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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Prevalence: a translation-invariant ``almost every'' on infinite-dimensional spaces

Authors: Brian R. Hunt, Tim Sauer and James A. Yorke
Journal: Bull. Amer. Math. Soc. 27 (1992), 217-238
MSC (2000): Primary 28C20; Secondary 46G12
MathSciNet review: 1161274
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Abstract: We present a measure-theoretic condition for a property to hold "almost everywhere" on an infinite-dimensional vector space, with particular emphasis on function spaces such as $ {C^k}$ and $ {L^p}$. Like the concept of "Lebesgue almost every" on finite-dimensional spaces, our notion of "prevalence" is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in terms of the class of all probability measures with compact support. Prevalence is a more appropriate condition than the topological concepts of "open and dense" or "generic" when one desires a probabilistic result on the likelihood of a given property on a function space. We give several examples of properties which hold "almost everywhere" in the sense of prevalence. For instance, we prove that almost every $ {C^1}$ map on $ {\mathbb{R}^n}$ has the property that all of its periodic orbits are hyperbolic.

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Article copyright: © Copyright 1992 American Mathematical Society