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Covering Finite Sets by Ergodic Images

Published online by Cambridge University Press:  20 November 2018

J. Michael Steele
J. Michael Steele*
Affiliation:
Department of Mathematics, University of British Colombia, 2075, Westbrook Place Vancouver, B.C., V6T 1W5
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Abstract

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For any ergodic transformation T a set A of measure less than ∈ is constructed with the property that for every finite set F there is a j = j(F) such that F ⊂ T-iA. The basic tool used to prove this is a purely combinatorial result which says there is a small subset of { l, 2, …, n } which can be shifted a small amount to cover any k set in {j: δn ≤j≤n}. Applications are given to the theory of combinatorial entropy.

Keywords

Ergodicaperiodicentropycombinatorial method
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 1 , 01 March 1978 , pp. 85 - 91
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Billingsley, P., Ergodic Theory and Information, Wiley, New York, 1965.Google Scholar
2. Halmos, P. R., Lectures on Ergodic Theory, Chelsea Publishing, New York, 1956.Google Scholar
3. Jones, L. K. and Krengel, U., On transformations without invariant measure, Adv. in Math., 12 (1974), 275-295.Google Scholar
4. Steele, J. M., Combinatorial Entropy and Uniform Limit Laws, Stanford University (Thesis), 1975.Google Scholar
5. Vapnik, V. N. and Chervonenkis, A. Ya., On the uniform convergence of relative frequencies of events to their probabilities, Theory Prob. and Appl., 16 (1971), 264-280.Google Scholar
6. Vapnik, V. N. and Chervonenkis, A. Ya., Teoriya Raspoznavaniya Obrazov, Nauka, Moscow, 1974.Google Scholar