Sequence entropy and the maximal pattern complexity of infinite words

TETURO KAMAE; LUCA ZAMBONI

doi:10.1017/s014338570200055x

Sequence entropy and the maximal pattern complexity of infinite words

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570200055x ◽

2002 ◽

Vol 22 (04) ◽

Cited By ~ 31

Author(s):

TETURO KAMAE ◽

LUCA ZAMBONI

Keyword(s):

Sequence Entropy ◽

Pattern Complexity ◽

Infinite Words

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On maximal pattern complexity of some automatic words

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385710000453 ◽

2010 ◽

Vol 31 (5) ◽

pp. 1463-1470 ◽

Cited By ~ 2

Author(s):

TETURO KAMAE ◽

PAVEL V. SALIMOV

Keyword(s):

Fixed Point ◽

Linear Function ◽

Finite Subset ◽

Constant Length ◽

Sequence Entropy ◽

Pattern Complexity ◽

Infinite Words ◽

Maximum Value ◽

Infinite Word

AbstractThe 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.

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Abelian maximal pattern complexity of words

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.51 ◽

2013 ◽

Vol 35 (1) ◽

pp. 142-151 ◽

Cited By ~ 1

Author(s):

TETURO KAMAE ◽

STEVEN WIDMER ◽

LUCA Q. ZAMBONI

Keyword(s):

Lower Bound ◽

Pattern Complexity ◽

Infinite Words ◽

Abelian Equivalence

AbstractIn this paper, we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection. We show that in the case of binary words, the bound is actually achieved and gives a characterization of recurrent aperiodic words.

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