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Houston Summer School on Dynamical Systems

May 20-28, 2015

For the third consecutive year, the Department of Mathematics at University of Houston will host a Summer School on Dynamical Systems, which will run from May 20-28, 2015. 

The school is designed for graduate students and will use short lecture courses, tutorial and discussion sessions, and student projects to explore some of the fundamental concepts of dynamical systems.   It will be accessible to students without a background in dynamics, but is also intended for students who have begun studying dynamics and wish to learn more about this field.

The school is funded by an NSF grant, which will cover travel and local expenses for all accepted participants.  For application instructions, see the bottom of this page.

Short course topics will include:
  • Decay of correlations in dynamical systems (University of Houston dynamics group: Vaughn Climenhaga, Matt Nicol, Will Ott, Andrew Torok)

    These lectures will introduce the notion of decay of correlations for a dynamical system and will describe three important methods for establishing a rate of decay: spectral gap (Perron--Frobenius theory); coupling techniques; and Birkhoff cones.

  • Uniform hyperbolicity (Boris Hasselblatt, Tufts University)

    Topological dynamics of hyperbolic sets (shadowing, expansivity, closing, spectral decomposition, specification, topological stability, structural stability, Markov approximation) from the Anosov Shadowing Theorem.  Ergodicity and multiple mixing of product-like measures (such as volume) from the Hopf argument. Exotic contact Anosov flows by Dehn--Foulon surgery. Optionally invariant manifold theory or equilibrium states.

  • Partial hyperbolicity (Keith Burns, Northwestern University)

    A diffeomorphism is partially hyperbolic if there is an invariant splitting of the tangent bundle into three subbundes, Es, Ec, and Eu, such that under iteration of the map vectors in Es shrink exponentially, vectors in Eu expand exponentially and any expansion/contraction of vectors in Ec is weaker. This is a natural setting to which many of the ideas from the study of uniformly hyperbolic (Anosov) systems extend. In particular Eberhard Hopf's proof of ergodicity can be made to work for a broad class of partially hyperbolic systems. The lectures  will give an outline of this theory.

  • Rotations and interval exchange transformations (Jon Chaika, University of Utah)

    Ergodicity and Weyl's criterion; Denjoy; continued fraction algorithms to prove Diophantine results via the Gauss map; Sturmian sequences; weak mixing of 3-IETs; some low complexity dynamics like bounds on ergodic measures.

In addition to the mini-courses, there will be a number of one-hour lectures introducing various applications and extensions of the theory presented in the short courses, such as non-uniform hyperbolicity and Young towers; applications to biology and neuroscience; non-stationary systems; open systems; martingales; statistical mechanics; applications to models in finance.

Students participating in the school should be familiar with the following prerequisite material: measure theory, basic functional analysis, basic theory of smooth manifolds.  There will be background sessions during the school to give a brief review of the most relevant parts of these topics.

Funding for this event is provided by NSF grant DMS-1500151.         NSF logo

Current schedule of lectures

Participant list

List of projects

References from spectral methods lecture

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