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

Abstract

Iterating an orientation-preserving piecewise isometry 𝑇 of 𝑛-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.

1. Introduction

Piecewise isometries (or PWIs in short) appear in a variety of contexts, including digital filters [1, 2], overflow oscillators [3], Hamiltonian systems, and dual billiards. Recently, some researchers investigated this area from a pure mathematics point of view, and PWIs can be treated as the natural generalizations of interval exchange transformations (IETs) discussed in details in [4–7].

In [8–10], some planar PWIs are studied systematically with a particular focus on geometry and symbolic dynamics. In [11–13], the singularity structure and the Devaney-chaos of 2-dimensional invertible PWIs are discussed by classifying singularity into three types with respect to their geometrical properties. In [14], the stability of periodic cells of planar piecewise rotations is investigated; the results showed that the periodic cells are stable if the planar piecewise rotation is irrational. Moreover, in [15, 16], the stability of periodic points of a general piecewise isometry of Euclidean space ℝ^𝑛 is discussed by using the Euclidean group structure of isometries, and by which we will investigate the structure of periodic cells of general PWIs in this paper. In our opinion, the Euclidean group of isometries is an important and helpful method to study the dynamics of general PWIs of Euclidean space ℝ^𝑑.

Iterating an orientation-preserving piecewise isometry 𝑇 of 𝑛-dimensional Euclidean space ℝ^𝑛, the phase space can be partitioned with full measure into the union of the rational set consisting of all periodically coded points, and its complement is called the exceptional set. For a planar irrational piecewise isometry (the rotation parameter 𝜃 is incommensurable with 𝜋), a periodic cell of all points with the same periodic coding is a disk [14, 17, 18]; hence, the rational set gives rise to a disk packing of the phase space 𝑀. A natural problem is whether any two periodic cells are tangent (see Figure 1).

(a)

(b)

Figure 1 

(a) The illustration to the invariant disks packing of the Sigma-Delta map for the parameter 𝜃=1.75. (b) The local magnification of the square [0,0.5]×[−0.25,0.25], we can see that the disk packing is very “loose” and there is no tangencies between any pair of invariant disks.

For the tangent-free property of disk packing induced by planar PWIs, some results have been presented in previous papers. All these previous results have suggested that the packing is typically very “loose”; that is, there are almost no tangencies between any two disks, and this supports the conjecture that the irrational set has positive measure for an irrational planar piecewise isometry. 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.

Theorem A (see, Fu et al.’s [17, 20]). There is at most a countable set of 𝜃 for which there are any tangencies between periodic disk in the disk packing induced by the overflow map (the Sigma-Delta map). Therefore, there is a full measure set of 𝜃 such that the disk packing induced by the Overflow map (the Sigma-Delta map) has no tangencies.

In [21], it is shown that the result above can be generalized to some quite general planar PWIs under some reasonable constraints (Theorem B below).

Theorem B (see, Fu et al.’s [21]). There is a countable set of 𝜃 in the parameter range for which there are no tangencies between the invariant periodic disks in the disk packing induced by a PWI; therefore, there is a full measure of 𝜃 such that the disk packing has no tangencies.

Although the previous results show that all tangencies may occur at a countable set of parameter values, the possibility that a dense set of parameters do have tangencies is not excluded. Naturally, we have the following questions.(Q1) Whether does the disk packing induced by a planar PWI has no tangencies if the rotation parameter 𝜃 is incommensurable with 𝜋?(Q2) How to characterize the tangent-free property between periodic cells for higher dimensional PWIs?

In this paper, we will focus on the above questions. For Question one (Q1), we will give a positive answer which is one special issue of higher dimensional PWIs if 𝑛 is even. For Question two (Q2), we must at first investigate the structure of the periodic cells induced by a general higher dimensional piecewise isometry. As mentioned in some previous papers [14, 17, 18], a periodic cell of planar PWIs is either a disk (the rotation parameter 𝜃 is incommensurable with 𝜋) or a polygonal region (the rotation parameter 𝜃 is commensurable with 𝜋). However, the structure of a periodic cell of a higher dimensional PWI is more complicated. In Section 3.5, we show that a periodic cell may be represented as the product space of two disks. For generality, we show in Theorem 3 that the closure of any periodic cell is symmetrical about the center (the unique fix point) if 𝑛 is even and the piecewise isometry is irrational. Moreover, we obtain the main result (Theorem 8, Section 3.2) about the tangent-free property of the periodic cell packing induced by higher dimensional PWIs.

In particular, we confirm that for any invertible planar irrational piecewise rotations, such as the Sigma-Delta map and the Overflow map, there are no tangencies between periodic disks in the disk packing. Our results are more general and more precise than some previous known results presented in [17, 20, 21] (e.g., Theorem A and Theorem B), and our proofs are simpler.

The rest of this paper is organized as follows. In Section 2, we will introduce the preliminaries about piecewise isometries, including the definitions of PWIs, codings, and cells. In Section 3, we investigate the periodic cells and their tangent-free properties, which are the main part of this paper. And some examples are also provided in this section, and it can be found that it is very easy to verify that these example systems have no tangencies according to our results. Furthermore, a 4-dimensional PWI is investigated in Section 3.5, and it is shown that any periodic cell is the product space of two disks, and there exist no tangencies between two periodic cells. Finally, in Section 4, we present some remarks and discussions about our research in this paper.

2. Preliminaries

The structure of the Euclidean groups plays an important role for the discussion of high-dimensional piecewise isometries. We briefly recall some notations about piecewise isometries and the structure of the Euclidean groups as papers [15, 22] in this part.

The Euclidean group 𝔼(𝑛)=𝕆(𝑛)⋉ℝ^𝑛 is the semidirect product of the orthogonal group 𝕆(𝑛) and ℝ^𝑛, where 𝕆(𝑛)={𝐴∈𝔾𝕃(𝑛,ℝ)∶𝐴𝐴^𝑡=𝐼𝑑}. Denote an element of 𝑔∈𝔼(𝑛) as a pair (𝐴,𝑣) where 𝐴∈𝕆(𝑛) and 𝑣∈ℝ^𝑛. The group multiplication is given by (𝐴₁,𝑣₁)⋅(𝐴₂,𝑣₂)=(𝐴₁𝐴₂,𝐴₁𝑣₂+𝑣₁). The subgroup 𝕊𝕆(𝑛)⊂𝕆(𝑛) called special orthogonal group consists of orientation-preserving transformations, defined by 𝕊𝕆(𝑛)={𝐴∈𝕆(𝑛)∣det(𝐴)=1}. The special Euclidean group 𝕊𝔼(𝑛) is the semidirect product of 𝕊𝕆(𝑛) and ℝ^𝑛.

