $d$-torus. Furthermore, we show that any oscillating sequence of order $d\geq 2$ is linearly disjoint from non-linear distal flows with Diophantine translations on the $d$-torus and any oscillating sequence of order $d\geq 2$ in the arithmetic sense is linearly disjoint from non-linear distal flows with rational translations on the $d$-torus.

#### Sep 14: Jun Hu (Brooklyn College and Graduate Center of CUNY) Teichmuller Spaces of Non-Discrete Subgroups of $PSL(2,{\mathbb R})$

The concept of Teichmuller space of a Fuchsian group can be extended to any non-discrete group of conformal isometries of the hyperbolic plane. Let G be a non-discrete subgroup of $PSL(2,{\mathbb R})$. We show that the Teichmuller space $T(G)$ of $G$ is not trivial if and only if $G$ is a non-discrete consisting of hyperbolic elements with two common fixed points. Furthermore, we show that if $T(G)$ is not trivial, then (i) $T(G)$ is conformally equivalent to a unit disk; (ii) the length spectrum is just a pseudometric, but it is a metric coinciding with the Teichmuller metric when restricted on a one-dimensional slice. This is a joint work with Francisco G. Jimenez-Lopez

#### Sep 21: Enrique Pujals (Graduate Center of CUNY) Robustly Transitive Dynamics

A dynamical system is said to be robustly transitive if the initial system and all nearby ones are transitive. We will discuss the meaning of this notion, the examples in different contexts, its significance and the different tentative to characterize them. In the end, we will discuss the particular context of linear dynamics in Banach spaces.

#### Sep 28: Umberto Leone Hryniewicz (IAS, Princeton) Existence of Global Cross-Sections: From Schwartzman Cycles to Holomorphic Curves

Using Schwartzman-Fried-Sullivan theory one can state sufficient (and generically necessary) conditions for a link of periodic orbits to bound a global cross-section. These conditions involve linking assumptions on the invariant measures, and in general are hard to check for concrete problems. I'd like to present improvements of such statements for Reeb flows. In this class of flows one obtains more applicable results using pseudo-holomorphic curves, some of which hold unconditionally. Then I will discuss how these techniques can be used to estimate the quadratic linking form (as defined by Ghys) on convex 3-dimensional energy levels. Part of these results are jointly with Pedro Salomao.

#### Oct 5: Christian Wolf (City College and Graduate Center of CUNY) Computability at Zero Temperature

In this talk, we discuss the computability of thermodynamic invariants at zero temperature for one-dimensional subshifts of finite type. In particular, we show that the residual entropy (i.e., the joint ground state entropy) is an upper semi-computable function on the space of continuous potentials, but it is not computable. Next, we consider locally constant potentials for which the zero-temperature measure is known to exist. We characterize the computability of the zero-temperature measure and its entropy for potentials that are constant on cylinders of a given length $k$. In particular, we show the existence of an open and dense set of locally constant potentials for which the zero-temperature measure can be computationally identified as an elementary periodic point measure. Finally, we show that our methods do not generalize to treat the case when $k$ is not given. The results presented in this talk are joint work with Michael Burr (Clemson University).

#### Oct 12: Yan Mary He (University of Toronto) Topology of the Shift Locus via Big Mapping Class Groups

The shift locus of (monic centered) complex polynomials of degree $d > 1$ is the set of polynomials whose filled-in Julia set contains no critical points. Traversing a loop in the shift locus gives rise to a holomorphic motion of Cantor Julia sets, which can be extended to a homeomorphism of the plane minus a Cantor set up to isotopy. Therefore there is a well-defined monodromy representation from the fundamental group of the shift locus to the mapping class group of the plane minus a Cantor set. In this talk, I will discuss the image and the kernel of this map as well as the presentation of the fundamental group. This is joint work with J. Bavard, D. Calegari, S. Koch and A. Walker.

