Complex Analysis and Dynamics Seminar

Department of Mathematics
Graduate Center of CUNY

Fridays 2:00 - 3:00 pm
Room 5417
Organizers: Ara Basmajian, Patrick Hooper, Jun Hu, and Saeed Zakeri

Past seminars:

Fall 2006, Spring 2007
Fall 2007, Spring 2008
Fall 2008, Spring 2009
Fall 2009, Spring 2010
Fall 2010, Spring 2011
Fall 2011, Spring 2012
Fall 2012, Spring 2013
Fall 2013, Spring 2014
Fall 2014, Spring 2015
Fall 2015, Spring 2016
Fall 2016

Spring 2017:

Feb 3: No seminar

Feb 10: Tamara Kucherenko (City College of CUNY)
Ground States at the Boundary of Rotation Sets

Ground states are accumulation points of equilibrium states when the temperature goes to zero. They play a fundamental role in statistical physics. We consider rotation sets associated with a continuous dynamical system on a compact metric space and a multi-dimensional continuous potential. We study the question for which boundary vectors of the rotation set one can realize an entropy maximizing measure as a ground state associated with a certain linear combination of the potential. We show that at an exposed point there always exists a ground state that maximizes entropy in its rotation class. We also construct examples of rotation sets (in any dimension ) that have exposed boundary points without a ground state in its rotation class. Finally, we consider non-exposed points and show that the following three phenomena exist: a) boundary points without an associated ground state; b) boundary points with a unique ground state that is not ergodic; c) the set of rotation vectors of ground states of a certain linear combination of the potential is a non-trivial line segment.

Feb 17: The seminar will feature two talks:

2:00-2:50 Liviana Palmisano (University of Bristol)
Foliations by Rigidity Classes

We prove that circle maps with a flat interval and degenerate geometry are an example of a dynamical system for which the topological classes do not coincide with the rigidity classes. Contrary to all well-known examples in one-dimensional dynamics (such as circle diffeomorphisms, unimodal interval maps at the boundary of chaos, critical circle maps), we show that the class of functions with Fibonacci rotation numbers is a $C^1$ manifold which is foliated with finite codimension rigidity classes. This is joint work with M. Martens.

3:00-3:50 Alexandre Paiva Barreto (Federal University of Sao Carlos)
Deformations of Hyperbolic Cone Structures

Unlike hyperbolic structures, which are rigid after Mostow Theorem, the hyperbolic cone structures can be deformed. The difficulty to understand these deformations lies in the possibility of degenerating the structure which occurs when the singular link intersects itself over the deformation. In this cases, the Hausdorff-Gromov limit of the deformation is just an Alexandrov space (of curvature bounded from below by $-1$) which may have dimension strictly smaller than $3$. In this talk we are interested in studying these deformations under the condition that the lengths of the singular link remain uniformly bounded over the deformation. This assumption avoids the undesirable case where the singularity becomes dense in the limiting Alexandrov space. The main theorem presented here will be used to bring some information to the following question proposed by W. Tursthon in 80's: Let $M$ be closed and orientable hyperbolic $3$-manifold and let $\Sigma$ be a simple closed geodesic in $M$. Can the hyperbolic structure of $M$ be deformed to the complete hyperbolic structure on $M-\Sigma$ through a path $M_{\alpha}$ of hyperbolic cone structures on $(M,\Sigma)$ with cone angles $\alpha \in [0,2\pi]$?

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