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 (2
i+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
pi=1 for some
i=0, 1, 2, …}.
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