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    Renormalizable two-parameter piecewise isometries. 
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    Renormalizable two-parameter piecewise isometries.

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    Accepted Version (424.2Kb)
    Volume
    26
    Pagination
    063119 - ?
    DOI
    10.1063/1.4954210
    Journal
    Chaos
    Issue
    6
    Metadata
    Show full item record
    Abstract
    We exhibit two distinct renormalization scenarios for two-parameter piecewise isometries, based on 2π/5 rotations of a rhombus and parameter-dependent translations. Both scenarios rely on the recently established renormalizability of a one-parameter triangle map, which takes place if and only if the parameter belongs to the algebraic number field K=Q(5) associated with the rotation matrix. With two parameters, features emerge which have no counterpart in the single-parameter model. In the first scenario, we show that renormalizability is no longer rigid: whereas one of the two parameters is restricted to K, the second parameter can vary continuously over a real interval without destroying self-similarity. The mechanism involves neighbouring atoms which recombine after traversing distinct return paths. We show that this phenomenon also occurs in the simpler context of Rauzy-Veech renormalization of interval exchange transformations, here regarded as parametric piecewise isometries on a real interval. We explore this analogy in some detail. In the second scenario, which involves two-parameter deformations of a three-parameter rhombus map, we exhibit a weak form of rigidity. The phase space splits into several (non-convex) invariant components, on each of which the renormalization still has a free parameter. However, the foliations of the different components are transversal in parameter space; as a result, simultaneous self-similarity of the component maps requires that both of the original parameters belong to the field K.
    Authors
    Lowenstein, JH; Vivaldi, F
    URI
    http://qmro.qmul.ac.uk/xmlui/handle/123456789/13355
    Collections
    • Applied Mathematics [140]
    Language
    eng
    Licence information
    “The final publication is available at http://scitation.aip.org/content/aip/journal/chaos/26/6/10.1063/1.4954210”
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