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On maximal pattern complexity of some automatic words

Published online by Cambridge University Press:  13 October 2010

TETURO KAMAE  and
PAVEL V. SALIMOV
TETURO KAMAE
Affiliation:
Satakedai 5-9-6, Suita 565-0855, Japan (email: kamae@apost.plala.or.jp)
PAVEL V. SALIMOV
Affiliation:
Sobolev Institute of Mathematics, prosp. Koptyuga 4, 630090 Novosibirsk, Russia (email: ch.cat.s.smile@gmail.com)
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Abstract

The pattern complexity of a word for a given pattern S, where S is a finite subset of {0,1,2,…}, is the number of distinct restrictions of the word to S+n (with n=0,1,2,…). The maximal pattern complexity of the word, introduced in the paper of T. Kamae and L. Zamboni [Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys.22(4) (2002), 1191–1199], is the maximum value of the pattern complexity of S with #S=k as a function of k=1,2,…. A substitution of constant length on an alphabet is a mapping from the alphabet to finite words on it of constant length not less than two. An infinite word is called a fixed point of the substitution if it stays the same after the substitution is applied. In this paper, we prove that the maximal pattern complexity of a fixed point of a substitution of constant length on {0,1} (as a function of k=1,2,…) is either bounded, a linear function of k, or 2k.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 5 , October 2011 , pp. 1463 - 1470
Copyright
Copyright © Cambridge University Press 2010

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