Let 𝑀 be a connected subset of Euclidean space ℝ^𝑛 and let ℳ={𝑀₀,𝑀₁,…,𝑀_𝑟−1} be a finite collection of connected open convex set. We call ℳ a partition of 𝑀 and each set 𝑀_𝑖 a partition atom if the following conditions hold:(1)𝑀=𝑀₁∪⋯∪𝑀_𝑟−1; (2)𝑀_𝑖∩𝑀_𝑗=∅ for 𝑖≠𝑗.A map 𝑇∶𝑀⧵𝒟↦𝑀 is called a piecewise isometry on 𝑀 if 𝑇(𝑥)=𝑇_𝑖(𝑥), 𝑥∈𝑀_𝑖, where 𝑇_𝑖∈𝔼(𝑛). And 𝒟=∪_𝑖≠𝑗(𝑀_𝑖∩𝑀_𝑗) is said to be the discontinuity set. For convenience, we denote by 𝔼(ℳ) the set of all piecewise isometries and by 𝕊𝔼(ℳ) the set of all orientation-preserving piecewise isometries on the partition ℳ.

In this paper, we consider the case where the orientation-preserving piecewise isometries (𝑇_𝑖∈𝕊𝔼(𝑛),𝑖=0,1,…,𝑟−1) and each of the partition atoms 𝑀_𝑖(𝑖=0,1,…,𝑟−1) is an open convex polyhedron of ℝ^𝑛 which is surrounded by 𝑚_𝑖 number of (𝑛−1)-dimensional hyperplanes taking the form 𝐧_𝑖𝑗⋅𝑥=𝑐_𝑖𝑗(𝑗=1,2,…,𝑚_𝑖), where 𝐧_𝑖𝑗 representing the unit normal vector of the hyperplane is an element of the projective space ℙ^𝑛−1 (i.e., ignoring the difference between ±𝐧), 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈ℝ^𝑛 is a column vector, and 𝑐_𝑖𝑗∈ℝ is a constant. We say that two unit vectors 𝐧 and 𝐧^′ are equal means that 𝐧=𝐧^′ or 𝐧=−𝐧^′. In [18], the authors characterized similarly the boundary of partition atoms for planar PWIs.

Let 𝜕ℳ=∪^𝑟_𝑖=0𝜕𝑀_𝑖 be the boundary of the partition ℳ and 𝒟^′=∪_𝑖≠𝑗(𝑇(𝑀_𝑖)∩𝑇(𝑀_𝑗)) the backward discontinuity set. A piecewise isometry is said to be invertible if 𝑇(𝑀_𝑖)∩𝑇(𝑀_𝑗)=∅(𝑖≠𝑗) and ∪^𝑟−1_𝑖=0𝑇(𝑀_𝑖)=𝑀. Obviously, all of the sets 𝜕𝑀, 𝒟, and 𝒟^′ consist of a finite number of (𝑛−1)-dimensional polyhedral regions for invertible PWIs.

In the following, for convenience, we say 𝐿 is a (𝑛−1)-dimensional hyperplane of 𝜕ℳ (or 𝒟,𝒟^′), if 𝐿⊂𝜕ℳ (or 𝒟,𝒟^′) is a (𝑛−1)-dimensional polyhedral region of 𝜕ℳ (or 𝒟,𝒟^′). For a (𝑛−1)-dimensional hyperplane 𝐿, let 𝐧(𝐿) be a unit normal vector of 𝐿 and 𝐍(ℳ)=∶∪_{𝐿⊂𝜕ℳ}{𝐧(𝐿)}=∪^𝑟−1_𝑖=0∪^𝑚𝑖_𝑗=1{𝐧_𝑖𝑗} the set of all unit normal vectors of hyperplanes of ℳ. Similarly, we define 𝐍(𝒟) and 𝐍(𝒟^′).

The partition ℳ={𝑀₀,𝑀₁,…,𝑀_𝑟−1} of 𝑀 associated with a piecewise isometry 𝑇 gives rise to a natural one-sided coding map. Let ℰ⁻=∪^∞_𝑘=0𝑇^−𝑘(𝒟) and 𝒜={0,1,…,𝑟−1} the alphabet set; a map 𝜒∶𝑀⧵ℰ⁻↦𝒜^ℕ is called a coding map, if for any point 𝑥∈𝑀⧵ℰ⁻, 𝜒(𝑥)_𝑗=𝜔_𝑗,iff𝑇^𝑗(𝑥)∈𝑀_𝜔𝑗,𝑗=0,1,2,…. (1) And we call the infinite sequence 𝜔=𝜔₀𝜔₁𝜔₂… the coding of the orbit {𝑇^𝑗(𝑥)}^∞_𝑗=0 (or the coding of the point 𝑥 for simplicity). An infinite sequence 𝜔=𝜔₀𝜔₁… is said to be admissible, if there exists a point 𝑥 such that 𝜒(𝑥)=𝛼. Similarly, a finite sequence 𝛽=𝛽₀𝛽₁⋯𝛽_𝑛−1 with length 𝑛 is said to be admissible, if there exists a point 𝑥 such that 𝛽 is the preword of 𝜒(𝑥). A coding 𝜔 is said to be periodic, if there exists a natural number 𝑝 such that 𝜔_𝑝+𝑘=𝜔_𝑘 for all 𝑘∈ℕ, and we denote the coding 𝜔 by 𝜔=𝒫(𝜔₀𝜔₁⋯𝜔_𝑝−1), where 𝒫(⋅) represents concatenation operation. A coding 𝜔 is said to be rational, if it is eventually periodic. Otherwise, the coding 𝜔 is said to be irrational. In fact, for an invertible piecewise isometry, a rational coding is periodic. Sometimes, we may denote 𝑗-dimensional coding as paper [23] for convenience. Namely, let the alphabet set 𝒜={𝛼₀,𝛼₁,…,𝛼_𝑟−1} and every element 𝛼_𝑖=(𝛼_𝑖1,𝛼_𝑖2,…,𝛼_𝑖𝑗) is a 𝑗-dimensional vector.

Definition 1. If a coding 𝜔 (finite or infinite) is admissible, we call the set of all points following the same coding 𝜔 a cell, denoted by 𝐶(𝜔).

The structure of cells of planar PWIs has been investigated by some researchers; some of the results are stated in the following proposition.

Proposition 2 (see [9, 14, 17, 18, 24]). For a planar piecewise isometry with convex polygonal partition, a cell 𝐶(𝜔) is a convex set. More precisely, it has interior if and only if 𝜔 is rational, and if 𝜔 is irrational, then the cell is either a point or a line segment.

