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

Authors: Rongzhong Yu; Xinchu Fu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

Iterating an orientation-preserving piecewise isometryT  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.


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## Body

## 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 ℝn 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 ℝd.Iterating an orientation-preserving piecewise isometryT of n-dimensional Euclidean space ℝn, 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) The illustration to the invariant disks packing of the Sigma-Delta map for the parameterθ=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.
(a)
(b)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 ofθ 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 ofθ 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 parameterθ 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 ifn 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 Section2, 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 group𝔼(n)=𝕆(n)⋉ℝn is the semidirect product of the orthogonal group 𝕆(n) and ℝn, where 𝕆(n)={A∈𝔾𝕃(n,ℝ):AAt=Id}. Denote an element of g∈𝔼(n) as a pair (A,v) where A∈𝕆(n) and v∈ℝn. 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 ℝn.LetM be a connected subset of Euclidean space ℝn and let ℳ={M0,M1,…,Mr-1} be a finite collection of connected open convex set. We call ℳ a partition of M and each set Mi a partition atom if the following conditions hold:(1)
M=M¯1∪⋯∪M¯r-1;(2)
Mi∩Mj=∅ for i≠j.A map T:M∖𝒟↦M is called a piecewise isometry on M if  T(x)=Ti(x), x∈Mi, where Ti∈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 (Ti∈𝕊𝔼(n),i=0,1,…,r-1) and each of the partition atoms Mi(i=0,1,…,r-1) is an open convex polyhedron of ℝn which is surrounded by mi number of (n-1)-dimensional hyperplanes taking the form nij·x=cij(j=1,2,…,mi), where nij representing the unit normal vector of the hyperplane is an element of the projective space ℙn-1 (i.e., ignoring the difference between ±n), x=(x1,x2,…,xn)T∈ℝn is a column vector, and cij∈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=∪i=0r∂Mi be the boundary of the partition ℳ and 𝒟′=∪i≠j(T(Mi)¯∩T(Mj)¯) the backward discontinuity set. A piecewise isometry is said to be invertible if T(Mi)∩T(Mj)=∅(i≠j) and ∪i=0r-1T(Mi)¯=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 sayL 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)}=∪i=0r-1∪j=1mi{nij} the set of all unit normal vectors of hyperplanes of ℳ. Similarly, we define N(𝒟) and N(𝒟′).The partitionℳ={M0,M1,…,Mr-1} of M associated with a piecewise isometry T gives rise to a natural one-sided coding map. Let ℰ-=∪k=0∞T-k(𝒟) and 𝒜={0,1,…,r-1} the alphabet set; a map χ:M∖ℰ-↦𝒜ℕ is called a coding map, if for any point x∈M∖ℰ-,
(1)[χ(x)]j=ωj,iffTj(x)∈Mωj,j=0,1,2,….
And we call the infinite sequence ω=ω0ω1ω2… the coding of the orbit {Tj(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⋯βn-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 ωp+k=ωk for all k∈ℕ, and we denote the coding ω by ω=𝒫(ω0ω1⋯ωp-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,…,αr-1} and every element αi=(αi1,αi2,…,αij) is a j-dimensional vector.Definition 1.
If a codingω (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 cellC(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.LetTω=Tωm-1·Tωm-2⋯Tω0 and Aω=Aωm-1·Aωm-2⋯Aω0 for every finite coding. A piecewise isometry T∈𝕊𝔼(ℳ) is said to be incommensurate if FixAω={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

Ifn is even and ω=ω0⋯ωm-1 is admissible, then the linear part Aω of the map Tω=(Aω,v) can be orthogonally diagonalized as diag{B1,B2,…,Bn/2} under one orthogonal basis, where Bj=(cosθj,sinθj;-sinθj,cosθj),  θj∈[0,2π). It is obvious that an even dimensional orientation-preserving PWI is incommensurate if and only if all θj are incommensurable with π in diagonalization representation of Aω for every finite word ω. Furthermore, we say the PWI is irrational if it is incommensurable and all θj are rationally independent in diagonalization representation of Aω for every finite word ω.Letω=𝒫(α) be an admissible periodic coding, where α=ω0ω1⋯ωm-1 is of length m, then the self-map Tα:C(ω)↦C(ω) can be represented as the product of n/2 number of planar rotations since the linear part Aα can be orthogonally diagonalized. Namely, every point of C(ω) can be rewritten as complex number coordinate z=(z1,z2,…,zn/2), and the map Tα can be represented as Tα=T~1×T~2×⋯×T~n/2, T~j(zj)=ei·θj+cj,j=1,2,…,n/2. We have the following result.Theorem 3.
