Summary
Let r≧2 be an integer and let ϕ: {0, 1}r→{0,1} be a function. Let T be the transformation on Ω={0, 1}zgiven by (Tω)(i)=ϕ(ω(ri), ω(ri +1), ..., ω(ri+r−1)) for all iεZ. For P in the class of strongly-mixing shiftinvariant measures on Ω, we investigate when P is invariant with respect to T and when T nP converges. For example if r is odd and ϕ(ω 0,..., ω r−1)=1 iff ∑ω>1/2r, the invariant measures are the Bernoulli measures with means 0, 1/2 or 1 and T nP must converge to one of these three measures. Other choices of ϕ can give more complicated behaviour.
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Research supported in part by the National and Engineering Research Council of Canada.
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O'Brien, G.L. Scaling transformations for {0, 1}-valued sequences. Z. Wahrscheinlichkeitstheorie verw Gebiete 53, 35–49 (1980). https://doi.org/10.1007/BF00531610
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DOI: https://doi.org/10.1007/BF00531610