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Local symmetry dynamics in one-dimensional aperiodic lattices: a numerical study

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Abstract

A unifying description of lattice potentials generated by aperiodic one-dimensional sequences is proposed in terms of their local reflection or parity symmetry properties. We demonstrate that the ranges and axes of local reflection symmetry possess characteristic distributional and dynamical properties, determined here numerically for certain lattice types. A striking aspect of such a property is given by the return maps of sequential spacings of local symmetry axes, which typically traverse few-point symmetry orbits. This local symmetry dynamics allows for a description of inherently different aperiodic lattices according to fundamental symmetry principles. Illustrating the local symmetry distributional and dynamical properties for several representative binary lattices, we further show that the renormalized axis-spacing sequences follow precisely the particular type of underlying aperiodic order, revealing the presence of dynamical self-similarity. Our analysis thus provides evidence that the long-range order of aperiodic lattices can be characterized in a compellingly simple way by its local symmetry dynamics.

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Notes

  1. Since we perform a numerical investigation of the considered aperiodic sequences, throughout the paper the notion of asymptotic convergence is not used in a mathematically strict sense, but refers to the observed behavior for large number of corresponding iterations.

  2. The first and second range subsequences would be \(L_0, L_{0+m}, L_{0+2m}, ...\) and \(L_{1}, L_{1+m}, L_{1+2m}, ...\), respectively. Thus, in order to consider the very first \(m\)-tet bundle \(L_0, L_1, ..., L_m\), one has to include, apart from the single-letter maximal palindromes with length \(L_1 = 1\), also the empty maximal palindromes \(\epsilon \) of length \(L_0 = 0\). The latter are the “interfaces” between two different letters \(A\) and \(B\) in a word. Since \(L_0\) and \(L_1\) are not plotted throughout our analysis, we describe the \(m\)-tets starting from the second bundle \(L_{0+m}, L_{1+m}, ..., L_{m+m}\).

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Acknowledgments

This research has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund. Further financial support by the Greek Scholarship Foundation IKY in the framework of an exchange program with Germany (IKYDA) is also acknowledged.

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Authors and Affiliations

  1. Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 , Hamburg, Germany

    C. Morfonios & P. Schmelcher

  2. The Hamburg Centre for Ultrafast Imaging, Universität Hamburg, Luruper Chaussee 149, 22761 , Hamburg, Germany

    P. Schmelcher

  3. Department of Physics, University of Athens, 15771 , Athens, Greece

    P. A. Kalozoumis & F. K. Diakonos

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  1. C. Morfonios
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Correspondence to C. Morfonios.

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Morfonios, C., Schmelcher, P., Kalozoumis, P.A. et al. Local symmetry dynamics in one-dimensional aperiodic lattices: a numerical study. Nonlinear Dyn 78, 71–91 (2014). https://doi.org/10.1007/s11071-014-1422-1

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