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Proceedings of the American Mathematical Society

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Measurable sensitivity

Authors: Jennifer James, Thomas Koberda, Kathryn Lindsey, Cesar E. Silva and Peter Speh
Journal: Proc. Amer. Math. Soc. 136 (2008), 3549-3559
MSC (2000): Primary 37A05; Secondary 37F10
Published electronically: May 30, 2008
MathSciNet review: 2415039
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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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Additional Information

Jennifer James
Affiliation: Department of Mathematics, Brandeis University, 415 South Street, Waltham, Massachusetts 02454

Thomas Koberda
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138-2901
MR Author ID: 842738
ORCID: 0000-0001-5465-2651

Kathryn Lindsey
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
MR Author ID: 842785

Cesar E. Silva
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
MR Author ID: 251612

Peter Speh
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307

Keywords: Measure-preserving, ergodic, sensitive dependence
Received by editor(s): December 8, 2006
Received by editor(s) in revised form: July 25, 2007
Published electronically: May 30, 2008
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 American Mathematical Society