In [25], it is shown that for an irrational piecewise rotation, if the admissible coding 𝜔 is irrational, then the cell 𝐶(𝜔) consists of only a point.

Let 𝑇_𝜔=𝑇_𝜔𝑚−1⋅𝑇_𝜔𝑚−2⋯𝑇_𝜔0 and 𝐴_𝜔=𝐴_𝜔𝑚−1⋅𝐴_𝜔𝑚−2⋯𝐴_𝜔0 for every finite coding. A piecewise isometry 𝑇∈𝕊𝔼(ℳ) is said to be incommensurate if Fix𝐴_𝜔={0} for every finite word 𝜔 in the alphabet 𝒜. As mentioned in [15], if 𝑛 is even then almost every piecewise isometry in 𝕊𝔼(ℳ) is incommensurate.

3. Periodic Cells and the Tangent-Free Property

Based on the fact revealed by Mendes and Nicol in [15] that if 𝑛 is odd, then almost every piecewise isometry in 𝕊𝔼(ℳ) has no recurrent points with rational coding (consequently, has no periodic points), and we will only discuss the periodic cells and the tangent-free property of the even dimensional piecewise isometries in this part.

3.1. Periodic Cells

If 𝑛 is even and 𝜔=𝜔₀⋯𝜔_𝑚−1 is admissible, then the linear part 𝐴_𝜔 of the map 𝑇_𝜔=(𝐴_𝜔,𝑣) can be orthogonally diagonalized as diag{𝐵₁,𝐵₂,…,𝐵_𝑛/2} under one orthogonal basis, where 𝐵_𝑗=(cos𝜃_𝑗,sin𝜃_𝑗;−sin𝜃_𝑗,cos𝜃_𝑗), 𝜃_𝑗∈[0,2𝜋). It is obvious that an even dimensional orientation-preserving PWI is incommensurate if and only if all 𝜃_𝑗 are incommensurable with 𝜋 in diagonalization representation of 𝐴_𝜔 for every finite word 𝜔. Furthermore, we say the PWI is irrational if it is incommensurable and all 𝜃_𝑗 are rationally independent in diagonalization representation of 𝐴_𝜔 for every finite word 𝜔.

Let 𝜔=𝒫(𝛼) be an admissible periodic coding, where 𝛼=𝜔₀𝜔₁⋯𝜔_𝑚−1 is of length 𝑚, then the self-map 𝑇_𝛼∶𝐶(𝜔)↦𝐶(𝜔) can be represented as the product of 𝑛/2 number of planar rotations since the linear part 𝐴_𝛼 can be orthogonally diagonalized. Namely, every point of 𝐶(𝜔) can be rewritten as complex number coordinate 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛/2), and the map 𝑇_𝛼 can be represented as 𝑇_𝛼=𝑇^∼₁×𝑇^∼₂×⋯×𝑇^∼_𝑛/2, 𝑇^∼_𝑗(𝑧_𝑗)=𝑒^{𝑖⋅𝜃𝑗}+𝑐_𝑗,𝑗=1,2,…,𝑛/2. We have the following result.

Theorem 3. Suppose 𝑛 is even, 𝑇∈𝕊𝔼(ℳ), then every periodic cell is convex and of positive Lebesgue measure. Furthermore, if 𝑇 is irrational then the closure of the periodic cell is centrosymmetric with respect to a unique fixed point.

Proof. The proof of the convexity of the cell is similar to the one given in [18]; here we almost repeat the procedure. Let 𝜔=𝒫(𝛼) be an admissible periodic coding, where 𝛼=𝜔₀𝜔₁⋯𝜔_𝑚−1 is of length 𝑚, then the cell 𝐶(𝜔) can be represented as 𝐶(𝜔)=(𝑥)=𝑥∶𝜒=𝜔𝑥∶𝑥∈𝑀_𝜔0,𝑇^𝑘(𝑥)∈𝑀_𝜔𝑘fo𝑟𝑘∈ℕ=𝑀_𝜔0∩𝑇⁻¹𝑀_𝜔1∩𝑇⁻²𝑀_𝜔1=∩⋯_∞𝑘=0𝑇^−𝑘𝑀_𝜔𝑘. (2) At the same time, we have 𝑇^−𝑘𝑀_𝜔𝑘=𝑇_𝜔⁻¹₁∘⋯∘𝑇_𝜔⁻¹_𝑘𝑀_𝜔𝑘, (3) where 𝑀_𝜔𝑘 is convex and each of the maps {𝑇_𝜔⁻¹₁,𝑇_𝜔⁻¹₂,…,𝑇_𝜔⁻¹_𝑘} is an isometry, then the set 𝑇^−𝑘(𝑀_𝜔𝑘) is convex, consequently, the cell 𝐶(𝜔) is convex.
Since the map 𝑇_𝛼∶𝐶(𝜔)↦𝐶(𝜔) is isometric and 𝑇_𝛼(𝐶(𝜔))=𝐶(𝜔), the cell 𝐶(𝜔) at least contains a fixed point denoted by 𝑝^∗. Let 1𝑑=2min_{0≤𝑖≤𝑝−1}𝜌𝑇^𝑖𝑝^∗,𝜕ℳ, (4) then 𝐵(𝑝^∗,𝑑)⊂𝐶(𝜔), which implies that the cell 𝐶(𝜔) is of positive Lebesgue measure.
Now, we prove the latter claim. Suppose, without loss of generality, the map 𝑇_𝛼 is represented as the product of 𝑛/2 planar rotations and every point of 𝐶(𝜔) has the form of complex coordinate. Furthermore, assume the unique fixed point equals 𝟎 of the map 𝑇_𝛼 since the piecewise isometry 𝑇 is irrational. We will show that 𝑧={𝑧₁,𝑧₂,…,𝑧_𝑛/2}∈𝐶(𝜔) implies that every point 𝑧^′={𝑧^′₁,𝑧^′₂,…,𝑧^′_𝑛/2}∈𝐶(𝜔) with ‖𝑧^′_𝑖‖=‖𝑧_𝑖‖(1≤𝑖≤𝑟₂), (i.e., 𝐓^𝑛/2⊂𝐶(𝜔)), where 𝐓^𝑛/2=𝑧₁𝑒^𝛼1,𝑧₂𝑒^𝛼2,…,𝑧_𝑟2𝑒^𝛼𝑛/2∣𝛼_𝑖∈[0,2𝜋),𝑖=1,2,…,𝑟₂ (5) is the (𝑛/2)-dimensional torus in the phase space. For 𝑧={𝑧₁,𝑧₂,…,𝑧_𝑛/2}∈𝐶(𝜔), and there exists a point 𝑧^⋆={𝑧^⋆₁,𝑧^⋆₂,…,𝑧^⋆_𝑛/2}∈𝐶(𝜔) such that for any 𝜖>0, 𝑧−𝑧^⋆<𝜖𝑛. (6) Let 𝐓₁^𝑛/2=𝑧^⋆₁𝑒^𝛼1,𝑧^⋆₂𝑒^𝛼2,…,𝑧^⋆_𝑟2𝑒^𝛼𝑛/2∣𝛼_𝑖∈[),𝑛0,2𝜋𝑖=1,2,…,2 (7) be another torus in the phase space. At the same time, the orbit Orb⁺(𝑧^⋆) under the map 𝑇_𝛼∶𝐶(𝜔)→𝐶(𝜔) is dense in the torus 𝐓₁^𝑛/2 because the map 𝑇 is irrational. For any point 𝑧^′=(𝑧^′₁,…,𝑧^′_𝑛/2)∈𝐓^𝑛/2,then 𝑧^′′=(‖𝑧^⋆₁/𝑧^′₁‖𝑧^′₁,…,‖𝑧^⋆_𝑖/𝑧^′_𝑖‖𝑧^′_𝑖,…,‖𝑧^⋆_𝑛/2/𝑧^′_𝑛/2‖𝑧^′_𝑛/2)∈𝐓₁^𝑛/2. Consequently, there exists a natural number 𝑘 such that 𝑇^𝑘_𝛼𝑧^⋆<𝜖−𝑧2. (8) Because 𝑧^′_𝑖−𝑧^⋆_𝑖𝑧^′_𝑖⋅𝑧^′_𝑖=𝑧^′_𝑖−𝑧^⋆_𝑖𝑧^′_𝑖⋅𝑧^′_𝑖=𝑧^′_𝑖−𝑧^⋆_𝑖=𝑧_𝑖−𝑧^⋆_𝑖≤𝑧_𝑖−𝑧^⋆_𝑖 (9) for all 𝑖=1,2,…,𝑛/2, from (9) and (6) we obtain 𝑧^′−𝑧≤^𝑛/2_𝑖=1𝑧^′_𝑖−𝑧^⋆_𝑖𝑧^′_𝑖⋅𝑧^′_𝑖≤^𝑛/2_𝑖=1𝑧_𝑖−𝑧^⋆_𝑖≤𝜖2. (10) Consequently, from (8) and (10), we have 𝑇^𝑘_𝛼𝑧^⋆−𝑧^′≤𝑇^𝑘_𝛼𝑧^⋆−𝑧+𝑧^′<𝜖−𝑧2+𝜖2=𝜖, (11) namely, 𝑧^′∈𝐶(𝜔). Consequently, the periodic cell is centrosymmetric with respect to the unique fixed point.

