Summary
If x=b 0 × b 1× . is a regular Morse sequence and sup \ i<+∞, then x has rank 2. There are regular Morse shifts with rank one. If x is a Kakutani sequence, then x has rank one iff x is not regular. If θ is a nonperiodic substitution of constant length on two symbols, then θ is of rank 2 iff θ is a continuous substitution. Every Morse sequence has a simple spectrum.
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Lemańczyk, M. The rank of regular morse dynamical systems. Z. Wahrscheinlichkeitstheorie verw Gebiete 70, 33–48 (1985). https://doi.org/10.1007/BF00532236
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DOI: https://doi.org/10.1007/BF00532236