Feb 1: Christian Wolf (City College and Graduate Center of CUNY)
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.
A†Topological Classification of Locally Constant Potentials
via Zero-Temperature Measures
Feb 8: Mike Todd (University of St. Andrews)
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.† Iíll 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
Iíll give some indication of the general approach there.
Stability of Measures in Interval Dynamics
Feb 15: Han Li (Wesleyan University)
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.
Masserís Conjecture on Equivalence of Integral Quadratic Forms
Feb 22: Fabio Tal (Universidade de S„o Paulo)
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:
Homeomorphisms of Surfaces with Zero Entropy
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)
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.
Inscribing Triangles and Rectangles in Jordan Curves
Mar 8: Enrique Pujals (Graduate Center of CUNY)
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).
Surface Dissipative Difffeomorphisms with Zero Entropy and Renormalization
March 15: Tao Chen (Laguardia Community College of CUNY)
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.
Dynamics of General Tangent Family
March 22: Yunping Jiang (Queens College and Graduate Center of CUNY)
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.
Shape of the Metric Entropy
April 26: Bryce Gollobit (Graduate Center of CUNY)
May 3: Linda Keen (Lehman College and Graduate Center of CUNY)
May 10: Saeed Zakeri (Queens College and Graduate Center of CUNY)