Remark 4. According to Theorem 3, since the set 𝐶(𝜔) is convex, then 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛/2)∈𝐶(𝜔) implies 𝑧^′=(𝑧^′₁,𝑧^′₂,…,𝑧^′_𝑛/2)∈𝐶(𝜔) with ‖𝑧^′_𝑖‖≤‖𝑧_𝑖‖ for all 1≤𝑖≤𝑛/2. Let Θ(𝑧)={𝑧^′∣‖𝑧^′_𝑖‖≤‖𝑧_𝑖‖,𝑖=1,…,𝑛/2} be the product of 𝑛/2 number of discs, then the closure of the periodic cell 𝐶(𝜔) is the union of some such sets as Θ(𝑧).

Let 𝛾_𝑖=max{‖𝑧_𝑖‖∣(𝑧₁,…,𝑧_𝑖,…,𝑧_𝑛/2)∈𝐶(𝜔)} for all 𝑖=1,2,…,𝑛/2, and Θ={𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛/2)∣‖𝑧_𝑖‖≤𝛾_𝑖,𝑖=1,2,…,𝑛/2}. According to the analysis above, we can obtain that 𝐶(𝜔)=Θ, which is the product of 𝑛/2 discs. We have the following corollary.

Corollary 5. Suppose 𝑛 is even, 𝑇∈𝕊𝔼(ℳ) is irrational, if there exists a point 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛/2) with ‖𝑧_𝑖‖=‖𝛾_𝑖‖(𝑖=1,…,𝑛/2) such that 𝑧∈𝐶(𝜔), then 𝐶(𝜔)=Θ is the product of 𝑛/2 number of discs.

Note that the result in the above theorem is a generalization of the results in [14, 17, 18], where it is shown that a periodic cell is a disk or a symmetric polygonal region. Obviously, for a 3-dimensional piecewise isometry with polygonal partition, a periodic cell must not be a sphere. In Section 3.5, we will show that a periodic cell of the product Goetz map, which is a 4-dimensional PWI, may be represented as 𝐶(𝜔)×𝐶(𝜛), where 𝜔, 𝜛 are two admissible periodic codings under the Goetz map. For higher-dimensional PWIs, the structure of a periodic cell is more complicated. In the following, we further investigate the periodic cells.

At the same time, for higher-dimensional PWIs, their periodic cells have some similar properties as the ones of planar PWIs.

Proposition 6. For a piecewise isometry 𝑇∈𝕊𝔼(ℳ), if 𝐶(𝜔) and 𝐶(𝜛) are two different periodic cells, then there must exist a natural number 𝑘 such that 𝑇^𝑘(𝐶(𝜔)) and 𝑇^𝑘(𝐶(𝜛)) are contained in different partition atoms 𝑀_𝑖 and 𝑀_𝑗, respectively; that is, they will be separated by an (𝑛−1)-dimensional boundary 𝐿⊂𝒟.

We must point out that two periodic cells are separated by an (𝑛−1)-dimensional boundary 𝐿⊂𝒟 as above means that (𝐧⋅𝑥^⋆−𝑐)⋅(𝐧⋅𝑥^⋆⋆−𝑐)<0, where 𝑥^⋆ and 𝑥^⋆⋆ are the centers of two periodic cells, respectively, and 𝐧⋅𝑥=𝑐 is the equation of hyperplane 𝐿. Note that Proposition 6 is a generalization of Lemma  1 in [17], and the proof is similar. More importantly, for an invertible piecewise isometry, we have another similar property stated as follows.

