A note on the coding of orbits in certain discontinuous maps (Q968753)

Let \((X,d)\) be a metric space and \(\{P_1,\dots,P_m\}\) a finite collection of open sets defining a topological partition of \(X\) i.e. such that \(P_i\cap P_j=\emptyset\), \(\forall i,j\) and \(X=\bigcup_1^m\bar{P}_i\). Let \(f:\bigcup_1^m\bar{P}_i\mapsto X\) be such that for every \(i\), \(f_i:=f\mid_{P_i}\) is continuous; since in general \(f\) does not admit a continuous extension to \(X\), let \(D=\bigcup_1^m\partial\bar{P}_i\) be the discontinuity set. Two situations are discussed: when \(f_i\) are homeomorphisms and \(f\) is thus a \textit{piecewise homeomorphism} and when \(f_i\) are isometries and \(f\) is thus a \textit{piecewise isometry}. The paper is concerned with the discontinuity set and its preimage which separate points in \(X\) according to their dynamical behavior. Those whose orbits are arbitrarily close to the discontinuity show such an evolution that might be characterized by a coding map. These maps are studied for piecewise homeomorphisms and piecewise isometries separately.