# 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.