Proposition 7. For a piecewise isometry 𝑇∈𝕊𝔼(ℳ), if 𝐶(𝜔) and 𝐶(𝜛) are two different periodic cells, then there must exist a natural number 𝑘 such that 𝑇^−𝑘(𝐶(𝜔)) and 𝑇^−𝑘(𝐶(𝜛)) are separated by an (𝑛−1)-dimensional polyhedral region 𝐿^′⊂𝒟^′.

Proof. Assume that the periodic codings 𝜔 and 𝜛 have periods of 𝑁₀ and 𝑁₁, respectively, and further assume, without loss of generality, that 𝑁₁≥𝑁₀. Consider the preimages of the pair of periodic cells 𝐶(𝜔) and 𝐶(𝜛), if 𝑇^−𝑘−1(𝐶(𝜔)) and 𝑇^−𝑘−1(𝐶(𝜔)) locate in different atoms, then 𝑇^−𝑘(𝐶(𝜔)) and 𝑇^−𝑘(𝐶(𝜔)) are separated by a hyperplane 𝐿^′⊂𝒟^′. In fact, it can be verified that such 𝑘 (less than [𝑁₀,𝑁₁], the least common multiple of 𝑁₀ and 𝑁₁) exists since 𝜔≠𝜛.

3.2. Tangencies between Periodic Cells

Similar to the planar cases [17, 20, 21], for a 𝑛-dimensional PWI (𝑛≥4 is even), we say that the set 𝑋=⋃𝐶(𝜔), the union of all periodic cells, is a periodic cell packing of the phase space 𝑀 by the 𝑑-dimensional PWI. We consider similarly the tangent property between any two periodic cells. Assume that the boundaries of the periodic cells 𝐶₁ and 𝐶₂ are smooth at the intersection point 𝑥, and we say the two periodic cells 𝐶₁ and 𝐶₂ are tangent to each other at the point 𝑥 if(1)int(𝐶₁)∩int(𝐶₂)=∅; (2)𝑥∈𝜕𝐶₁∩𝜕𝐶₂; (3)𝐧₁(𝑥)=±𝐧₂(𝑥), where 𝐧_𝑖(𝑥)(𝑖=1,2) represent the unit normal vectors of surface 𝜕𝐶_𝑖 at the point 𝑥.The above three conditions are obvious; the fact that any two different periodic cells do not overlap implies Condition (1); and Conditions (2) and (3) guarantee they are tangent at 𝑥. Let 𝒯𝐶₁,𝐶₂=𝑥∣𝑥isatangentpointof𝐶₁and𝐶₂ (12) be the mutually tangent set. We say the periodic cells 𝐶₁ and 𝐶₂ are tangent, if 𝒯(𝐶₁,𝐶₂) is nonempty. We say the packing is tangent-free, if any two periodic cells are not tangent.

In fact, the mutually tangent set 𝒯(𝐶₁,𝐶₂) may be a single point or an open straight line segment for a planar piecewise isometry, and it may be a 𝑘-dimensional (𝑘=0,1,…,𝑛−1) hyperplane for a general piecewise isometry of 𝑛-dimensional Euclidean space ℝ^𝑛. In the following, we give another main result, which will be proven later in Section 3.2 after preparing some propositions.

Theorem 8. Suppose 𝑛 is even, 𝑇∈𝕊𝔼(ℳ) is invertible and incommensurate. If for any admissible finite coding 𝜔=𝜔₀𝜔₁⋯𝜔_𝑘 and any 𝐧, 𝐧^′ of 𝐍(𝒟^′), one has 𝐴_𝜔⋅𝐧≠𝐧^′, then there are no tangencies between periodic cells in the periodic cells packing induced by 𝑇.

For a class of PWIs, the backward discontinuity set consists of one hyperplane or finitely many parallel hyperplanes. Namely, 𝐍(𝒟^′) consists of only one element without considering the sign. Then we get the following corollary.

Corollary 9. Suppose 𝑛 is even, 𝑇∈𝕊𝔼(ℳ) is invertible and incommensurate. If 𝐍(𝒟^′) consists of only one element, then there are no tangencies between periodic cells in the periodic cells packing induced by 𝑇.

For convenience of expression, we give some notations introduced in [19]. Two admissible 𝑛-periodic codings 𝜔 and 𝜛 are said to be equivalent, denoted by 𝜔∼𝜛, if there exists a natural number 𝑖(0≤𝑖≤𝑛) such that 𝜎^𝑖(𝜔)=𝜛, where 𝜎 is the shift map. Two points of 𝑀 are said to be equivalent, if their corresponding symbolic sequences are equivalent. Similarly, we can define the equivalence of two cells 𝐶(𝜔) and 𝐶(𝜛). We denote by GO(𝐶(𝜔)) the set of all the cells which are equivalent to 𝐶(𝜔) under the above equivalence relation and call GO(𝐶(𝜔)) the great orbit of 𝐶(𝜔). According to Propositions 6 and 7, we have the following result.

Proposition 10. Suppose 𝑛 is even, 𝑇∈𝕊𝔼(ℳ) is invertible, if two different periodic cells 𝐶(𝜔) and 𝐶(𝜛) are tangent to each other, then there exist two hyperplanes 𝐿₁⊂𝒟^′, 𝐿₂⊂𝒟 and periodic cells 𝐶₁,𝐶₂∈GO(𝐶(𝜔)),𝐶^′₁,𝐶^′₂∈GO(𝐶(𝜛)) such that 𝐶_𝑖 and 𝐶^′_𝑖(𝑖=1,2) are tangent to each other, and the mutually tangent set 𝒯(𝐶_𝑖,𝐶^′_𝑖) is contained in 𝐿_𝑖 for 𝑖=1,2, respectively.

From the above result, we further have the following.

Proposition 11. Suppose 𝑛 is even, 𝑇∈𝕊𝔼(ℳ) is invertible, if two different periodic cells 𝐶(𝜔) and 𝐶(𝜛) are tangent to each other, then there exist two vectors 𝐧₁ and 𝐧₂ of 𝐍(ℳ) and an admissible coding 𝛼=𝛼₀⋯𝛼_𝑘−1 with length 𝑘 such that 𝐴_𝛼⋅𝐧₁=𝐧₂.

