Spectrum of a substitution minimal set

概要

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K. Jacobs ([1]) reported as an example of Toeplitz type sequences that 0 0 0 0 0 0 0 0 0 0 0 0 0 … 1 1 1 1 1 1 … 0 0 0 … 1 1 … 0 … =0100010101000100010001010 … is strictly ergodic and has a rational pure point spectrum. This sequence has the following properties:(i) It is a shift of the sequence 001000101010001… which is invariant under the substitution 0→0010, 1→1010 of length 4.(ii) The (2i+1)-th symbol of it is 0 for i=0, 1, 2, ….In this paper, we prove that if some general conditions like (i) (ii) above are satisfied for a sequence over some finite alphabet, then it is strictly ergodic and has a rational pure point spectrum. That is, our main results are the followings:I. If M is a minimal set associated with a substitution of some constant length, then M is strictly ergodic.II. Let M be a strictly ergodic set associated with a substitution of length pk, where p is a prime number and k is any positive integer. Assume that for some (or, equivalently, any) α∈M, there exist integers h≥0 and r≥1, such that (iph+r)-th symbol of α is the same for i=0, 1, 2, …. Then, M has a rational pure point spectrum {ω; ω<Sup>pi</Sup>=1 for some i=0, 1, 2, …}.
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