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Houston Summer School on Dynamical Systems

May 13-21, 2013

Pictures from the school:  (Original announcement is below)










The Department of Mathematics at University of Houston will host a Summer School on Dynamical Systems from May 13-21, 2013.  The school is designed for advanced undergraduate and early graduate students, and will use short lecture courses, tutorial and discussion sessions, and student projects to introduce some of the fundamental concepts of dynamical systems in a manner that is accessible to students who do not necessarily have prior experience with this area of mathematics.


Travel reimbursements and lodging will be provided for all non-UH students.  All students will receive a stipend to cover food and other local expenses.  The application deadline has passed and we have contacted all accepted participants. 



All lectures will be given by members of the research group in Dynamical Systems and Ergodic Theory at UH.  A brief description is below, together with a tentative schedule.  The mini-courses will each consist of a series of 90-minute lectures; tutorial and discussion sessions will be held in conjunction with the lectures, and student projects will be incorporated into the program.


List of topics

The following plan of lectures is tentative and is subject to change as the school goes on, based on the interest and background of the audience, and the progress made in the initial lectures.  The topics covered in lectures may vary slightly from those listed here.  See below for the schedule of when the lectures will be given.

  • Background (BG).  These lectures will review the fundamental concepts from dynamical systems that are needed for the other courses at the school.

    1. Review of random variables, modes of convergence, and classical results from probability theory for IID processes, such as the law of large numbers and the central limit theorem.  Interpretation of observations of a dynamical system as a random process, using an invariant measure.

    2. First steps beyond the IID case: Markov chains and stationary probabilities; the Perron-Frobenius theorem.  Interpretation of the transition matrix as a transfer operator.

    3. Symbolic coding of dynamical systems.  Markov partitions for uniformly hyperbolic systems.  Connections to statistical mechanics.  Abundance of invariant measures and need to select a distinguished measure.  Absolutely continuous invariant measures for certain interval maps.

    4. Tools from functional analysis.  Commonly used function spaces, compactness results: Arzela-Ascoli, Helly.  The need to work with multiple norms.  Bounded variation functions on the interval.

    5. Selecting an appropriate "physical" measure for higher-dimensional maps; SRB measure.  Entropy, topological pressure, and equilibrium states.

    6. Markov chains with countably many states.  Connection to non-uniformly hyperbolic systems via Young towers.

  • Spectral theory (ST)One tool for proving statistical properties of dynamical systems is to understand the spectrum of the transfer operator.  These lectures will explore this approach in various settings.

    1. Spectral methods for Markov chains, definition of transfer operator.  Transition matrix as transfer operator acting on locally constant functions.  Spectral gap for action on Holder continuous functions, and exponential decay of correlations as a corollary.

    2. Abstract results in spectral approach to dynamics; use of spectral gap to deduce exponential decay of correlations via results of Ionescu Tulcea and Marinescu.

    3. Existence of absolutely continuous invariant probability measures for piecewise expanding interval maps using Lasota-Yorke inequality and action of the transfer operator on the space of bounded variation functions.

    4. Exponential decay of correlations and central limit theorem for interval maps.

  • Convex cones (CC).  Spectral methods do not give constructive estimates on the rate of decay of correlations; such estimates can be obtained using convex cones and the Hilbert (projective) metric.

    1. Definition of Hilbert metric, use of convex cones to obtain an explicit estimate on the rate of convergence in the Perron-Frobenius theorem.

    2. Construction of convex cones for piecewise expanding interval maps, and deduction of an explicit estimate for rate of decay of correlations following work of Liverani.

  • Probabilistic methods (PM).  Two lectures on various probabilistic methods applied to dynamical systems, such as coupling techniques and martingales.  The first lecture will focus on finite-state Markov chains (and hence uniformly hyperbolic systems); the second will give applications to non-uniformly hyperbolic systems via countable-state Markov chains and Young towers.


Schedule

Details of the schedule are subject to change.  Registration will take place on the 6th floor of PGH building, all lectures will take place in AH 302 (next door).  Time slots labeled "Proj" are open for work on student projects, further discussion of lecture topics, extra tutorials, etc.  The schedule for the second week will be determined during the school.


Mon

Tue
Wed
Thu
Fri
9-10
Register
9-10:30
BG 4
ST 2
ST 3
BG 6
10-11:30
BG 1
11-12:30
Proj
PM 1
BG 5
ST 4
1-2:30
BG 2
2-3:30
ST 1
Proj
CC 2
Proj
3-4:30
BG 3
4-5:30
CC 1
Free
Proj
PM 2



References and external material

The book "Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics", by William Parry and Mark Pollicott, will be referenced in some of the lectures.  An online copy of the book can be found here.


Application and registration

The application process for this year's school is closed.  If you are interested in receiving information about a possible similar event next year, please contact the organizers.






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