Proof. Without loss of generality, we assume that the mutually tangent set is contained in the hyperplane 𝐿^′⊂𝒟^′ (i.e., 𝒯(𝐶(𝜔),𝐶(𝜛))⊂𝐿^′) with unit normal vector 𝐧^′₁ according to Proposition 10. Then, there is a hyperplane 𝐿₁⊂𝜕ℳ with unit normal vector 𝐧₁ such that 𝐴_𝛼0⋅𝐧₁=𝐧^′₁. From Proposition 6, there exists a natural number 𝑘 such that the periodic cells 𝑇^𝑘−1(𝐶(𝜔)) and 𝑇^𝑘−1(𝐶(𝜛)) are in the different partition atoms 𝑀_𝑖 and 𝑀_𝑗, respectively, and 𝑇^𝑘−1(𝒯(𝐶(𝜔),𝐶(𝜛)))⊂𝐿⊂𝒟, where 𝐿=𝑀_𝑖∩𝑀_𝑗. Let 𝐧₂ be the unit normal vector of the hyperplane 𝐿, then there exists a finite admissible coding 𝛼₁𝛼₂⋯𝛼_𝑘−1 such that 𝐴_𝛼𝑘−1⋯𝐴_𝛼1⋅𝐧^′₁=𝐧₂. Consequently, 𝐴_𝛼⋅𝐧₁=𝐴_𝛼𝑘−1⋯𝐴_𝛼1𝐴_𝛼0⋅𝐧₁=𝐧₂. (13) The proof is therefore complete.

We are now ready to give the proof of Theorem 8.

Proof of Theorem 8. Firstly, we will reveal that there exists a natural number 𝑘₀ such that for any 𝐧 of 𝐍(ℳ), there exists an admissible finite coding 𝛼=𝛼₀⋯𝛼_𝑘−1 with length 𝑘≤𝑘₀ such that 𝐴_𝛼⋅𝐧∈𝐍(𝒟^′). Since every partition atom is surrounded by finite (𝑛−1)-dimensional hyperplanes, then 𝐍(ℳ) consists of finite elements, and let 𝑘₀=♯𝐍(ℳ) be the cardinal number. Because the isometry is invertible, for any (𝑛−1)-dimensional hyperplanes 𝐿⊂𝜕ℳ, we can get 𝑇(𝐿)⊂𝜕ℳ∪𝒟^′. If there exists an admissible finite coding 𝛼=𝛼₀⋯𝛼_𝑘−1 with length 𝑘>𝑘₀ such that 𝑇^𝑖(𝐿)⊂𝜕ℳ for all 𝑖=0,…,𝑘−1, then 𝐴(𝑖)⋅𝐧∈𝐍(ℳ) for all 𝑖=0,…,𝑘−1, where 𝐴(𝑖)=𝐴_𝛼0⋯𝐴_𝛼𝑖−1. Since the piecewise isometry is incommensurate, all 𝐴(𝑖)⋅𝐧 are different from each other. This implies that ♯𝐍(ℳ)≥𝑘>𝑘₀, a contradiction.
Suppose that there exist two periodic cells 𝐶₁ and 𝐶₂ which are tangent to each other, then, in view of Proposition 11, there is two unit normal vectors 𝐧₁ of a hyperplane 𝐿^′⊂𝒟^′ and 𝐧 of a hyperplane 𝐿⊂𝒟, and an admissible finite coding 𝛽=𝛽₀𝛽₁⋯𝛽_𝑘−1, such that 𝐴_𝛽⋅𝐧₁=𝐧. (14) Via the typical invertibility condition, we can obtain that there exists an admissible finite coding 𝛼=𝛼₀𝛼₁⋯𝛼_𝑙−1 and a unit normal vector 𝐧₂∈𝐍(𝒟^′) such that 𝐴_𝛼⋅𝐧=𝐧₂. (15) According to (14) and (15), we have 𝐴_𝛽⋅𝐴_𝛼⋅𝐧₁=𝐧₂. (16) Let 𝜔=𝛼𝛽=𝛼₀⋯𝛼_𝑙−1𝛽₀⋯𝛽_𝑘−1 with length 𝑘+𝑙, then we get 𝐴_𝜔⋅𝐧₁=𝐧₂ for two elements of 𝐍(𝒟^′), which contradicts the results as above. Therefore, there are no tangencies between periodic cells.

3.3. Applications to Planar PWIs

For planar piecewise rotations, the corresponding results are very explicit. Let 𝜌_𝑗=𝑒^{𝑖⋅𝜃𝑗}(𝑗=0,1,…,𝑟−1) be the linear part of the map 𝑇|_𝑀𝑗, then we have the following corollary.

Corollary 12. Suppose ℳ⊂ℝ², 𝑇∈𝕊𝔼(ℳ) is invertible, if 𝜌^𝑘0₀𝜌^𝑘1₁⋯𝜌^𝑘𝑟−1_𝑟−1=1 with 𝑘_𝑖≥0(0≤𝑖≤𝑟−1) implies that 𝑘_𝑖=0, and if 𝐍(𝒟^′) consists of only one element, then the invariant disk packing is tangent-free.

We note that the above corollary is just a simple case of Corollary 9.

In particular, if an invertible planar piecewise rotation has the common irrational rotation angle 𝜃, that is, any restricted map 𝑇|_𝑀𝑗 has the same linear part 𝜌=𝑒^𝑖⋅𝜃, then we have the following corollary, which strengthens the results in [17, 20, 21].

Corollary 13. If an invertible planar piecewise rotation has the common irrational rotation angle 𝜃(𝜃/𝜋∈ℚ^𝐶) and 𝐍(𝒟^′) consists of only one element, then the invariant disk packing induced by the map is tangent-free.

3.4. Goetz Map

Now we consider a map called the (bounded) Goetz map 𝑇_𝜃∶𝑀↦𝑀 with parameter 𝜃∈(𝜋/2,2𝜋/3) as follows: 𝑇_𝜃(𝑧)=𝑒^{2(𝜋−𝜃)𝑖}⋅𝑧+𝑐₁if𝑧∈𝑀₁;𝑒^{(𝜋−𝜃)𝑖}⋅𝑧+𝑐₂if𝑧∈𝑀₂. (17) Here, 𝑐₁=1/sin𝜃, 𝑐₂=((1+2cos𝜃)/sin𝜃)𝑒^−𝜃𝑖, and the partition atoms 𝑀₁ and 𝑀₂ can be written as follows (see Figure 2): 𝑀₁=[𝐴,𝐵,𝐶]=−2cot𝜃⋅𝑒^{(𝜋−𝜃)𝑖}1,0,,𝑀sin𝜃₂=[]=1𝐵,𝐶,𝐷,𝐸0,,1sin𝜃⋅sin𝜃1+𝑒^−3𝜃𝑖,1+2cos𝜃sin𝜃⋅𝑒^−𝜃𝑖. (18)(19)

Figure 2 

The piecewise isometric map (17) has two partition atoms 𝑀1 and 𝑀2, and the linear parts of the restricted maps are 𝜌1=𝑒2(𝜋−𝜃)𝑖 and 𝜌2=𝑒(𝜋−𝜃)𝑖. At the same time, the map is invertible, that is, 𝑇(𝑀1)∪𝑇(𝑀2)=𝑀.

