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Stability of piecewise rotations and affine maps. (English) Zbl 1031.37039

Consider a compact \(X\subseteq{\mathbb R}^{2}\). The set \({\mathbb R}^{2}\) is identified with \({\mathbb C}\). For \(P\subseteq{\mathbb C}\) a function \(S:P\to{\mathbb C}\) is called a rotation, if there exist \(\rho_{P}\), \(z_{P}\in{\mathbb C}\) with \(|\rho_{P}|=1\) such that \(Tx=\rho_{P}x+z_{P}\) for every \(x\in P\). A map \(T:X\to X\) is called a piecewise rotation, if there exists a finite partition \({\mathcal P}\) of \(X\) such that \(T|_{P}\) is a rotation for every \(P\in{\mathcal P}\). If \(\prod_{P\in{\mathcal P}}\rho_{P}^{k_{P}}= 1\) for integers \(k_{P}\geq 0\) implies \(k_{P}=0\) for all \(P\in{\mathcal P}\), then the piecewise rotation \(T\) is called irrational. The author defines a very natural topology on the set of all piecewise rotations on \(X\). In a standard way \(T\) is semi-conjugate to a subshift of the one-sided shift on \(r\) symbols via a coding, if \({\mathcal P}\) consists of \(r\) elements. Consider a one-sided sequence \(\omega\) of \(r\) symbols, and define \(\langle\omega\rangle_{T}\) as the set of all \(x\in X\) whose coding equals \(\omega\).
Denote by \(\lambda\) the two-dimensional Lebesgue measure. It is proved that \(\lim_{\widetilde{T}\to T} \lambda (\langle\omega\rangle_{\widetilde{T}})= \lambda (\langle\omega\rangle_{T})\), if \(T\) is an irrational piecewise rotation. Moreover, if \(T\) is an irrational piecewise rotation and \(\lambda (\langle\omega\rangle_{T})>0\), then \(\langle\omega\rangle_{\widetilde{T}}\) converges to \(\langle\omega\rangle_{T}\) in the Hausdorff metric, if \(\widetilde{T}\to T\).
Define \(B_{T}\) as the set of all \(x\in X\) whose orbit intersects the boundary of an element of \({\mathcal P}\). The author proves that \(\limsup_{\widetilde{T}\to T} \lambda (B_{\widehat{T}})\leq \lambda (B_{T})\) for every irrational piecewise rotation. This implies that the map \(T\mapsto\lambda (B_{T})\) is continuous on a dense \(G_{\delta}\) subset of the space of piecewise rotations on \(X\).
Reviewer: Peter Raith (Wien)

MSC:

37E99 Low-dimensional dynamical systems
37B10 Symbolic dynamics
37C75 Stability theory for smooth dynamical systems
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
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