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In mathematics, a piecewise isometry is a dynamical system that generalizes an interval exchange transformation to higher dimensions. The phase space consists of a subset of $$\R^N$$, with $N \geq 2 $ and is partitioned into finitely many polytopes called atoms. The action of the map on each atom is an orientation preserving euclidean isometry.

Formal definition
Let $$X$$ be a subset of $$\R^N$$ and let $$P=\{P_1,...,P_d\}$$, with $$d>1$$, be a finite partition of $$X$$ into convex sets (or atoms), that is $\bigcup_{1\leq j\leq d}X_j=X$ and $$X_i \cap X_j = \emptyset$$ for $$ i \neq j$$. A piecewise isometry (PWI) is a map $$T:X \rightarrow X$$ such that its restriction to each atom $$P_j$$ is a an euclidean isometry.

When $$N=2$$, the map $$T$$ is called a planar piecewise isometry. In particular orientation preserving PWIs are the most widely studied. In this case it is convenient to take advantage of the isomorphism $$\R^2 \simeq \C$$ and write $$T$$ in complex notation as$$T(z)=e^{i\theta_j}z+\lambda_j, \quad \textrm{for} \ z \in X_j,$$for some $$\theta_j \in [0,2\pi)$$ and $$\lambda_j \in \C$$, for $$j=1,...,d$$.

Properties
Bijective PWIs preserve Lebesgue measure. It was proved by Buzzi that piecewise isometries have zero topological entropy. Generic properties of PWIs are still not widely understood. However studied examples indicate that PWIs can exhibit remarkable renormalization dynamics.

Applications
Planar piecewise isometries have been found in models used for signal processing and digital filters, printing processes , granular mixing , Hamiltonian systems and outer billiards.