The piecewise isometric map is invertible, and the backward discontinuity set 𝒟^′ consists of only the line segment 𝐴𝐹; furthermore, all of the boundary line segments of the partition atoms 𝑀₁ and 𝑀₂ will turn on to the line segment 𝐴𝐹 under finite iterations of the map 𝑇; that is, the isometric map 𝑇 satisfies the typical invertibility condition. At the same time, the linear parts of the restricted maps are 𝜌₁=𝑒^{2(𝜋−𝜃)𝑖} and 𝜌₂=𝑒^{(𝜋−𝜃)𝑖}, and obviously, when 𝜃/𝜋 is irrational, the equality 𝜌^𝑘1₁𝜌^𝑘2₂=1,𝑘₁≥0,𝑘₂≥0 (20) holds if and only if 𝑘₁=𝑘₂=0. Then, from Corollary 12, we have the following.

Proposition 14. For the piecewise isometry defined by (17), if the parameter 𝜃 is incommensurable with 𝜋, then the invariant disk packing is tangent-free. (See Figure 3).

Figure 3 

The illustration to invariant disk packing induced by the map (17), where the parameter 𝜃=1.78. We can observe that all the periodic disks are tangent-free.

3.5. The Product Goetz Map

Now we investigate the so-called product Goetz map 𝐹_𝜃=𝑇_𝜃×𝑇_𝜃∶𝑀×𝑀↦𝑀×𝑀 which is a 4-dimensional PWI with partition ℳ×ℳ={𝑀_𝑖×𝑀_𝑗,𝑖=1,2}. The partition atom 𝑀_𝑖×𝑀_𝑗 is surrounded by 𝑛_𝑖+𝑛_𝑗3-dimensional hyperplanes in ℂ², where 𝑛_𝑖(𝑛_𝑗) represents the number of hyperplanes which surround 𝑀_𝑖(𝑀_𝑗). The boundary of partition atom 𝑀_𝑖×𝑀_𝑗 may be represented as (𝜕𝑀_𝑖×𝑀_𝑗)∪(𝑀_𝑖×𝜕𝑀_𝑗), so a point 𝑧=(𝑧₁,𝑧₂)∈𝜕(𝑀_𝑖×𝑀_𝑗) if and only if 𝑧₁∈𝜕𝑀_𝑖, 𝑧₂∈𝑀_𝑗 or 𝑧₁∈𝑀_𝑖, 𝑧₂∈𝜕𝑀_𝑗.

Let 𝜒∶𝑀→𝑀 be the coding map of the Goetz map taking the form 𝜒(𝑧)_𝑖=𝜔_𝑖∈{1,2}iff𝑇^𝑖_𝜃(𝑧)∈𝑀_𝜔𝑖for𝑖=0,1,…. (21) As mentioned in Section 2, we denote the coding of any point 𝑧 of the phase space 𝑀×𝑀 by 𝜄(𝑧) with [𝜄(𝑧)]_𝑖=𝜅_𝑖=(𝜔_𝑖,𝜛_𝑖) if 𝐹^𝑖_𝜃(𝑧)∈𝑀_𝜔𝑖×𝑀_𝜛𝑖, where 𝜔_𝑖,𝜛_𝑖=1,2 for all 𝑖∈ℕ∪{0}.

Proposition 15. Let Ω be the set of all admissible codings of the Goetz map 𝑇_𝜃, then a 2-dimensional coding 𝜅 is admissible under the product Goetz map if and only if 𝜅=𝜔×𝜛, where 𝜔,𝜛∈Ω and [𝜔×𝜛]_𝑖=(𝜔_𝑖,𝜛_𝑖). Moreover, if the rotation parameter 𝜃 is incommensurable with 𝜋 and an admissible coding 𝜅=𝜔×𝜛 is periodic, then 𝑝(𝜅)=[𝑝(𝜔),𝑝(𝜛)], the least common multiple of 𝑝(𝜔) and 𝑝(𝜛), and the periodic cell 𝐶(𝜅) of the product Goetz map may be represented as 𝐶(𝜅)=𝐶(𝜔)×𝐶(𝜛), where 𝑝(⋅) represents the period of a periodic coding.

Proof. Obviously, if a 2-dimensional coding 𝜅=(𝜔₀,𝜛₀)(𝜔₁,𝜛₁)… is admissible, then there exists 𝑧=(𝑧₁,𝑧₂) such that 𝜄(𝑧)=𝜅. Consequently, 𝑇^𝑖_𝜃(𝑧₁)∈𝑀_𝜔𝑖 and 𝑇^𝑖_𝜃(𝑧₂)∈𝑀_𝜛𝑖 for all 𝑖=0,1,…. This implies that the codings 𝜔, 𝜛 are admissible under the map 𝑇_𝜃, that is, 𝜔,𝜛∈Ω. At the same time, let 𝜔,𝜛∈Ω be two admissible codings, then there exist two points 𝑧₁, 𝑧₂ such that 𝜒(𝑧₁)=𝜔 and 𝜒(𝑧₂)=𝜛. Let 𝑧=(𝑧₁,𝑧₂) be a point of the partition atom 𝑀_𝜔0×𝑀_𝜛0, then 𝐹^𝑖_𝜃(𝑧)=(𝑇^𝑖_𝜃(𝑧₁),𝑇^𝑖_𝜃(𝑧₂))∈𝑀_𝜔𝑖×𝑀_𝜛𝑖; that is, 𝜅=𝜔×𝜛 is admissible under the map 𝐹_𝜃.
It is obvious that 𝑝(𝜅)=[𝑝(𝜔),𝑝(𝜛)]. Now we check 𝐶(𝜅)=𝐶(𝜔)×𝐶(𝜛). For convenience, we only consider the case for 𝜔=𝜛=1^∞ (the other cases can be verified similarly). Let 𝑧^⋆ be the center, and 𝑟 the radius of the periodic disk 𝐶(𝜔)(𝐶(𝜛)). On one hand, for every point 𝑧=(𝑧₁,𝑧₂) with ‖𝑧_𝑖−𝑧^⋆‖<𝑟, 𝑖=1,2, we have 𝑇^𝑛_𝜃(𝑧_𝑖)∈𝑀₁ for 𝑖=1,2,𝑛∈ℕ∪{0} since 𝑧_𝑖∈𝐶(𝜔)=𝐶(𝜛), then 𝐹^𝑛_𝜃(𝑧)∈𝐶(𝜅) for all 𝑛∈ℕ∪{0}. Thus, 𝑧∈𝐶(𝜔)×𝐶(𝜛). On the other hand, for every point 𝑧=(𝑧₁,𝑧₂) with ‖𝑧₁−𝑧^⋆‖>𝑟 or ‖𝑧₂−𝑧^⋆‖>𝑟, we will show that 𝑧∉𝐶(𝜅). Without loss of generality, we suppose that ‖z₁−𝑧^⋆‖>𝑟, then there exists a natural number 𝑛₀ such that 𝑇^𝑛0_𝜃(𝑧₁)∉𝑀₁; this implies that 𝐹^𝑛0_𝜃(𝑧)={𝑇^𝑛0_𝜃(𝑧₁),𝑇^𝑛0_𝜃(𝑧₂)}∉𝐶(𝜅).

