Colloquium

Learning about reality from observation

by James Yorke, Distinguished University Professor of Mathematics and Physics, Institute for Physical Sciences and Technology (IPST), University of Maryland, College Park

Abstract

(See http://www.math.umd.edu/~ottfor a copy of the paper written with Will Ott.)

2400 years ago Plato asked what we can learn from seeing only shadowy imagesof reality. In the 1930's Whitney studied "typical" images of manifolds inR^m and asked when the image was homeomorphic to the original.Let A be a closed set in Rⁿ and let f :Rⁿ ----> R^m be a "typical" smoothmap where n > m. (Plato considered only the case n = 3, m = 2.)Whitney's question has natural extensions. If f(A) is a bounded set, can we conclude the same about A? When can weconclude the two sets have the same cardinality or the same dimension (fortypical f )? (To simplify or clarify thosequestions, you might assume f is a "typical"linear map in the sense of Lebesgue measure.)

In the 1980's Takens, Ruelle, Eckmann, Sano and Sawada extended thisinvestigation to the typical images of attractors of dynamical systems. Theyasked when typical images are similar to the original. Now assume further thatA is a compact invariant set for a map f on Rⁿ. Whencan we say that A and f(A) are similar,based only on knowledge of the images in R^m of trajectories in A? For example, under what conditions on f(A) (andthe induced dynamics thereon) are A and f(A) homeomorphic? Are their Lyapunov exponents the same? Or, more precisely, which of their Lyapunov exponents are the same? This talk (and corresponding paper) addresses these questions with respect to both the generalclass of smooth mappings f and the subclass ofdelay coordinate mappings.