Supposen 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⋯ωm-1 is of length m, then the cell C(ω) can be represented as
(2)C(ω)={x:χ(x)=ω}={x:x∈Mω0,Tk(x)∈Mωkfork∈ℕ}=Mω0∩T-1(Mω1)∩T-2(Mω1)∩⋯=⋂k=0∞‍T-k(Mωk).
At the same time, we have
(3)T-k(Mωk)=Tω1-1∘⋯∘Tωk-1(Mωk),
where Mωk is convex and each of the maps {Tω1-1,Tω2-1,…,Tωk-1} is an isometry, then the set T-k(Mωk) is convex, consequently, the cell C(ω) is convex.
Since the mapTα:C(ω)↦C(ω) is isometric and Tα(C(ω))=C(ω), the cell C(ω) at least contains a fixed point denoted by p*. Let
(4)d=12min0≤i≤p-1{ρ(Ti(p*),∂ℳ)},
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 mapTα 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 Tα since the piecewise isometry T is irrational. We will show that z={z1,z2,…,zn/2}∈C(ω)¯ implies that every point z′={z1′,z2′,…,zn/2′}∈C(ω)¯ with ∥zi′∥=∥zi∥(1≤i≤r2), (i.e., Tn/2⊂C(ω)¯), where
(5)Tn/2={(∥z1∥eα1,∥z2∥eα2,…,∥zr2∥eαn/2)∣αi∈[0,2π),i=1,2,…,r2(∥z1∥eα1,∥z2∥eα2,…,∥zr2∥eαn/2)}
is the (n/2)-dimensional torus in the phase space. For z={z1,z2,…,zn/2}∈C(ω)¯, and there exists a point z⋆={z1⋆,z2⋆,…,zn/2⋆}∈C(ω) such that for any ϵ>0,
(6)∥z-z⋆∥<ϵn.
Let
(7)T1n/2={n2(∥z1⋆∥eα1,∥z2⋆∥eα2,…,∥zr2⋆∥eαn/2)∣αi∈[0,2π),i=1,2,…,n2}
be another torus in the phase space. At the same time, the orbit Orb+(z⋆) under the map Tα:C(ω)→C(ω) is dense in the torus T1n/2 because the map T is irrational. For any point z′=(z1′,…,zn/2′)∈Tn/2,then z′′=(∥z1⋆/z1′∥z1′,…,∥zi⋆/zi′∥zi′,…,∥zn/2⋆/zn/2′∥zn/2′)∈T1n/2. Consequently, there exists a natural number k such that
(8)∥Tαk(z⋆)-z∥<ϵ2.
Because
(9)∥zi′-∥zi⋆zi′∥·zi′∥=∥∥zi′∥-∥zi⋆∥∥zi′∥·zi′∥=|∥zi′∥-∥zi⋆∥|=|∥zi∥-∥zi⋆∥|≤∥zi-zi⋆∥
for all i=1,2,…,n/2, from (9) and (6) we obtain
(10)∥z′-z∥≤∑i=1n/2‍∥zi′-∥zi⋆zi′∥·zi′∥≤∑i=1n/2‍∥zi-zi⋆∥≤ϵ2.
Consequently, from (8) and (10), we have
(11)∥Tαk(z⋆)-z′∥≤∥Tαk(z⋆)-z∥+∥z′-z∥<ϵ2+ϵ2=ϵ,
namely, z′∈C(ω)¯. Consequently, the periodic cell is centrosymmetric with respect to the unique fixed point.Remark 4.
