Trace Maps, Invariants, and Some of Their Applications

Baake, Michael; Grimm, Uwe; Joseph, Dieter

doi:10.1142/s021797929300247x

Cited by 74 publications

(99 citation statements)

References 36 publications

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“…In this description the Landauer resistance is given again by R T = |P 12 | 2 with P the product of Λ's, namely P n = Λ(F n )...Λ(1). In this case the invariant I of the underlying dynamical map turns out to be 13 …”

Section: Schrödinger Equ With Fibonacci and Phason-defected δ-Potmentioning

confidence: 99%

Role of phason defects on the conductance of a one-dimensional quasicrystal

Moulopoulos¹,

Roche²

1996

Phys. Rev. B

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We have studied the influence of a particular kind of phason-defect on the Landauer resistance of a Fibonacci chain. Depending on parameters, we sometimes find the resistance to decrease upon introduction of defect or temperature, a behavior that also appears in real quasicrystalline materials. We demonstrate essential differences between a standard tight-binding model and a full continuous model. In the continuous case, we study the conductance in relation to the underlying chaotic map and its invariant. Close to conducting points, where the invariant vanishes, and in the majority of cases studied, the resistance is found to decrease upon introduction of a defect. Subtle interference effects between a sudden phason-change in the structure and the phase of the wavefunction are also found, and these give rise to resistive behaviors that produce exceedingly simple and regular patterns.

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Section: Schrödinger Equ With Fibonacci and Phason-defected δ-Potmentioning

confidence: 99%

Role of phason defects on the conductance of a one-dimensional quasicrystal

Moulopoulos¹,

Roche²

1996

Phys. Rev. B

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“…The corresponding substitution matrix R is obtained from the standard abelianization process, compare [40] and refs. therein, and reads…”

Section: The Silver Mean Substitution Rulementioning

confidence: 99%

A Critical Ising Model on the Labyrinth

Baake

Grimm

Baxter

1995

Perspectives on Solvable Models

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A zero-field Ising model with ferromagnetic coupling constants on the so-called Labyrinth tiling is investigated. Alternatively, this can be regarded as an Ising model on a square lattice with a quasi-periodic distribution of up to eight different coupling constants. The duality transformation on this tiling is considered and the self-dual couplings are determined. Furthermore, we analyze the subclass of exactly solvable models in detail parametrizing the coupling constants in terms of four rapidity parameters. For those, the self-dual couplings correspond to the critical points which, as expected, belong to the Onsager universality class.

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“…In physics, such systems occur e.g. as models for one-dimensional quasicrystals with two composites [30,31].…”

Section: Trace Mapsmentioning

confidence: 99%

“…In Table 3 the generators of the group of matrices with integer coefficients and determinant -+1 and their trace maps are presented [13,31,32]. …”

Section: Trace Mapsmentioning

confidence: 99%

Reversing k-symmetries in dynamical systems

Lamb

Quispel

1994

Physica D: Nonlinear Phenomena

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We generalize the concept of (reversing) symmetries of a dynamical system, i.e. we study dynamical systems that possess symmetry properties only if considered on a proper time scale. In particular (considering dynamical systems with discrete time), the kth iterate of a map may possess more (reversing) symmetries than the map itself. In this way the concepts of (reversing) symmetries and (reversing) symmetry groups are generalized to (reversing) k-symmetries and (reversing) k-symmetry groups.Furthermore, a method is studied for finding orbits that are (k-) symmetric with respect to reversing (k-)-symmetries. Firstly an existing method for finding orbits that are symmetric with respect to one reversing symmetry is extended to the case of more than one reversing symmetry and secondly a generalization of this method to the case of reversing k-symmetries is introduced.Some physically relevant examples of dynamical systems possessing reversing k-symmetries are discussed briefly.

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Trace Maps, Invariants, and Some of Their Applications

Cited by 74 publications

References 36 publications

Role of phason defects on the conductance of a one-dimensional quasicrystal

Role of phason defects on the conductance of a one-dimensional quasicrystal

A Critical Ising Model on the Labyrinth

Reversing k-symmetries in dynamical systems

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