Measurable sensitivity

James, Jennifer; Koberda, Thomas; Lindsey, Kathryn; Silva, Cesar; Speh, Peter

doi:10.1090/S0002-9939-08-09294-0

Measurable sensitivity
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by Jennifer James, Thomas Koberda, Kathryn Lindsey, Cesar E. Silva and Peter Speh PDF

Proc. Amer. Math. Soc. 136 (2008), 3549-3559 Request permission

Abstract:

We introduce the notions of measurable and strong measurable sensitivity, which are measure-theoretic versions of the conditions of sensitive dependence on initial conditions and strong sensitive dependence on initial conditions, respectively. Strong measurable sensitivity is a consequence of light mixing, implies that a transformation has only finitely many eigenvalues, and does not exist in the infinite measure-preserving case. Unlike the traditional notions of sensitive dependence, measurable and strong measurable sensitivity carry up to measure-theoretic isomorphism, thus ignoring the behavior of the transformation on null sets and eliminating dependence on the choice of metric.

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