Dr. Fern Y. Hunt Computing and Applied Mathematics Laboratory National Institute of Standards and Technology Bldg. 101A-238 Gaithersburg, MD20899
This presentation is intended for an audience of mathematicians working in other fields who might wonder why they should listen to a talk in the area of ergodic theory of dynamical systems. Our purpose is to show that invariant measures can give precision to our intuitive sense of a "distinction" between randomness and deterministic behavior. In particular we discuss how the pioneers of this field, H. Poincare and E. Hopf developed a coherent framework for investigating the emergence of randomness in erstwhile deterministic systems. As an example we discuss the coin tossing problem, as analyzed by J.Keller and extended by P.Diaconis and E.Engler. The statement that the probability of tossing a coin and having it land on a surface showing heads is 1/2; stated so easily and assented to so quickly in a probability class, is true because the randomness of the coin arises from imprecision or uncertainty in the initial position and velocity of the coin. The value of 1/2 is not a consequence of the coin's symmetry but is a limiting value that is approached as either the initial vertical or initial angular velocity of the coin is large -as it is in practice.
With this motivation, we go on to discuss the author's recent work on invariant measures of finite dimensional maps. Just as the value 1/2 reflects the long time behavior of the coin's motion, so the invariant measure of an ergodic map reflects the limiting distribution of possible positions after iterating a point for a long time. The ergodic theory point of view allows us to consider the possibility that the initial position of this point is uncertain. Rather than study maps directly we consider maps perturbed by white noise instead. Proving properties like ergodicity of invariant measures even with special conditions on the map is hard, thus we look at the invariant measures of nearby maps hoping that they can tell us something about the invariant measures of the unperturbed maps. Hence the amplitude of the perturbing noise is assumed to be small. We show that under mild contraction conditions for the map, there does exist a unique invariant measure which is approached exponentially under iteration-starting from an arbitrary initial distribution of positions. Thus points iterated by the random map will at large times have the same distribution, whose support we may associate with the attractor of the randomized map.
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