According to Theorem3, since the set C(ω)¯ is convex, then z=(z1,z2,…,zn/2)∈C(ω)¯ implies z′=(z1′,z2′,…,zn/2′)∈C(ω)¯ with ∥zi′∥≤∥zi∥ for all 1≤i≤n/2. Let Θ(z)={z′∣∥zi′∥≤∥zi∥,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).Letγi=max{∥zi∥∣(z1,…,zi,…,zn/2)∈C(ω)¯} for all i=1,2,…,n/2, and Θ={z=(z1,z2,…,zn/2)∣∥zi∥≤γi,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.
Supposen is even, T∈𝕊𝔼(ℳ) is irrational, if there exists a point z=(z1,z2,…,zn/2) with ∥zi∥=∥γi∥(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 isometryT∈𝕊𝔼(ℳ), if C(ω) and C(ϖ) are two different periodic cells, then there must exist a natural number k such that Tk(C(ω)) and Tk(C(ϖ)) are contained in different partition atoms Mi and Mj, 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 isometryT∈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 codingsω 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 ni(x)(i=1,2) represent the unit normal vectors of surface ∂Ci 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
(12)𝒯(C1,C2)={x∣xisatangentpointofC1andC2}
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 set𝒯(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 ℝn. In the following, we give another main result, which will be proven later in Section 3.2 after preparing some propositions.Theorem 8.
Supposen is even, T∈𝕊𝔼(ℳ) is invertible and incommensurate. If for any admissible finite coding ω=ω0ω1⋯ωk and any n,  n′ of N(𝒟′), one has Aω·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.
Supposen 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 σi(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.
Supposen 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)),C1′,C2′∈GO(C(ϖ)) such that Ci and Ci′(i=1,2) are tangent to each other, and the mutually tangent set 𝒯(Ci,Ci′) is contained in Li for i=1,2, respectively.From the above result, we further have the following.Proposition 11.
Supposen 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⋯αk-1 with length k such that Aα·n1=n2.Proof.
Without loss of generality, we assume that the mutually tangent set is contained in the hyperplaneL′⊂𝒟′ (i.e., 𝒯(C(ω),C(ϖ))⊂L′) with unit normal vector n1′ according to Proposition 10. Then, there is a hyperplane L1⊂∂ℳ with unit normal vector n1 such that Aα0·n1=n1′. From Proposition 6, there exists a natural number k such that the periodic cells Tk-1(C(ω)) and Tk-1(C(ϖ)) are in the different partition atoms Mi and Mj, respectively, and Tk-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⋯αk-1 such that Aαk-1⋯Aα1·n1′=n2. Consequently,
(13)Aα·n1=Aαk-1⋯Aα1Aα0·n1=n2.
The proof is therefore complete.We are now ready to give the proof of Theorem8.Proof of Theorem8.
Firstly, we will reveal that there exists a natural numberk0 such that for any n of N(ℳ), there exists an admissible finite coding α=α0⋯αk-1 with length k≤k0 such that Aα·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⋯αk-1 with length k>k0 such that Ti(L)⊂∂ℳ for all i=0,…,k-1, then A(i)·n∈N(ℳ) for all i=0,…,k-1, where A(i)=Aα0⋯Aαi-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 cellsC1 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⋯βk-1, such that
(14)Aβ·n1=n.
Via the typical invertibility condition, we can obtain that there exists an admissible finite coding α=α0α1⋯αl-1 and a unit normal vector n2∈N(𝒟′) such that
(15)Aα·n=n2.
According to (14) and (15), we have
(16)Aβ·Aα·n1=n2.
Let ω=αβ=α0⋯αl-1β0⋯βk-1 with length k+l, then we get Aω·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. Letρj=ei·θj(j=0,1,…,r-1) be the linear part of the map T|Mj, then we have the following corollary.Corollary 12.
Supposeℳ⊂ℝ2, T∈𝕊𝔼(ℳ) is invertible, if ρ0k0ρ1k1⋯ρr-1kr-1=1 with ki≥0(0≤i≤r-1) implies that ki=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 Corollary9.In particular, if an invertible planar piecewise rotation has the common irrational rotation angleθ, that is, any restricted map T|Mj has the same linear part ρ=ei·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 angleθ(θ/π∈ℚC) 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 mapTθ:M↦M with parameter θ∈(π/2,2π/3) as follows:
(17)Tθ(z)={e2(π-θ)i·z+c1ifz∈M1;e(π-θ)i·z+c2ifz∈M2.
