Piecewise isometries have zero topological entropy (Q2757009)

Let \(T\) be a piecewise isometry, this means \(X\) is a subset of~\({\mathbb R}^{d}\), there exists a finite collection~\({\mathcal Z}\) of pairwise disjoint open polytopes such that the closure of their union equals~\(X\), and \(T:X\to X\) satisfies that \(T|_{Z}\) is an isometry for every \(Z\in{\mathcal Z}\). The map~\(T\) need not be invertible. In this paper it is proved that the topological entropy of~\(T\) equals zero. This result holds also if \(T\) is a contracting piecewise affine map. However, the author provides a counterexample to this result, if \(T\) is contracting, but not piecewise affine.NEWLINENEWLINENEWLINEAlthough the result seems to be intuitively clear, it has been only conjectured. Only much weaker results were proved before (for example the special cases \(d=1\) and \(d=2\)).NEWLINENEWLINENEWLINEThe proof relies on an idea developped in \textit{M.~Tsujii} [Invent. Math. 143, 349-373 (2001; Zbl 0969.37012)]. Besides of solving a problem, which has been open for long time, this paper is also written very well.