James Yorke, University of Maryland
Learning about reality from observation
2400 years ago Plato asked what we can learn from seeing only shadowy images
of reality. In the 1930's Whitney studied "typical" images of
manifolds in Rm and asked when the image was homeomorphic to
the original. Let A be a closed set in Rn and let
f:Rn---->Rm
be a "typical" smooth map 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 we conclude the two sets have the
same cardinality or the same dimension (for typical f)?
(To simplify or clarify those questions, 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 this
investigation to the typical images of attractors of dynamical systems. They
asked when typical images are similar to the original. Now assume further that A
is a compact invariant set for a map f on Rn. When can
we say that A and f(A) are similar, based only on knowledge
of the images in Rm of trajectories in A? For example,
under what conditions on f(A) (and the 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 general class
of smooth mappings f and the subclass of delay coordinate mappings.
Click here for a copy of the paper
written with Will Ott.