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):
(18)M1=[A,B,C]=[-2cotθ·e(π-θ)i,0,1sinθ],(19)M2=[B,C,D,E]=[0,1sinθ,1sinθ·(1+e-3θi),1+2cosθsinθ·e-θi].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 set𝒟′ 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=e2(π-θ)i and ρ2=e(π-θ)i, and obviously, when θ/π is irrational, the equality
(20)ρ1k1ρ2k2=1,k1≥0,k2≥0
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 mapFθ=Tθ×Tθ:M×M↦M×M which is a 4-dimensional PWI with partition ℳ×ℳ={Mi×Mj,i=1,2}. The partition atom Mi×Mj is surrounded by ni+nj3-dimensional hyperplanes in ℂ2, where ni(nj) represents the number of hyperplanes which surround Mi(Mj). The boundary of partition atom Mi×Mj may be represented as (∂Mi×Mj)∪(Mi×∂Mj), so a point z=(z1,z2)∈∂(Mi×Mj) if and only if z1∈∂Mi,  z2∈M¯j or z1∈M¯i,  z2∈∂Mj.Letχ:M→M be the coding map of the Goetz map taking the form
(21)[χ(z)]i=ωi∈{1,2}iffTθi(z)∈Mωifori=0,1,….
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=κi=(ωi,ϖi) if Fθi(z)∈Mωi×Mϖi, where ωi,ϖi=1,2 for all i∈ℕ∪{0}.Proposition 15.
LetΩ be the set of all admissible codings of the Goetz map Tθ, 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=(ωi,ϖi). 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 a2-dimensional coding κ=(ω0,ϖ0)(ω1,ϖ1)… is admissible, then there exists z=(z1,z2) such that ι(z)=κ. Consequently, Tθi(z1)∈Mωi and Tθi(z2)∈Mϖi for all i=0,1,…. This implies that the codings ω, ϖ are admissible under the map Tθ, 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 Mω0×Mϖ0, then Fθi(z)=(Tθi(z1),Tθi(z2))∈Mωi×Mϖi; that is, κ=ω×ϖ is admissible under the map Fθ.
It is obvious thatp(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 ∥zi-z⋆∥<r, i=1,2, we have Tθn(zi)∈M1 for i=1,2,n∈ℕ∪{0} since zi∈C(ω)=C(ϖ), then Fθn(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 Tθn0(z1)∉M1; this implies that Fθn0(z)={Tθn0(z1),Tθn0(z2)}∉C(κ).In fact, we can find thatD1⊂C(κ)⊂D2 for κ=(1,1)∞, where
(22)D1={(z1,z2)∣∥z1-z⋆∥+∥z1-z⋆∥<r},D2={(z1,z2)∣∥z1-z⋆∥+∥z1-z⋆∥<22r}.
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 any3-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), Fθk(z)∈L1∪L2 for some k≤k0. Let
(23)A(i,j)=(cosαi-sinαi00sinαicosαi0000cosαj-sinαj00sinαjcosαj)
for i,j=1,2, where α1=2π-θ,  α2=π-θ, then A(i,j) is the linear part of the restricted map Fθ|Mi×Mj. If the rotation parameter θ is incommensurable with π, then for any admissible coding κ=(ω0,ϖ0)(ω1,ϖ1)… we have Aκ·n(Li)≠n(Lj) for all k∈ℕ and i,j=1,2. Therefore, from Theorem 8, we have the following proposition.Proposition 16.
Ifθ is incommensurable with π, then the periodic cell packing induced by the product Goetz map Fθ is tangent-free.


## 3.1. Periodic Cells

Ifn is even and ω=ω0⋯ωm-1 is admissible, then the linear part Aω of the map Tω=(Aω,v) can be orthogonally diagonalized as diag{B1,B2,…,Bn/2} under one orthogonal basis, where Bj=(cosθj,sinθj;-sinθj,cosθj),  θj∈[0,2π). It is obvious that an even dimensional orientation-preserving PWI is incommensurate if and only if all θj are incommensurable with π in diagonalization representation of Aω for every finite word ω. Furthermore, we say the PWI is irrational if it is incommensurable and all θj are rationally independent in diagonalization representation of Aω for every finite word ω.Letω=𝒫(α) be an admissible periodic coding, where α=ω0ω1⋯ωm-1 is of length m, then the self-map Tα:C(ω)↦C(ω) can be represented as the product of n/2 number of planar rotations since the linear part Aα can be orthogonally diagonalized. Namely, every point of C(ω) can be rewritten as complex number coordinate z=(z1,z2,…,zn/2), and the map Tα can be represented as Tα=T~1×T~2×⋯×T~n/2, T~j(zj)=ei·θj+cj,j=1,2,…,n/2. We have the following result.Theorem 3.
