Symbolic Dynamics

Morse, Marston; Hedlund, Gustav A.

doi:10.2307/2371264

Cited by 498 publications

(325 citation statements)

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“…This implies that the decoration of the edge y j y j+1 is also a, with j + 1 taken modulo q. Since the edges y i y i+1 and y j y j+1 have different colours, we find that (p, q) is not a period of w. Hence, by Theorem 4, we see that |w| = n − r − 1 ≤ p + q − 2, proving (6).…”

Section: Hereditary Properties Of Low Complexitymentioning

confidence: 65%

Hereditary properties of words

Balogh

Bollobás

2005

RAIRO-Theor. Inf. Appl.

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Abstract. Let P be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to P is also in P. Extending the classical Morse-Hedlund theorem, we show that either P contains at least n + 1 words of length n for every n or, for some N , it contains at most N words of length n for every n. More importantly, we prove the following quantitative extension of this result: if P has m ≤ n words of length n then, for every k ≥ n + m, it contains at most (m + 1)/2 (m + 1)/2 words of length k.Mathematics Subject Classification. 05C.

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Section: Hereditary Properties Of Low Complexitymentioning

confidence: 65%

Hereditary properties of words

Balogh

Bollobás

2005

RAIRO-Theor. Inf. Appl.

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“…, η(s + N ) occurs infinitely often in η and there is M > 0 so that between any two occurrences, there are at most M symbols. Note that η is a minimal orbit under the action of the shift operator σ [MorHed38]. It is shown in [MorHed38] that periodic and almost periodic sequences provide all periodic and non-periodic minimal sequences.…”

Section: Symmetric Homoclinic Orbits In Reversible Systemsmentioning

confidence: 99%

“…Note that η is a minimal orbit under the action of the shift operator σ [MorHed38]. It is shown in [MorHed38] that periodic and almost periodic sequences provide all periodic and non-periodic minimal sequences. An almost periodic itinerary is called symmetric if its σ-orbit is accumulated by σ-orbits of symmetric periodic itineraries.…”

Section: Symmetric Homoclinic Orbits In Reversible Systemsmentioning

confidence: 99%

Multiple homoclinic orbits in conservative and reversible systems

Homburg

Knobloch

2005

Trans. Amer. Math. Soc.

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Abstract. We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times.In conservative systems one finds one parameter families of suspended horseshoes, parameterized by the level of the first integral. A somewhat similar picture occurs in reversible systems, with two homoclinic orbits that are both symmetric. The lack of a first integral implies that complete horseshoes do not exist. We provide a description of orbits that necessarily do exist.A second possible configuration in reversible systems occurs if a non-symmetric homoclinic orbit exists and forms a bellows together with its symmetric image. We describe the nonwandering set in an unfolding. The nonwandering set is shown to simultaneously contain one-parameter families of periodic orbits, hyperbolic periodic orbits of different index, and heteroclinic cycles between these periodic orbits.

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“…Intersections of stable and unstable manifolds give rise to homoclinic and heteroclinic orbits [1], which have fixed past and future asymptotes. Moreover, using regions bounded by the stable and unstable manifolds as Markov partitions [14,15], generic behaviors of homoclinic tangles give rise to Smale's horseshoe structures and symbolic dynamics [16,17], in which the motions of points from non-wandering sets [2,3] under successive mappings are topologically conjugate to a Bernoulli shift on their symbolic strings [18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Exact relations between homoclinic and periodic orbit actions in chaotic systems

Tomsovic

2018

Phys. Rev. E

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Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify geometric relations between homoclinic and unstable periodic orbits, and derive exact formulae expressing the periodic orbit classical actions in terms of corresponding homoclinic orbit actions plus certain phase space areas. The exact relations provide a basis for approximations of the periodic orbit actions as action differences between homoclinic orbits with well-estimated errors. This make possible the explicit study of relations between periodic orbits, which results in an analytic expression for the action differences between long periodic orbits and their shadowing decomposed orbits in the cycle expansion.

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Symbolic Dynamics

Cited by 498 publications

References 0 publications

Hereditary properties of words

Hereditary properties of words

Multiple homoclinic orbits in conservative and reversible systems

Exact relations between homoclinic and periodic orbit actions in chaotic systems

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