Research

I work under the supervision of Professor Mariusz Urbanski in the field of Dynamical Systems. My main research project studies discrete isometric actions on infinite-dimensional hyperbolic spaces. I am currently studying the geometry of fractal limit sets that arise from such actions, via extensions of the theory of Graph Directed Markov Systems created by Professors Mauldin and Urbanski. The near future will be spent investigating various directions relating the exotic geometry, dynamics, analysis and representation theory involved.

My research interests include Conformal and Holomorphic Dynamical Systems, Fuchsian and Kleinian Group actions, Hyperbolic Geometry, Ergodic Theory, Fractal Sets, Diophantine Approximation and Statistical Physics with an emphasis on Gas Lattices and the Thermodynamic Formalism.

Slides/Notes

Dynamics and geometry in infinite-dimensional hyperbolic spaces

Ergodic Theory Workshop, University of North Carolina - Chapel Hill, March 24, 2012.

Slides

Rigidity in infinite-dimensional hyperbolic spaces

Geometry and Analysis on Fractal Spaces, 2012 AMS Spring Western Sectional Meeting
University of Hawaii at Manoa, March 4, 2012

Slides

Infinite-dimensional models of hyperbolic space and related analogues of dynamics and discrete groups

Einstein Chair Mathematics Seminar, Graduate Center, CUNY.

Wednesday 22^nd Feb, 2012, 2:00pm – 4:00pm, Rm. 4419.

Abstract: We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite-dimensional separable Hilbert space. We generalize most results of negative curvature and Gromov-hyperbolic settings to get to their geometric core and have greater scope for applications. Many of the essential ideas are already present when working in Hilbert space, although one must be careful with boundaries and non-geodesic scenarios. There are many examples that explain what is fundamentally different from the classical finite-dimensional setting. For starters, in infinite dimensions properly discontinuous actions are no longer strongly discrete (finitely many orbit points in arbitrary balls) and though a Poincare-type summation over the orbits being finite implies strong discreteness always, the reverse fails in infinite-dimensions. The existence of fixed points of isometries and their structure will be discussed - here one discovers interesting parabolic behaviour that's absent in finite dimensions. We characterize convex-cobounded groups in terms of radial points in the limit set and go on to characterize groups whose limit sets are compact. Schottky groups whose limit sets are Cantor sets provide a variety of interesting phenomena where extensions of the classical thermodynamic formalism (a la Bowen) prove strong results about the geometry and dynamical properties of their limit sets. We prove a generalization of the Bishop-Jones theorem, equating the Hausdorff dimension of the radial limit set with the Poincare exponent. Time permitting, we sketch the proof of the Ahlfors-Thurston theorem and develop Patterson-Sullivan theory for divergence type groups. Here there are examples of convergence type groups that do not admit a conformal measure. To end, we discuss a few problems/applications. Almost everything will be developed from scratch with an attempt to present the underlying geometric ideas behind the proofs – graduate students are very welcome.

Kleinian Limit Sets in Hilbert Spaces
AMS Session on Dynamic Systems and Ergodic Theory, AMS/MAA Joint Math Meetings, Boston, 2012.

1:30pm Saturday January 7, 2012. Republic Ballroom A, 2nd Floor, Sheraton.

Abstract

On a theorem of Bishop and Jones

RTG Logic and Dynamics Seminar, University of North Texas, 2011.

Abstract: Bishop and Jones, in a remarkable paper from Acta '84, proved that for any Kleinian group acting on a finite-dimensional hyperbolic space the Poincare exponent is equal to the Hausdorff dimension of the radial/conical limit set. In joint work with Bernd Stratmann and Mariusz Urbanski we generalize this result to strongly discrete groups acting on infinite-dimensional hyperbolic space. Although the original proof of Bishop and Jones crucially uses the the compactness of the sphere at infinity as well as the fact that finite-dimensional spaces are "doubling", i.e. there is a uniformly bounded number of disjoint balls of a fixed radius inside a ball of twice the radius, our proof avoids such dependence. We first prove a rather general mass-redistribution result that works for complete metric spaces and then use the group action to carefully construct a tree in hyperbolic space to which we apply the former result.

Kleinian Groups in Hilbert Spaces

45th Spring Topology and Dynamics Conference, University of Texas at Tyler, 2011.

Abstract

What is a Continued Fraction?

Informal Mathematics Research Problem Session, University of North Texas, 2010.

See the UNT RTG in Dynamics and Logic webpages http://www.math.unt.edu/rtg/activities.html

Notes

Publications and Preprints

(With M. Urbanski) "The Geometry of Baire Spaces", Preprint 2008.

Published version: Dynamical Systems, Vol. 26, No. 4, 2011, 537-567.

To link to this article: http://www.tandfonline.com/doi/abs/10.1080/14689367.2011.628010

This paper was the outgrowth of my M.S. degree supervised by Professor Urbanski. It was inspired by beautiful ideas in the short paper by Sullivan - Differentiable Structures in Fractal-like Sets, Determined by Intrinsic Scaling Functions on Dual Cantor Sets that were further extended in the book by Przytycki and Urbanski, Conformal Fractals: Ergodic Theory Methods.

Abstract - We introduce the concept of Baire embeddings and we classify them up to C^(1+epsilon) conjugacies. We show that two such embeddings are C^(1+epsilon)-equivalent if and only if they have exponentially equivalent geometries. Next, we introduce the class of IFS-like Baire embeddings and we also show that two Holder equivalent IFS-like Baire embeddings are C^(1+epsilon) conjugate if and only if their scaling functions are the same. In the remaining sections we introduce metric scaling functions and we show that the logarithm of such a metric scaling function and the logarithm of Sullivans scaling function multiplied by the Hausdorff dimension of the Baire embedding are cohomologous up to a constant. This permits us to conclude that if the Bowen measures coincide for two IFS-like Baire embeddings, then the embeddings are bi-Lipschitz conjugate.