Supposen 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⋯ωm-1 is of length m, then the cell C(ω) can be represented as
(2)C(ω)={x:χ(x)=ω}={x:x∈Mω0,Tk(x)∈Mωkfork∈ℕ}=Mω0∩T-1(Mω1)∩T-2(Mω1)∩⋯=⋂k=0∞‍T-k(Mωk).
At the same time, we have
(3)T-k(Mωk)=Tω1-1∘⋯∘Tωk-1(Mωk),
where Mωk is convex and each of the maps {Tω1-1,Tω2-1,…,Tωk-1} is an isometry, then the set T-k(Mωk) is convex, consequently, the cell C(ω) is convex.
Since the mapTα:C(ω)↦C(ω) is isometric and Tα(C(ω))=C(ω), the cell C(ω) at least contains a fixed point denoted by p*. Let
(4)d=12min0≤i≤p-1{ρ(Ti(p*),∂ℳ)},
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 mapTα 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 Tα since the piecewise isometry T is irrational. We will show that z={z1,z2,…,zn/2}∈C(ω)¯ implies that every point z′={z1′,z2′,…,zn/2′}∈C(ω)¯ with ∥zi′∥=∥zi∥(1≤i≤r2), (i.e., Tn/2⊂C(ω)¯), where
(5)Tn/2={(∥z1∥eα1,∥z2∥eα2,…,∥zr2∥eαn/2)∣αi∈[0,2π),i=1,2,…,r2(∥z1∥eα1,∥z2∥eα2,…,∥zr2∥eαn/2)}
is the (n/2)-dimensional torus in the phase space. For z={z1,z2,…,zn/2}∈C(ω)¯, and there exists a point z⋆={z1⋆,z2⋆,…,zn/2⋆}∈C(ω) such that for any ϵ>0,
(6)∥z-z⋆∥<ϵn.
Let
(7)T1n/2={n2(∥z1⋆∥eα1,∥z2⋆∥eα2,…,∥zr2⋆∥eαn/2)∣αi∈[0,2π),i=1,2,…,n2}
be another torus in the phase space. At the same time, the orbit Orb+(z⋆) under the map Tα:C(ω)→C(ω) is dense in the torus T1n/2 because the map T is irrational. For any point z′=(z1′,…,zn/2′)∈Tn/2,then z′′=(∥z1⋆/z1′∥z1′,…,∥zi⋆/zi′∥zi′,…,∥zn/2⋆/zn/2′∥zn/2′)∈T1n/2. Consequently, there exists a natural number k such that
(8)∥Tαk(z⋆)-z∥<ϵ2.
Because
(9)∥zi′-∥zi⋆zi′∥·zi′∥=∥∥zi′∥-∥zi⋆∥∥zi′∥·zi′∥=|∥zi′∥-∥zi⋆∥|=|∥zi∥-∥zi⋆∥|≤∥zi-zi⋆∥
for all i=1,2,…,n/2, from (9) and (6) we obtain
(10)∥z′-z∥≤∑i=1n/2‍∥zi′-∥zi⋆zi′∥·zi′∥≤∑i=1n/2‍∥zi-zi⋆∥≤ϵ2.
Consequently, from (8) and (10), we have
(11)∥Tαk(z⋆)-z′∥≤∥Tαk(z⋆)-z∥+∥z′-z∥<ϵ2+ϵ2=ϵ,
namely, z′∈C(ω)¯. Consequently, the periodic cell is centrosymmetric with respect to the unique fixed point.Remark 4.
