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Iterating an orientation-preserving piecewise isometry

Piecewise isometries (or PWIs in short) appear in a variety of contexts, including digital filters [

In [

Iterating an orientation-preserving piecewise isometry

(a) The illustration to the invariant disks packing of the Sigma-Delta map for the parameter

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 [

There is at most a countable set of

In [

There is a countable set of

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.

Whether does the disk packing induced by a planar PWI has no tangencies if the rotation parameter

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

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 [

The rest of this paper is organized as follows. In Section

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 [

The Euclidean group

Let

In this paper, we consider the case where the orientation-preserving piecewise isometries (

Let

In the following, for convenience, we say

The partition

If a coding

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

For a planar piecewise isometry with convex polygonal partition, a cell

In [

Let

Based on the fact revealed by Mendes and Nicol in [

If

Let

Suppose

The proof of the convexity of the cell is similar to the one given in [

Since the map

Now, we prove the latter claim. Suppose, without loss of generality, the map

According to Theorem

Let

Suppose

Note that the result in the above theorem is a generalization of the results in [

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

For a piecewise isometry

We must point out that two periodic cells are separated by an

For a piecewise isometry

Assume that the periodic codings

Similar to the planar cases [

In fact, the mutually tangent set

Suppose

For a class of PWIs, the backward discontinuity set consists of one hyperplane or finitely many parallel hyperplanes. Namely,

Suppose

For convenience of expression, we give some notations introduced in [

Suppose

From the above result, we further have the following.

Suppose

Without loss of generality, we assume that the mutually tangent set is contained in the hyperplane

We are now ready to give the proof of Theorem

Firstly, we will reveal that there exists a natural number

Suppose that there exist two periodic cells

For planar piecewise rotations, the corresponding results are very explicit. Let

Suppose

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

In particular, if an invertible planar piecewise rotation has the common irrational rotation angle

If an invertible planar piecewise rotation has the common irrational rotation angle

Now we consider a map called the (bounded) Goetz map

The piecewise isometric map (

The piecewise isometric map is invertible, and the backward discontinuity set

For the piecewise isometry defined by (

The illustration to invariant disk packing induced by the map (

Now we investigate the so-called product Goetz map

Let

Let

Obviously, if a

It is obvious that

In fact, we can find that

We consider below the tangent-free property between any two periodic cells. Obviously, for any

If

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.

Let

Let

Generally, for a PWI of even-dimensional Euclidean space, we guess that every periodic cell may be written as the topological product space of

As mentioned in Section

For the Sigma-Delta map and the Overflow map with

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.