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Stability of piecewise rotations and affine maps. (English) Zbl 1031.37039
Consider a compact $X\subseteq{\Bbb R}^{2}$. The set ${\Bbb R}^{2}$ is identified with ${\Bbb C}$. For $P\subseteq{\Bbb C}$ a function $S:P\to{\Bbb C}$ is called a rotation, if there exist $\rho_{P}$, $z_{P}\in{\Bbb C}$ 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 ${\Cal P}$ of $X$ such that $T|_{P}$ is a rotation for every $P\in{\Cal P}$. If $\prod_{P\in{\Cal P}}\rho_{P}^{k_{P}}= 1$ for integers $k_{P}\geq 0$ implies $k_{P}=0$ for all $P\in{\Cal P}$, 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 ${\Cal P}$ 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 ${\Cal P}$. 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 $X$.

37E99Low-dimensional dynamical systems
37B10Symbolic dynamics
37C75Stability theory
37B05Transformations and group actions with special properties
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