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Uniform Expansion and Topological Entropy in Smooth Ergodic Theory

Alex Blumenthal

Abstract.
I will be presenting results from a paper of Philippe Thieullen [1].
Consider a smooth, discrete-time dynamical system on a compact
manifold. There is an 'easy' bound for the topological entropy of this
system in terms of the Lipschitz constant of the map and the dimension
of the manifold. This speaks to a more general principle in smooth
ergodic theory, which dictates that entropy may only arise because of
the presence of expansion in the system. By analogy with Lyapunov
exponents, though, it is reasonable that the highest degree of
expansion (as given by the Lipschitz constant) might not characterize
expansion in all directions, but rather in only some subspace of
directions. Pursuing this idea, we will define a 'uniform' version of
Lyapunov exponents, for which the Lipschitz constant is merely the
largest such exponent. These 'uniform' exponents may be defined with
no additional difficulty in general Banach spaces, for differentiable
maps with compact or asymptotically compact derivatives. At the end,
we'll show how to obtain sharper bounds for the topological entropy
that only involves the positive uniform exponents of the system.
[1] Philippe Thieullen. Entropy and Hausdorff Dimension for
Infinite-Dimension Dynamical Systems. Journal of Dynamics and
Differential Equations, Vol. 4:1, (1992), 127-159.
Paper URL:
www.math.u-bordeaux1.fr/~pthieull/Recherche/Publications/ThieullenP1992_01.pdf