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Integer Cantor sets and an order-two ergodic theorem

Published online by Cambridge University Press:  19 September 2008

Albert M. Fisher
Albert M. Fisher
Affiliation:
University of Göttingen, Lotzestr. 13, D3400 Göttingen, Germany
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Abstract

Let denote the orbit closure, under the left shift σ, of the sequence… (all zeroes)… 101000101000000000101 … corresponding to the integer Cantor set . We prove that with respect to the infinite invariant measure ρ, which is the unique normalized non-atomic invariant measure on M, for every f ∈ L1(M, ρ), for ρ-a.e. x∈ M

where d = log 2/log 3, and c is the almost-sure value of the right-hand order-two density of the middle-third Cantor set. The proof uses renormalization to a scaling flow, plus identification of (M, σ) as a tower over the Kakutani-von Neumann dyadic odometer.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 1 , March 1993 , pp. 45 - 64
Copyright
Copyright © Cambridge University Press 1993

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