#### Oct 19: Suddhasattwa Das (NYU)Kernel Integral Operators and Ergodic Theory

The operator theoretic framework for dynamical systems studies the dynamics induced on some functional space like $C^0(M)$ or $L^2(\mu)$, instead of the trajectories on the underlying phase space $M$. It transforms the dynamics under any nonlinear flow $\Phi^t$ into a linear map on some Banach/Hilbert space. The dynamics is induced by the Koopman operator $U^t$, which acts on functions by time shifts, namely, $(U^t f)(x) = f (\Phi^t x)$. The spectral properties of $U^t$ have important implications on the actual dynamics. A common choice of domain for $U^t$ is $L^2(\mu)$ for some invariant measure $\mu$. It is a Hilbert space on which $U^t$ is a unitary group. In this talk, I will discuss connections between three different topics : (i) eigenfunctions and spectrum of $U^t$, (ii) ergodic convergence of the product system $\Phi^t \times \Phi^t$; and (iii) kernel functions $k:M \times M \to {\mathbb R}$.

#### Oct 26: Linda Keen (Lehman College and Graduate Center of CUNY)Cycle Doubling, Merging And Renormalization in the Tangent Family

In this talk, based on joint work with Tao Chen and Yunping Jiang, we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $\{ T_t(z) = it \tan z \}_{0 < t \leq \pi}$. Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as t increases, in addition to standard “period doubling,” we find “period merging” where two attracting cycles of period $2n$ “merge” into one attracting cycle of period 2n+1, and “cycle doubling” where an attracting cycle of period $2n+1$ becomes two attracting cycles of the same period. Describing these new phenomena involves adapting the concepts of renormalization” and “holomorphic motions” to our context. The parameters where these bifurcations occur limit at an “infinitely renormalizable tangent map $T_{t_\infty}$ with a “strange attractor” that has a Cantor set structure.

#### Nov 9: Kealey Dias (BCC, CUNY) Quadratic Differentials, Measured Foliations, and Metric Graphs on the Punctured Plane

A meromorphic quadratic differential on the Riemann sphere with two poles of finite order greater than $2$, induces horizontal and vertical measured foliations comprising foliated strips and half-planes. This defines a map from the space of these quadratic differentials to the space of pairs of such foliations. We describe a global parametrization of the space of the induced foliations, and determine the image of the map. To do so we consider an associated space of combinatorial objects on the surface, namely, metric graphs that are the leaf spaces of the foliations. This is joint work with Subhojoy Gupta and Maria Trnkova.

#### Nov 30: Paul Apisa (Yale University) Using Flat Geometry to Understand the Dynamics of Every Point - Hausdorff Dimension, Divergence, and Teichmuller Geodesic Flow!

The moduli space of Riemann surfaces admits a Kobayashi hyperbolic metric called the Teichmuller metric. The geodesic flow in this metric can be concretely understood in terms of a linear action on flat surfaces represented as polygons in the plane. In this talk, we will study the dynamics of this geodesic flow using the geometry of flat surfaces.
Given such a flat surface there is a circle of directions in which one might travel along Teichmuller geodesics. We will describe work showing that for every (not just almost every!) flat surface the set of directions in which Teichmuller geodesic flow diverges on average - i.e. spends asymptotically zero percent of its time in any compact set - is 1/2.
In the first part of the talk, we will recall the work of Masur, which connects divergence of Teichmuller geodesic flow with the dynamics of straight line flow on flat surfaces. In the second part of the talk, we will describe the lower bound (joint with H. Masur) and how it uses flat geometry to prove a quantitative recurrence result for Teichmuller geodesic flow. In the third and final part of the talk, we will describe the upper bound (joint with H. al-Saqban, A. Erchenko, O. Khalil, S. Mirzadeh, and C. Uyanik), which adapts the work of Kadyrov, Kleinbock, Lindenstrauss, and Margulis to the Teichmuller geodesic flow setting using Margulis functions.