## Maximal pattern complexity for discrete systems.(English)Zbl 1014.37003

This paper is the continuation of the authors’ paper [Ergodic Theory Dyn. Syst. 22, No. 4, 1191-1199 (2002; ; Zbl 1014.37004)]. They continue the study of the maximal pattern complexity. They first give a new proof of the fact that every Sturmian word is pattern Sturmian. This proof is purely combinatorial. Then they give a simple criterion to be pattern Sturmian and exhibit a new class of recurrent pattern Sturmian words which do not arise from rotations.
In the end of the paper, they study the maximal pattern complexity of various discrete dynamical systems. They first show that, for each irrational rotation on the circle, there exists a two fold partition of the circle with respect to which the system generated has full maximal pattern complexity with probability one. Then, they investigate the maximal pattern complexity of the self-similar dynamical system generated by the Rauzy substitution : $$1\to 12$$, $$2\to 13$$, $$3\to 1$$. They prove that the maximal pattern complexity of the fixed point of the Rauzy substitution has exponential growth.

### MSC:

 37B10 Symbolic dynamics 37B40 Topological entropy 39A12 Discrete version of topics in analysis 37E10 Dynamical systems involving maps of the circle

Zbl 1014.37004
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