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

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.


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][5][6][7].
In [8][9][10], some planar PWIs are studied systematically with a particular focus on geometry and symbolic dynamics.In [11][12][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  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  .
Iterating an orientation-preserving piecewise isometry  of -dimensional Euclidean space R  , 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  is incommensurable with ), 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 .A natural problem is whether any two periodic cells are tangent (see Figure 1).
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 we can see that the disk packing is very "loose" and there is no tangencies between any pair of invariant disks.
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  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  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  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  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  is incommensurable with ?
(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  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  is incommensurable with ) or a polygonal region (the rotation parameter  is commensurable with ).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  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.

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() = O() ⋉ R  is the semidirect product of the orthogonal group O() and R  , where O() = { ∈ GL(, R) :   = }.Denote an element of  ∈ E() as a pair (, V) where  ∈ O() and V ∈ R  .The group multiplication is given by Let  be a connected subset of Euclidean space R  and let M = { 0 ,  1 , . . .,  −1 } be a finite collection of connected open convex set.We call M a partition of  and each set   a partition atom if the following conditions hold: (1)  =  1 ∪ ⋅ ⋅ ⋅ ∪  −1 ; (2)   ∩   = 0 for  ̸ = .
A map  :  \ D  →  is called a piecewise isometry on  if () =   (),  ∈   , where   ∈ E().And D = ∪  ̸ = (  ∩   ) 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 orientationpreserving piecewise isometries (  ∈ SE(),  = 0, 1, . . .,  − 1) and each of the partition atoms   ( = 0, 1, . . .,  − 1) is an open convex polyhedron of R  which is surrounded by   number of ( − 1)-dimensional hyperplanes taking the form n  ⋅  =   ( = 1, 2, . . .,   ), where n  representing the unit normal vector of the hyperplane is an element of the projective space P −1 (i.e., ignoring the difference between ±n),  = ( 1 ,  2 , . . .,   )  ∈ R  is a column vector, and   ∈ 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.
In the following, for convenience, we say  is a ( − 1)dimensional hyperplane of M (or D, D  ), if  ⊂ M (or D, D  ) is a ( − 1)-dimensional polyhedral region of M (or D, D  ).For a ( − 1)-dimensional hyperplane , let n() be a unit normal vector of  and =1 {n  } the set of all unit normal vectors of hyperplanes of M. Similarly, we define N(D) and N(D  ).
The partition M = { 0 ,  1 , . . .,  −1 } of  associated with a piecewise isometry  gives rise to a natural one-sided coding map.Let And we call the infinite sequence  =  0  1  2 . . . the coding of the orbit {  ()} ∞ =0 (or the coding of the point  for simplicity).An infinite sequence  =  0  1 . . . is said to be admissible, if there exists a point  such that () = .Similarly, a finite sequence  =  0  1 ⋅ ⋅ ⋅  −1 with length  is said to be admissible, if there exists a point  such that  is the preword of ().A coding  is said to be periodic, if there exists a natural number  such that  + =   for all  ∈ N, and we denote the coding  by  = P( 0  1 ⋅ ⋅ ⋅  −1 ), where P(⋅) represents concatenation operation.A coding  is said to be rational, if it is eventually periodic.Otherwise, the coding  is said to be irrational.In fact, for an invertible piecewise isometry, a rational coding is periodic.Sometimes, we may denote -dimensional coding as paper [23] for convenience.Namely, let the alphabet set A = { 0 ,  1 , . . .,  −1 } and every element   = ( 1 ,  2 , . . .,   ) is a -dimensional vector.
Definition 1.If a coding  (finite or infinite) is admissible, we call the set of all points following the same coding  a cell, denoted by ().
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 () is a convex set.More precisely, it has interior if and only if  is rational, and if  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  is irrational, then the cell () consists of only a point. Let is said to be incommensurate if Fix   = {0} for every finite word  in the alphabet A. As mentioned in [15], if  is even then almost every piecewise isometry in SE(M) is incommensurate.