In fact, we can find that 𝐷₁⊂𝐶(𝜅)⊂𝐷₂ for 𝜅=(1,1)^∞, where 𝐷₁=𝑧₁,𝑧₂∣𝑧₁−𝑧^⋆+𝑧₁−𝑧^⋆,𝐷<𝑟₂=𝑧₁,𝑧₂∣𝑧₁−𝑧^⋆+𝑧₁−𝑧^⋆<√22𝑟. (22) More precisely, the boundary 𝜕𝐶(𝜅) of the periodic cell 𝐶(𝜅) consists of 𝐶(𝜔)×𝜕𝐶(𝜛) and 𝜕𝐶(𝜔)×𝐶(𝜛); however, the 2-dimensional torus 𝜕𝐶(𝜔)×𝜕𝐶(𝜛) is merely a fraction of 𝜕𝐶(𝜅).

We consider below the tangent-free property between any two periodic cells. Obviously, for any 3-dimensional boundary 𝐿⊂𝜕(ℳ×ℳ), the unit normal vector may be represented as 𝐧(𝐿)=(−sin𝛼,cos𝛼,0,0) or 𝐧(𝐿)=(0,0,−sin𝛼,cos𝛼), where tan𝛼 is a slope of a line segment of 𝜕ℳ. Moreover, the backward discontinuity set 𝒟^′ consists of two 3-dimensional polyhedral regions 𝐿₁=𝐴𝐹×𝑀 and 𝐿₂=𝑀×𝐴𝐹, with 𝐧(𝐿₁)=(−sin𝛽,cos𝛽,0,0), 𝐧(𝐿₂)=(0,0,−sin𝛽,cos𝛽), where tan𝛽 is the slope of the line segment 𝐴𝐹. By the analysis in Section 3.4, we can obtain that there exists a natural number 𝑘₀ such that for every point 𝑧∈𝜕(ℳ×ℳ), 𝐹^𝑘_𝜃(𝑧)∈𝐿₁∪𝐿₂ for some 𝑘≤𝑘₀. Let 𝐴_(𝑖,𝑗)=cos𝛼_𝑖−sin𝛼_𝑖00sin𝛼_𝑖cos𝛼_𝑖0000cos𝛼_𝑗−sin𝛼_𝑗00sin𝛼_𝑗cos𝛼_𝑗 (23) for 𝑖,𝑗=1,2, where 𝛼₁=2𝜋−𝜃, 𝛼₂=𝜋−𝜃, then 𝐴_(𝑖,𝑗) is the linear part of the restricted map 𝐹_𝜃|_{𝑀𝑖×𝑀𝑗}. If the rotation parameter 𝜃 is incommensurable with 𝜋, then for any admissible coding 𝜅=(𝜔₀,𝜛₀)(𝜔₁,𝜛₁)… we have 𝐴_𝜅⋅𝐧(𝐿_𝑖)≠𝐧(𝐿_𝑗) for all 𝑘∈ℕ and 𝑖,𝑗=1,2. Therefore, from Theorem 8, we have the following proposition.

Proposition 16. If 𝜃 is incommensurable with 𝜋, then the periodic cell packing induced by the product Goetz map 𝐹_𝜃 is tangent-free.

4. Discussions

As we know, for planar irrational piecewise rotations, a periodic cell is a disk except from possible countable points on the boundary, while for higher-dimensional PWIs, we do not think that a periodic cell is a sphere. In fact, it is easy to see that, for odd dimensional PWIs with polygonal partition, a periodic cell is not a sphere. By the same way as the product Goetz map, we have the following results.

Proposition 17. Let 𝑇∈𝕊𝔼(ℳ) and 𝐹∈𝕊𝔼(𝒳) be two PWIs and 𝐻=𝑇×𝐹∶𝑀×𝑋→𝑀×𝑋, every periodic cell 𝐶(𝜅) of 𝐻 can be written as 𝐶(𝜔)×𝐶(𝜛), where 𝜔 and 𝜛 are two admissible periodic codings of 𝑇 and 𝐹, respectively.

Proposition 18. Let 𝑇^𝑖∈𝕊𝔼(ℳ^𝑖)(𝑖=1,2,…,𝑘) be 𝑘 planar PWIs and 𝐻=𝑇¹×𝑇²×⋯×𝑇^𝑘, if each of them is an irrational rotation, then every periodic cell 𝐶 of the map 𝐻 can be represented as 𝐶=𝐶₁×𝐶₂×⋯×𝐶_𝑘, where 𝐶_𝑖 is a periodic disk of the map 𝑇^𝑖 for all 𝑖=1,2,…,𝑘. Moreover, the periodic cell packing induced by the PWI 𝐻 is tangent-free.

Generally, for a PWI of even-dimensional Euclidean space, we guess that every periodic cell may be written as the topological product space of 𝑛/2 disks under a continuous translation. To confirm this, further research is needed.

As mentioned in Section 1, we can confirm here that the disk packing induced by the Sigma-Delta map [20, 26] and the overflow map [17] are tangent-free for all the parameters 𝜃 which are incommensurable with 𝜋. In fact, it is now easy to check that the two piecewise isometries satisfy the conditions of Corollary 13. So we just state the results as follows.

Proposition 19. For the Sigma-Delta map and the Overflow map with 𝜃/𝜋 irrational, the induced disk packings are tangent-free.

Acknowledgments

This research was jointly supported by NSFC Grant 11072136 and Shanghai University Leading Academic Discipline Project (A.13-0101-12-004), and a grant of “The First-class Discipline of Universities in Shanghai.” Rongzhong Yu was also supported by a science and technology Project of Jiangxi Province, Department of Education (GJJ12617 and GJJ13714). The authors would also like to thank the anonymous referees for their critical but very helpful comments and suggestions.