Elsevier

Physica D: Nonlinear Phenomena

Volume 240, Issue 21, 15 October 2011, Pages 1785-1790
Physica D: Nonlinear Phenomena

On codings and dynamics of planar piecewise rotations

https://doi.org/10.1016/j.physd.2011.08.001Get rights and content

Abstract

In this paper, we investigate codings and dynamics of the piecewise rotations belonging to a sub-class of piecewise isometries. We show that the irrational set is empty for some piecewise rational rotations under some assumptions, while a piecewise irrational rotation has at least one irrational coding by extending the definition of the coding map onto the entire phase space. We further prove that the cell corresponding to irrational coding is a single point set for a piecewise irrational rotation.

Highlights

► The irrational set is empty for some piecewise rational rotations. ► A piecewise irrational rotation has at least one irrational coding. ► The cell corresponding to irrational coding is a single point set for a piecewise irrational rotation.

Introduction

Piecewise isometries (PWIs) are natural generalizations of interval exchange transformations, which arise in many electronic systems in information technology, both digital and analogue, such as DC–DC buck and boost converters, lossless overflow oscillations in digital filters, bandpass Sigma–Delta modulators, dual billiards, kicked Hamiltonian systems, and so on [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. It is well known that both theories and techniques for interval exchange transformations are hard to be generalized to PWIs, so that, some complex dynamical behaviors of PWIs cannot be described properly in general. In spite of the difficulties, there have been many efforts to study such systems in the past leading to some useful results [2], [12], [13], [14], [15], [16], [7], [17], [18]. In [12], some basic properties of a class of orientation-preserving PWIs are revealed, and a framework is proposed for classifying PWIs with a polygonal partition over a planar region. In [19], the Euclidean PWIs are discussed systematically with focus on geometry and symbolic dynamics, and in [20] the stability of piecewise rotations are discussed. In [21], [17], the stability of periodic points of PWIs defined on the Euclidean space is investigated. The singularity structure and the Devaney-chaos of 2-dimensional invertible PWIs are discussed by classifying singularities into three types with respect to their geometrical properties in [22], [23], [24]. It is proved in [13] that piecewise isometries in arbitrary dimensional Euclidean space have zero topological entropy.

In general, a PWI (in particular, a piecewise rotation) partitions phase space into a number of periodically coded cells, called the rational set, and the remainder is referred to as the exceptional set. These periodic cells are some convex sets consisting of polygons or disks. More precisely, for a piecewise isometry with a rational rotation, every periodic cell is a polygon, while for a piecewise isometry with an irrational rotation, the corresponding periodic cell is a disk. In [25], [26], [27], the tangency between the periodic cells is studied, and it is shown that almost all disk packings for certain families of PWIs have no tangencies. And in [28] it is shown that for a class of PWIs that are irrational rotations, to any disk only a finite number of tangencies are possible.

Coding of orbits for PWIs is investigated in some papers [29], [18], [19]. As a matter of fact, not all piecewise isometries have periodic codings (e.g., a piecewise translation defined on a rectangular region ), and not all piecewise isometries have irrational codings. In [30], [18], the existence of periodic codings is investigated, showing that the singular set determines the existence of periodic codings.

In this paper, we further study codings, especially irrational codings, for planar piecewise rotations. In Proposition 3, we show that a piecewise rotation with a rational rotation angle θ=2πp(p=2,3,4) has no irrational codings (i.e., the irrational set is empty) under some assumptions. This result implies that all non-singular points are periodic for the piecewise rotation (Corollary 3). Our main results are summarized in Theorem 3.1, Theorem 3.2. We show in Theorem 3.1 that a piecewise irrational rotation has irrational codings, and in Theorem 3.2 we show that any cell C(α) corresponding to an irrational coding α is a single point set for a piecewise irrational rotation.

Section snippets

Preliminaries

Definition 2.1

Let XRd be a set, and denote a collection of open subsets of X by M={M1,M2,M3,,Mm}, where X=i=1mM¯i (sometimes, denoted by π(M)). A pair {f,M} is called a piecewise isometry if

  • 1.

    MiMj= for ij,i,j{1,2,,m}, and each Mi is called a partition atom of X;

  • 2.

    the restriction f|M¯i to each partition atom Mi,i=1,2,,m, is an isometric map, i.e., f|M¯i=fi, where {f1,f2,,fm} is a collection of isometric maps.

In particular, if each restriction f|M¯i is a rotation, we say that the pair {f,M} is a

Main results and proofs

Let J be the irrational set of all points with irrational codings J={z|χ¯(z) is irrational}.Theorem 3.1 shows that the irrational set J is non-empty for a piecewise irrational rotation. More importantly, Theorem 3.2 verifies that the cell of any coding αJ is a single point set for a piecewise irrational rotation. Proposition 3, Proposition 4 show that the irrational set J is empty for some piecewise rotations with proper rotation angles and translation vectors.

Discussions

In this paper, we have focused our discussion on the irrational set J which contains all the points whose codings are irrational. The main results in this paper are given in Theorem 3.1, Theorem 3.2. Theorem 3.1 claims the existence of irrational codings for a piecewise irrational rotation by extending the definition of the coding map onto the entire phase space. Theorem 3.2 shows that the cell of an irrational coding contains only a single point for a piecewise irrational rotation.

In [25], [32]

Acknowledgments

This research was jointly supported by NSFC grant 11072136 and Natural Science (Youth) Foundation of Jiangxi Province grant 2007GQS0142. RZY was also supported by a grant from the Innovation Foundation for Graduate Students of Shanghai University. It was also supported by Shanghai Leading Academic Discipline Project (S30104). We are grateful to Prof. Guanrong Chen for his helpful comments. The authors also thank the anonymous referees for their constructive suggestions.

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