Maximal pattern complexity of two-dimensional words

Kamae, Teturo; Rao, Hui; Xue, Yumei

doi:10.1016/j.tcs.2006.02.023

Cited by 9 publications

(12 citation statements)

References 13 publications

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“…The n-dimensional rotation words have been studied by many authors [1,3,6,12]. Following the argument of [6], we show that (see Section 7): Theorem 1.3.…”

Section: Pattern Sturmian Wordsmentioning

confidence: 73%

“…Following the argument of [6], we show that (see Section 7): Theorem 1.3. Irrational rotation words are pattern Sturmian words.…”

Section: Pattern Sturmian Wordsmentioning

confidence: 78%

“…The proof of the general case (see Section 5) is an analog of [6] which deals with two-dimensional words.…”

Section: Maximal Pattern Complexity Of N-dimensional Wordmentioning

confidence: 99%

“…Kamae et al [6] proved that two-dimensional rotation words are pattern Sturmian. The n-dimensional rotation words are defined as follows.…”

Section: Pattern Sturmian Wordsmentioning

confidence: 99%

“…Following the terminology of Kamae et al [6], a rational direction u is of type 1, if α u is eventually 1; is of type 0 if α u is eventually 0; is of mixed type otherwise. It is proved in [6] that: These words have been studied in [4,6].…”

Section: Restricting α To a Raymentioning

confidence: 99%

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Maximal pattern complexity of higher dimensional words

Rao

Zhou

et al. 2010

Journal of Combinatorial Theory, Series A

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This paper studies the pattern complexity of n-dimensional words. We show that an n-recurrent but not n-periodic word has pattern complexity at least 2k, which generalizes the result of [T. Kamae, H. Rao, Y.-M. Xue, Maximal pattern complexity of two dimension words, Theoret. Comput. Sci. 359 (1-3) (2006) 15-27] on two-dimensional words. Analytic directions of a word are defined and its topological properties play a crucial role in the proof. Accordingly n-dimensional pattern Sturmian words are defined. Irrational rotation words are proved to be pattern Sturmian. A new class of higher dimensional words, the simple Toeplitz words, are introduced. We show that they are also pattern Sturmian words.

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“…The n-dimensional rotation words have been studied by many authors [1,3,6,12]. Following the argument of [6], we show that (see Section 7): Theorem 1.3.…”

Section: Pattern Sturmian Wordsmentioning

confidence: 73%

“…Following the argument of [6], we show that (see Section 7): Theorem 1.3. Irrational rotation words are pattern Sturmian words.…”

Section: Pattern Sturmian Wordsmentioning

confidence: 78%

“…The proof of the general case (see Section 5) is an analog of [6] which deals with two-dimensional words.…”

Section: Maximal Pattern Complexity Of N-dimensional Wordmentioning

confidence: 99%

“…Kamae et al [6] proved that two-dimensional rotation words are pattern Sturmian. The n-dimensional rotation words are defined as follows.…”

Section: Pattern Sturmian Wordsmentioning

confidence: 99%

Section: Restricting α To a Raymentioning

confidence: 99%

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Maximal pattern complexity of higher dimensional words

Rao

Zhou

et al. 2010

Journal of Combinatorial Theory, Series A

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Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals

Damanik

Embree

Gorodetski

2015

Progress in Mathematics

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We consider discrete Schrödinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schrödinger operators are well understood. In particular, it is known that for every Sturmian potential, the associated Schrödinger operator has zero-measure spectrum and purely singular continuous spectral measures. We conjecture that the same statements hold in the more general class of pattern Sturmian potentials. We prove partial results in support of this conjecture. In particular, we confirm the conjecture for all pattern Sturmian potentials that belong to the family of Toeplitz sequences. Date

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