Sunday, May 18

7:00 PM-9:30 PM Superior A - Level C

Ulam's Conjecture and the Approximation of Invariant Measures

Continued successful interaction between computational observations of random-like behavior and mathematically rigorous description of dynamical systems will depend on the development of methods for computing the invariant measures of invariant sets whose accuracy can be assessed. This minisymposium will be devoted to a discussion of a method first proposed by Stanislaw Ulam for approximating the absolutely continuous invariant measures of finite-dimensional maps. There will be a discussion of convergence rates in the absolutely continuous case, and presentation of recent results on approximation of measures and attractors where the Birkhoff Ergodic theorem holds almost everywhere (Lebesgue). Our purpose is to present an easily implemented alternative to the computation of invariant measures (histograms) by box counting-a procedure for which there are to our knowledge no general estimates. We show how the method can be used to calculate Lyapunov exponents. Highlighted will be extensions of the original method to multi-dimensional maps and set-valued maps. Convergence rates of the method will be discussed in the absolutely continuous case, and results on the approximation of attractors and invariant measures will be presented for the singular case.

Organizer: Fern Y. Hunt
National Institute of Standards and Technology

7:00 Computing Physical Measures of Mixing Multidimensional Systems with an Application to Lyapunov Exponents
Gary Froyland, The University of Western Australia, Australia
7:30 A Finite Element Method for the Frobenius-Perron Operator Equation
Jiu Ding, University of Southern Mississippi; and Aihui Zhou, Academia Sinica, People's Republic of China
8:00 Approximating Attractors and Chain Transitive Invariant Sets with Ulam's Method
Fern Y. Hunt, Organizer
8:30 Markov Finite Approximation of Frobenius-Perron Operators and Invariant Measures for Set-Valued Dynamical Systems
Walter Miller, Howard University
9:00 Cone Conditions and Error Bounds for Ulam's Method
Rua D. A. Murray, University of Cambridge, United Kingdom

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TMP, 4/3/97
MMD, 4/11/97 TJF, 4/15/97