A natural class of generalized Fibonacci chains

Wen, Zhi-Ying; Wijnands, F.H.G.M.; Lamb, Jeroen S. W.

doi:10.1088/0305-4470/27/11/019

Cited by 21 publications

(9 citation statements)

References 35 publications

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“…The structure thus obtained can be alternatively built as sections of periodic patterns in a twodimensional space. Along the same line, Wen et al 24 have proposed following a class of substitution rules to generate quasiperiodic chains which share some typical properties with the Fibonacci chain. The substitution matrix is…”

Section: ͑2͒mentioning

confidence: 99%

“…24,25 To obtain a FC(n) sequence by the projection method, we consider a two-dimensional periodic lattice with square unit cells of dimension 1ϫ1 whose edges are aligned with the X and Y axes, respectively. Then, construct a projection line oriented at angle with respect to the X axis and let tanϭ n .…”

Section: ͑11͒mentioning

confidence: 99%

“…The relationship between these two sides has been studied recently. [20][21][22][23][24][25] In this paper, we restrict ourselves to the binary sequences. A binary substitution is formally defined by its simultaneous action on two letters A and B, which replaces each letter with a finite word, of the form…”

Section: Introductionmentioning

confidence: 99%

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Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices

Liu

Zhou

et al. 1997

Phys. Rev. B

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One-dimensional Fibonacci-class quasilattices are proposed and studied, which are constructed by the substitution rules B→B nϪ1 A, A→B nϪ1 AB. We have proved that this class of binary lattices is self-similar and also quasiperiodic. By the use of the renormalization-group technique, it has been proved that for all Fibonacciclass lattices the electronic energy spectra are perfect self-similar, and the branching rules of spectra are obtained. We analytically prove that each energy gap can be simply labeled by a characteristic integer, i.e., for the Fibonacci-class lattices there is a universal gap-labeling theorem ͓Phys. Rev. B 46, 9216 ͑1992͔͒.

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Section: ͑2͒mentioning

confidence: 99%

Section: ͑11͒mentioning

confidence: 99%

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Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices

Liu

Zhou

et al. 1997

Phys. Rev. B

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“…This generalization takes into account the fact that a map L may possess fewer symmetries and reversing symmetries than its kth iterate L k . k-Symmetry has been observed to occur in various dynamical systems of physical interest such as web maps modelling the dynamics of a kicked charged particle in a magnetic eld 17,22,42], a classical and quantum kicked top 16], and trace maps 33,41]. In particular, general mechanisms have pointed out the occurrence and relevance of k-symmetry in resonantly driven dynamical systems 24,25].…”

Section: Introductionmentioning

confidence: 99%

Local bifurcations ink-symmetric dynamical systems

Lamb¹

1996

Nonlinearity

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A map U : IR d 7 ! IR d is called a (reversing) k-symmetry of the dynamical system represented by the map L : IR d 7 ! IR d if k is the smallest positive integer for which U is a (reversing) symmetry of the kth iterate of L. We study generic local bifurcations of xed points that are invariant under the action of a compact Lie group that is a reversing k-symmetry group of the map L, on the basis of a normal form approach. We derive normal forms relating the local bifurcations of k-symmetric maps to local steady-state bifurcations of symmetric ows of vector elds. Alternatively, we also discuss the derivation of normal forms entirely within the framework of Taylor expansions of maps. We illustrate our results with some examples.

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“…Self-similarity of an SSL allows us to use the trace map (TM) for researches of the electronic states on it [2,3]. The character of the TM as a non-linear map depends crucially on whether it is conservative or not [7,11,12]. It is known that SSLs with conservative TMs form a special subset, Π c I , of Π s I [8,12].…”

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confidence: 99%

Universalities in one-electron properties of limit quasiperiodic lattices

Endou¹,

Niizeki²,

Fujita³

2004

J. Phys. A: Math. Gen.

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We investigate one-electron properties of one-dimensional self-similar structures called limit quasiperiodic lattices. The trace map of such a lattice is nonconservative in contrast to the quasi-periodic case, and we can determine the structure of its attractor. It allows us to obtain the three new features of the present system: 1) The multi-fractal characters of the energy spectra are universal.2) The supports of the f (α)-spectra extend over the whole unit interval, [0, 1]. 3) There exist marginal critical states.

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A natural class of generalized Fibonacci chains

Cited by 21 publications

References 35 publications

Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices

Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices

Local bifurcations ink-symmetric dynamical systems

Universalities in one-electron properties of limit quasiperiodic lattices

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