On the permutations generated by Sturmian words

Макаров, М. А.

doi:10.1007/s11202-009-0076-6

Cited by 17 publications

(30 citation statements)

References 7 publications

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“…Note that the same irrational rotation R σ was used above to define a class of Sturmian equidistributed permutations. Sturmian permutations have been studied in [19]; in particular, it is known that their complexity is p β (n) ≡ n (i.e., p β (n) = n for all n).…”

Section: Sturmian Words and Sturmian Permutationsmentioning

confidence: 99%

Minimal complexity of equidistributed infinite permutations

Avgustinovich

Frid

Puzynina

2017

European Journal of Combinatorics

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An infinite permutatation is a linear ordering of the set of natural numbers. An infinite permutation can be defined by a sequence of real numbers where only the order of elements is taken into account. In the paper we investigate a new class of equidistributed infinite permutations, that is, infinite permutations which can be defined by equidistributed sequences. Similarly to infinite words, a complexity p(n) of an infinite permutation is defined as a function counting the number of its subpermutations of length n. For infinite words, a classical result of Morse and Hedlund, 1938, states that if the complexity of an infinite word satisfies p(n) ≤ n for some n, then the word is ultimately periodic. Hence minimal complexity of aperiodic words is equal to n + 1, and words with such complexity are called Sturmian. For infinite permutations this does not hold: There exist aperiodic permutations with complexity functions growing arbitrarily slowly, and hence there are no permutations of minimal complexity. We show that, unlike for permutations in general, the minimal complexity of an equidistributed permutation α is p α (n) = n. The class of equidistributed permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words.

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Section: Sturmian Words and Sturmian Permutationsmentioning

confidence: 99%

Minimal complexity of equidistributed infinite permutations

Avgustinovich

Frid

Puzynina

2017

European Journal of Combinatorics

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“…Such a permutation is said to be valid over the alphabet Σ q . Valid permutations have been investigated by Makarov [11,12,13,14]. It is not difficult to see that there exist infinite permutations which are not valid, including for example the monotonic ones.…”

Section: Infinite Permutationsmentioning

confidence: 99%

On automatic infinite permutations

Frid¹,

Zamboni²

2011

RAIRO-Theor. Inf. Appl.

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An infinite permutation α is a linear ordering of N. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we try to extend to permutations the notion of automaticity. As we shall show, the standard definitions which are equivalent in the case of words are not equivalent in the context of permutations. We investigate the relationships between these definitions and prove that they constitute a chain of inclusions. We also construct and study an automaton generating the Thue-Morse permutation.Une permutation infinie est un ordre total sur N. Nousétudions les caractéristiques des permutations infinies qui sont analoguesà celles des mots infinis, et nous montrons que certaines d'entre elles se comportent de la même façon, et d'autres pas. Dans cet article nous essayons d'étendre la notion d'automaticité de mots aux permutations. Cependant il arrive que des définitionséquivalentes pour les mots ne sont paséquivalentes pour les permutations. Nousétudions la relation entre ces définitions en démontrant qu'ils constituent une chaîne d'inclusions. En outre, nous discutons les automates engendrant la permutation de Thue-Morse.

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“…The allowed and forbidden patterns of many maps from dynamical systems have been studied during the last several years, including the left shift on words [9,15], signed shifts on the unit interval [1,2,6,7], beta shifts [11], negative beta-shifts [13] and the logistic maps [12].…”

Section: Introductionmentioning

confidence: 99%

Allowed Patterns of Symmetric Tent Maps via Commuter Functions

Archer

LaLonde²

2017

SIAM J. Discrete Math.

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We introduce a new technique to study pattern avoidance in dynamical systems, namely the use of a commuter function between non-conjugate dynamical systems. We investigate the properties of such a commuter function, specifically h :where Tµ denotes a symmetric tent map of height µ. We make use of this commuter function to prove strict inclusion of the set of allowed patterns of Tµ in the set of allowed patterns of T 1 .Date: November 15, 2018. 2010 Mathematics Subject Classification. 05A05, 37E05 (primary) and 37E15 (secondary).

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On the permutations generated by Sturmian words

Cited by 17 publications

References 7 publications

Minimal complexity of equidistributed infinite permutations

Minimal complexity of equidistributed infinite permutations

On automatic infinite permutations

Allowed Patterns of Symmetric Tent Maps via Commuter Functions

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