Periodic Cells and the Tangent-Free Property
Based on the fact revealed by Mendes and Nicol in [15] that if  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.Let = P() be an admissible periodic coding, where  =  0  1 ⋅ ⋅ ⋅  −1 is of length , then the self-map   : ()  → () can be represented as the product of /2 number of planar rotations since the linear part   can be orthogonally diagonalized.Namely, every point of () can be rewritten as complex number coordinate  = ( 1 ,  2 , . . .,  /2 ), and the map   can be represented as   = T1 × T2 × ⋅ ⋅ ⋅ × T/2 , T (  ) =  ⋅  +   ,  = 1, 2, . . ., /2.We have the following result.
Theorem 3. Suppose  is even,  ∈ SE(M), then every periodic cell is convex and of positive Lebesgue measure.Furthermore, if  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  = P() be an admissible periodic coding, where  =  0  1 ⋅ ⋅ ⋅  −1 is of length , then the cell () can be represented as (2) At the same time, we have where    is convex and each of the maps Since the map   : ()  → () is isometric and   (()) = (), the cell () at least contains a fixed point denoted by  * .Let then ( * , ) ⊂ (), which implies that the cell () is of positive Lebesgue measure.Now, we prove the latter claim.Suppose, without loss of generality, the map   is represented as the product of /2 planar rotations and every point of () has the form of complex coordinate.Furthermore, assume the unique fixed point equals 0 of the map   since the piecewise isometry  is irrational.We will show that  = {  ‖ ≤ ‖  ‖,  = 1, . . ., /2} be the product of /2 number of discs, then the closure of the periodic cell () is the union of some such sets as Θ().
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 () × (), where ,  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  ∈ SE(M), if () and () are two different periodic cells, then there must exist a natural number  such that   (()) and   (()) are contained in different partition atoms   and   , respectively; that is, they will be separated by an ( − 1)-dimensional boundary  ⊂ D.
We must point out that two periodic cells are separated by an ( − 1)-dimensional boundary  ⊂ D as above means that (n ⋅  ⋆ − ) ⋅ (n ⋅  ⋆⋆ − ) < 0, where  ⋆ and  ⋆⋆ are the centers of two periodic cells, respectively, and n ⋅ =  is the equation of hyperplane .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.

Tangencies between Periodic
Cells.Similar to the planar cases [17,20,21], for a -dimensional PWI ( ≥ 4 is even), we say that the set  = ⋃ (), the union of all periodic cells, is a periodic cell packing of the phase space  by the dimensional PWI.We consider similarly the tangent property between any two periodic cells.Assume that the boundaries of the periodic cells  1 and  2 are smooth at the intersection point , and we say the two periodic cells  1 and  2 are tangent to each other at the point  if (1) int( 1 ) ∩ int( 2 ) = 0; (2)  ∈  1 ∩  2 ; (3) n 1 () = ±n 2 (), where n  () ( = 1, 2) represent the unit normal vectors of surface   at the point .
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 .Let be the mutually tangent set.We say the periodic cells  1 and  2 are tangent, if T( 1 ,  2 ) is nonempty.We say the packing is tangent-free, if any two periodic cells are not tangent.
In fact, the mutually tangent set T( 1 ,  2 ) may be a single point or an open straight line segment for a planar piecewise isometry, and it may be a -dimensional ( = 0, 1, . . .,  − 1) hyperplane for a general piecewise isometry of -dimensional Euclidean space R  .In the following, we give another main result, which will be proven later in Section 3.2 after preparing some propositions.Theorem 8. Suppose  is even,  ∈ SE(M) is invertible and incommensurate.If for any admissible finite coding  =  0  1 ⋅ ⋅ ⋅   and any n, n  of N(D  ), one has   ⋅ n ̸ = n  , then there are no tangencies between periodic cells in the periodic cells packing induced by .
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  is even,  ∈ 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 .
For convenience of expression, we give some notations introduced in [19].Two admissible -periodic codings  and  are said to be equivalent, denoted by  ∼ , if there exists a natural number  (0 ≤  ≤ ) such that   () = , where  is the shift map.Two points of  are said to be equivalent, if their corresponding symbolic sequences are equivalent.Similarly, we can define the equivalence of two cells () and ().We denote by GO(()) the set of all the cells which are equivalent to () under the above equivalence relation and call GO(()) the great orbit of ().According to Propositions 6 and 7, we have the following result.From the above result, we further have the following.Proposition 11.Suppose  is even,  ∈ SE(M) is invertible, if two different periodic cells () and () are tangent to each other, then there exist two vectors n 1 and n 2 of N(M) and an admissible coding Proof.Without loss of generality, we assume that the mutually tangent set is contained in the hyperplane   ⊂ D  (i.e., T((), ()) ⊂   ) with unit normal vector n  1 according to Proposition 10.Then, there is a hyperplane  1 ⊂M with unit normal vector n 1 such that   0 ⋅ n 1 = n  1 .From Proposition 6, there exists a natural number  such that the periodic cells  −1 (()) and  −1 (()) are in the different partition atoms   and   , respectively, and  −1 (T((), ())) ⊂ ⊂ D, where  =   ∩  .Let n 2 be the unit normal vector of the hyperplane , then there exists a finite admissible coding 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  0 such that for any n of N(M), there exists an admissible finite coding  =  0 ⋅ ⋅ ⋅  −1 with length  ≤  0 such that   ⋅ n ∈ N(D  ).Since every partition atom is surrounded by finite ( − 1)-dimensional hyperplanes, then N(M) consists of finite elements, and let  0 = ♯N(M) be the cardinal number.Because the isometry is invertible, for any ( − 1)-dimensional hyperplanes  ⊂ M, we can get () ⊂ M ∪ D  .If there exists an admissible finite coding  =  0 ⋅ ⋅ ⋅  −1 with length  >  0 such that   () ⊂ M for all  = 0, . . ., −1, then () ⋅ n ∈ N(M) for all  = 0, . . ., −1, where () =   0 ⋅ ⋅ ⋅   −1 .Since the piecewise isometry is incommensurate, all () ⋅ n are different from each other.This implies that ♯N(M) ≥  >  0 , a contradiction.Suppose that there exist two periodic cells  1 and  2 which are tangent to each other, then, in view of Proposition 11, there is two unit normal vectors n 1 of a hyperplane   ⊂ D  and n of a hyperplane  ⊂ D, and an admissible finite coding  =  0  1 ⋅ ⋅ ⋅  −1 , such that Via the typical invertibility condition, we can obtain that there exists an admissible finite coding  =  0  1 ⋅ ⋅ ⋅  −1 and a unit normal vector n 2 ∈ N(D  ) such that According to ( 14) and ( 15), we have Let =  =  0 ⋅ ⋅ ⋅  −1  0 ⋅ ⋅ ⋅  −1 with length +, then we get   ⋅n 1 = n 2 for two elements of N(D  ), which contradicts the results as above.Therefore, there are no tangencies between periodic cells.

