Density of invariant disk packings for planar piecewise isometries

Yu, Rong-Zhong; Fu, Xinchu; Shui, Shuliang

doi:10.1080/14689360601054759

Cited by 6 publications

(3 citation statements)

References 12 publications

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“…In the following, we give another main result, which will be proven later in Section 3.2 after preparing some propositions. For convenience of expression, we give some notations introduced in [19]. Two admissible -periodic codings and are said to be equivalent, denoted by ∼ , if there exists a natural number (0 ≤ ≤ ) such that ( ) = , where is the shift map.…”

Section: Tangencies Between Periodic Cells Similar To the Planar Casesmentioning

confidence: 99%

“…In [18] it is shown that the disk packings induced by invariant periodic cells cannot contain certain Apollonian packings, namely, the Arbelos. It is revealed in [19] that for planar irrational piecewise rotations, only finitely many tangencies are possible to any disk. In [17,20], the tangent-free property of disk packing induced by a one-parameter family of PWIs (the Sigma-Delta map and the Overflow map) is investigated, and the results (as summarized in Theorem A below) showed that tangencies between disks in this packing are rare, and this was proven by discussing analytic functions of the parameters.…”

Section: Introductionmentioning

confidence: 99%

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Tangent-Free Property for Periodic Cells Generated by Some General Piecewise Isometries

2013

Discrete Dynamics in Nature and Society

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Iterating an orientation-preserving piecewise isometryTofn-dimensional Euclidean space, the phase space can be partitioned with full measure into the union of the rational set consisting of periodically coded points, and the complement of the rational set is usually called the exceptional set. The tangencies between the periodic cells have been studied in some previous papers, and the results showed that almost all disk packings for certain families of planar piecewise isometries have no tangencies. In this paper, the authors further investigate the structure of any periodic cells for a general piecewise isometry of even dimensional Euclidean space and the tangencies between the periodic cells. First, we show that each periodic cell is a symmetrical body to a center if the piecewise isometry is irrational; this result is a generalization of the results in some previously published papers. Second, we show that the periodic cell packing induced by an invertible irrational planar piecewise rotation, such as the Sigma-Delta map and the overflow map, has no tangencies. And furthermore, we generalize the result to general even dimensional Euclidean spaces. Our results generalize and strengthen former research results on this topic.

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Section: Tangencies Between Periodic Cells Similar To the Planar Casesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tangent-Free Property for Periodic Cells Generated by Some General Piecewise Isometries

2013

Discrete Dynamics in Nature and Society

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“…We note that a stronger result than Theorem 4.4 has recently been given in [12], but for a more restrictive class of piecewise isometries than considered here: for such a PWI, any given disc in the associated disc packing can be tangent to at most finitely many other discs.…”

mentioning

confidence: 99%

Packings induced by piecewise isometries cannot contain the arbelos

Trovati¹,

Ashwin²,

Byott³

2008

Discrete &Amp; Continuous Dynamical Systems - A

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Abstract. Planar piecewise isometries with convex polygonal atoms that are piecewise irrational rotations can naturally generate a packing of phase space given by periodic cells that are discs. We show that such packings cannot contain certain subpackings of Apollonian packings, namely those belonging to a family of Arbelos subpackings. We do this by showing that the unit complex numbers giving the directions of tangency within such an isometric-generated packing lie in a finitely generated subgroup of the circle group, whereas this is not the case for the Arbelos subpackings. In the opposite direction, we show that, given an arbitrary disc packing of a polygonal region, there is a piecewise isometry whose regular cells approximate the given packing to any specified precision.

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