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Pre-image entropy

Published online by Cambridge University Press:  04 July 2005

WEN-CHIAO CHENG  and
SHELDON E. NEWHOUSE
WEN-CHIAO CHENG
Affiliation:
Department of Mathematics, National Chung Cheng University, Chia-Yi, Taiwan 621, R.O.C. (e-mail: [email protected])
SHELDON E. NEWHOUSE
Affiliation:
Department of Mathematics, Michigan State University, E. Lansing, MI 48824, USA (e-mail: [email protected])
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Abstract

We define and study new invariants called pre-image entropies which are similar to the standard notions of topological and measure-theoretic entropies. These new invariants are only non-zero for non-invertible maps, and they give a quantitative measurement of how far a given map is from being invertible. We obtain analogs of many known results for topological and measure-theoretic entropies. In particular, we obtain product rules, power rules, analogs of the Shannon–Breiman–McMillan theorem, and a variational principle.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 25 , Issue 4 , August 2005 , pp. 1091 - 1113
Copyright
2005 Cambridge University Press

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