According to Theorem3, since the set C(ω)¯ is convex, then z=(z1,z2,…,zn/2)∈C(ω)¯ implies z′=(z1′,z2′,…,zn/2′)∈C(ω)¯ with ∥zi′∥≤∥zi∥ for all 1≤i≤n/2. Let Θ(z)={z′∣∥zi′∥≤∥zi∥,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).Letγi=max{∥zi∥∣(z1,…,zi,…,zn/2)∈C(ω)¯} for all i=1,2,…,n/2, and Θ={z=(z1,z2,…,zn/2)∣∥zi∥≤γi,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.
Supposen is even, T∈𝕊𝔼(ℳ) is irrational, if there exists a point z=(z1,z2,…,zn/2) with ∥zi∥=∥γi∥(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 isometryT∈𝕊𝔼(ℳ), if C(ω) and C(ϖ) are two different periodic cells, then there must exist a natural number k such that Tk(C(ω)) and Tk(C(ϖ)) are contained in different partition atoms Mi and Mj, 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 isometryT∈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 codingsω 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 ni(x)(i=1,2) represent the unit normal vectors of surface ∂Ci 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
(12)𝒯(C1,C2)={x∣xisatangentpointofC1andC2}
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 set𝒯(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 ℝn. In the following, we give another main result, which will be proven later in Section 3.2 after preparing some propositions.Theorem 8.
Supposen is even, T∈𝕊𝔼(ℳ) is invertible and incommensurate. If for any admissible finite coding ω=ω0ω1⋯ωk and any n,  n′ of N(𝒟′), one has Aω·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.
Supposen 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 σi(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.
Supposen 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)),C1′,C2′∈GO(C(ϖ)) such that Ci and Ci′(i=1,2) are tangent to each other, and the mutually tangent set 𝒯(Ci,Ci′) is contained in Li for i=1,2, respectively.From the above result, we further have the following.Proposition 11.
Supposen 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⋯αk-1 with length k such that Aα·n1=n2.Proof.
Without loss of generality, we assume that the mutually tangent set is contained in the hyperplaneL′⊂𝒟′ (i.e., 𝒯(C(ω),C(ϖ))⊂L′) with unit normal vector n1′ according to Proposition 10. Then, there is a hyperplane L1⊂∂ℳ with unit normal vector n1 such that Aα0·n1=n1′. From Proposition 6, there exists a natural number k such that the periodic cells Tk-1(C(ω)) and Tk-1(C(ϖ)) are in the different partition atoms Mi and Mj, respectively, and Tk-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⋯αk-1 such that Aαk-1⋯Aα1·n1′=n2. Consequently,
(13)Aα·n1=Aαk-1⋯Aα1Aα0·n1=n2.
The proof is therefore complete.We are now ready to give the proof of Theorem8.Proof of Theorem8.
Firstly, we will reveal that there exists a natural numberk0 such that for any n of N(ℳ), there exists an admissible finite coding α=α0⋯αk-1 with length k≤k0 such that Aα·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⋯αk-1 with length k>k0 such that Ti(L)⊂∂ℳ for all i=0,…,k-1, then A(i)·n∈N(ℳ) for all i=0,…,k-1, where A(i)=Aα0⋯Aαi-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 cellsC1 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⋯βk-1, such that
(14)Aβ·n1=n.
Via the typical invertibility condition, we can obtain that there exists an admissible finite coding α=α0α1⋯αl-1 and a unit normal vector n2∈N(𝒟′) such that
(15)Aα·n=n2.
According to (14) and (15), we have
(16)Aβ·Aα·n1=n2.
Let ω=αβ=α0⋯αl-1β0⋯βk-1 with length k+l, then we get Aω·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. Letρj=ei·θj(j=0,1,…,r-1) be the linear part of the map T|Mj, then we have the following corollary.Corollary 12.
Supposeℳ⊂ℝ2, T∈𝕊𝔼(ℳ) is invertible, if ρ0k0ρ1k1⋯ρr-1kr-1=1 with ki≥0(0≤i≤r-1) implies that ki=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 Corollary9.In particular, if an invertible planar piecewise rotation has the common irrational rotation angleθ, that is, any restricted map T|Mj has the same linear part ρ=ei·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 angleθ(θ/π∈ℚC) 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 mapTθ:M↦M with parameter θ∈(π/2,2π/3) as follows:
(17)Tθ(z)={e2(π-θ)i·z+c1ifz∈M1;e(π-θ)i·z+c2ifz∈M2.