Applications to Planar PWIs.
For planar piecewise rotations, the corresponding results are very explicit.Let   =  ⋅  ( = 0, 1, . . .,  − 1) be the linear part of the map |   , then we have the following corollary.
implies that   = 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 , that is, any restricted map |   has the same linear part  =  ⋅ , then we have the following corollary, which strengthens the results in [17,20,21].
Here,  1 = 1/ sin ,  2 = ((1 + 2 cos )/ sin ) − , and the partition atoms  1 and  2 can be written as follows (see Figure 2): The piecewise isometric map is invertible, and the backward discontinuity set D  consists of only the line segment ; furthermore, all of the boundary line segments of the partition atoms  1 and  2 will turn on to the line segment  under finite iterations of the map ; that is, the isometric map  satisfies the typical invertibility condition.At the same time, the linear parts of the restricted maps are  1 =  2(−) and  2 =  (−) , and obviously, when / is irrational, the equality holds if and only if  1 =  2 = 0.Then, from Corollary 12, we have the following.
Proposition 14.For the piecewise isometry defined by (17), if the parameter  is incommensurable with , then the invariant disk packing is tangent-free.(See Figure 3).
We consider below the tangent-free property between any two periodic cells.Obviously, for any 3-dimensional boundary  ⊂ (M × M), the unit normal vector may be represented as n() = (− sin , cos , 0, 0) or n() = (0, 0, − sin , cos ), where tan  is a slope of a line segment of M.Moreover, the backward discontinuity set D  consists of two 3-dimensional polyhedral regions  1 =  ×  and  2 =  × , with n( 1 ) = (−sin , cos , 0, 0), n( 2 ) = (0, 0, − sin , cos ), where tan  is the slope of the line segment .By the analysis in Section 3.4, we can obtain that there exists a natural number  0 such that for every point  ∈ (M × M),    () ∈

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  ∈ SE(M) and  ∈ SE(X) be two PWIs and  =  ×  :  ×  →  × , every periodic cell () of  can be written as () × (), where  and  are two admissible periodic codings of  and , respectively.Proposition 18.Let   ∈ SE(M  ) ( = 1, 2, . . ., ) be  planar PWIs and  =  1 ×  2 × ⋅ ⋅ ⋅ ×   , if each of them is an irrational rotation, then every periodic cell  of the map  can be represented as  =  1 ×  2 × ⋅ ⋅ ⋅ ×   , where   is a periodic disk of the map   for all  = 1, 2, . . ., .Moreover, the periodic cell packing induced by the PWI  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 /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  which are incommensurable with .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 / irrational, the induced disk packings are tangentfree.

Figure 1 :
Figure 1: (a) The illustration to the invariant disks packing of the Sigma-Delta map for the parameter  = 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.

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

3. 5 .
The Product Goetz Map.Now we investigate the socalled product Goetz map   =   ×   :  ×   →  ×  which is a 4-dimensional PWI with partition

Figure 3 :
Figure 3: The illustration to invariant disk packing induced by the map(17), where the parameter  = 1.78.We can observe that all the periodic disks are tangent-free.
3.1.Periodic Cells.If  is even and  =  0 ⋅ ⋅ ⋅  −1 is admissible, then the linear part   of the map   = (  , V) can be orthogonally diagonalized as diag{ 1 ,  2 , . . .,  /2 } under one orthogonal basis, where   = (cos   , sin   ; − sin   , cos   ),   ∈ [0, 2).It is obvious that an even dimensional orientation-preserving PWI is incommensurate if and only if all   are incommensurable with  in diagonalization representation of   for every finite word .Furthermore, we say the PWI is irrational if it is incommensurable and all   are rationally independent in diagonalization representation of   for every finite word .