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On Extending Rohlin's Theorem to Sequences

On Extending Rohlin's Theorem to Sequences MARTIN H. ELLISt AND HARRIET J. FELL Let (X, @, M) denote an atomless probability space such that $ contains a countable collection of sets which separates the points of X. If T is an aperiodic invertible bimeasurable nonsingular transformation of X then for each increasing sequence of integers {n } and for each e > 0 there is a set B e 08 with m(B) < e and \J T"'B = X (see Ellis and Friedman [1]). One might go further and ask when Rohlin's theorem can be extended to increasing sequences of natural numbers {«,}. Theorems 1 and 2 below have been found by M. Keane and P. Michel [3]. Our proof of Theorem 2 is quite different from theirs which can be found in [4]. DEFINITION. The Rohlin number of T with respect to {n ,..., ^. ^ is defined by: r x ( 7 R {{n ,...,n _ }) = supm [j T B T Q r x Be9t where T"'B n T >B = 0 (0 ^ i < j ^ r-1). Rohlin's Theorem says that for every aperiodic T, for all neN, R {{1, 2,..., n}) = 1. We will show that R http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On Extending Rohlin's Theorem to Sequences

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/16.1.25
Publisher site
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Abstract

MARTIN H. ELLISt AND HARRIET J. FELL Let (X, @, M) denote an atomless probability space such that $ contains a countable collection of sets which separates the points of X. If T is an aperiodic invertible bimeasurable nonsingular transformation of X then for each increasing sequence of integers {n } and for each e > 0 there is a set B e 08 with m(B) < e and \J T"'B = X (see Ellis and Friedman [1]). One might go further and ask when Rohlin's theorem can be extended to increasing sequences of natural numbers {«,}. Theorems 1 and 2 below have been found by M. Keane and P. Michel [3]. Our proof of Theorem 2 is quite different from theirs which can be found in [4]. DEFINITION. The Rohlin number of T with respect to {n ,..., ^. ^ is defined by: r x ( 7 R {{n ,...,n _ }) = supm [j T B T Q r x Be9t where T"'B n T >B = 0 (0 ^ i < j ^ r-1). Rohlin's Theorem says that for every aperiodic T, for all neN, R {{1, 2,..., n}) = 1. We will show that R

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1984

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