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):
(18)M1=[A,B,C]=[-2cotθ·e(π-θ)i,0,1sinθ],(19)M2=[B,C,D,E]=[0,1sinθ,1sinθ·(1+e-3θi),1+2cosθsinθ·e-θi].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 set𝒟′ 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=e2(π-θ)i and ρ2=e(π-θ)i, and obviously, when θ/π is irrational, the equality
(20)ρ1k1ρ2k2=1,k1≥0,k2≥0
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 mapFθ=Tθ×Tθ:M×M↦M×M which is a 4-dimensional PWI with partition ℳ×ℳ={Mi×Mj,i=1,2}. The partition atom Mi×Mj is surrounded by ni+nj3-dimensional hyperplanes in ℂ2, where ni(nj) represents the number of hyperplanes which surround Mi(Mj). The boundary of partition atom Mi×Mj may be represented as (∂Mi×Mj)∪(Mi×∂Mj), so a point z=(z1,z2)∈∂(Mi×Mj) if and only if z1∈∂Mi,  z2∈M¯j or z1∈M¯i,  z2∈∂Mj.Letχ:M→M be the coding map of the Goetz map taking the form
(21)[χ(z)]i=ωi∈{1,2}iffTθi(z)∈Mωifori=0,1,….
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=κi=(ωi,ϖi) if Fθi(z)∈Mωi×Mϖi, where ωi,ϖi=1,2 for all i∈ℕ∪{0}.Proposition 15.
LetΩ be the set of all admissible codings of the Goetz map Tθ, 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=(ωi,ϖi). 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 a2-dimensional coding κ=(ω0,ϖ0)(ω1,ϖ1)… is admissible, then there exists z=(z1,z2) such that ι(z)=κ. Consequently, Tθi(z1)∈Mωi and Tθi(z2)∈Mϖi for all i=0,1,…. This implies that the codings ω, ϖ are admissible under the map Tθ, 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 Mω0×Mϖ0, then Fθi(z)=(Tθi(z1),Tθi(z2))∈Mωi×Mϖi; that is, κ=ω×ϖ is admissible under the map Fθ.
It is obvious thatp(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 ∥zi-z⋆∥<r, i=1,2, we have Tθn(zi)∈M1 for i=1,2,n∈ℕ∪{0} since zi∈C(ω)=C(ϖ), then Fθn(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 Tθn0(z1)∉M1; this implies that Fθn0(z)={Tθn0(z1),Tθn0(z2)}∉C(κ).In fact, we can find thatD1⊂C(κ)⊂D2 for κ=(1,1)∞, where
(22)D1={(z1,z2)∣∥z1-z⋆∥+∥z1-z⋆∥<r},D2={(z1,z2)∣∥z1-z⋆∥+∥z1-z⋆∥<22r}.
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 any3-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), Fθk(z)∈L1∪L2 for some k≤k0. Let
(23)A(i,j)=(cosαi-sinαi00sinαicosαi0000cosαj-sinαj00sinαjcosαj)
for i,j=1,2, where α1=2π-θ,  α2=π-θ, then A(i,j) is the linear part of the restricted map Fθ|Mi×Mj. If the rotation parameter θ is incommensurable with π, then for any admissible coding κ=(ω0,ϖ0)(ω1,ϖ1)… we have Aκ·n(Li)≠n(Lj) for all k∈ℕ and i,j=1,2. Therefore, from Theorem 8, we have the following proposition.Proposition 16.
Ifθ is incommensurable with π, then the periodic cell packing induced by the product Goetz map Fθ 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.
LetT∈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.
LetTi∈𝕊𝔼(ℳi)(i=1,2,…,k) be k planar PWIs and H=T1×T2×⋯×Tk, if each of them is an irrational rotation, then every periodic cell C of the map H can be represented as C=C1×C2×⋯×Ck, where Ci is a periodic disk of the map Ti 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 ofn/2 disks under a continuous translation. To confirm this, further research is needed.As mentioned in Section1, 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 withθ/π irrational, the induced disk packings are tangent-free.



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