# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Stability of piecewise rotations and affine maps. (English) Zbl 1031.37039

Consider a compact $X\subseteq {ℝ}^{2}$. The set ${ℝ}^{2}$ is identified with $ℂ$. For $P\subseteq ℂ$ a function $S:P\to ℂ$ is called a rotation, if there exist ${\rho }_{P}$, ${z}_{P}\in ℂ$ 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 $𝒫$ of $X$ such that ${T|}_{P}$ is a rotation for every $P\in 𝒫$. If ${\prod }_{P\in 𝒫}{\rho }_{P}^{{k}_{P}}=1$ for integers ${k}_{P}\ge 0$ implies ${k}_{P}=0$ for all $P\in 𝒫$, 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 $𝒫$ consists of $r$ elements. Consider a one-sided sequence $\omega$ of $r$ symbols, and define ${〈\omega 〉}_{T}$ as the set of all $x\in X$ whose coding equals $\omega$.

Denote by $\lambda$ the two-dimensional Lebesgue measure. It is proved that ${lim}_{\stackrel{˜}{T}\to T}\lambda \left({〈\omega 〉}_{\stackrel{˜}{T}}\right)=\lambda \left({〈\omega 〉}_{T}\right)$, if $T$ is an irrational piecewise rotation. Moreover, if $T$ is an irrational piecewise rotation and $\lambda \left({〈\omega 〉}_{T}\right)>0$, then ${〈\omega 〉}_{\stackrel{˜}{T}}$ converges to ${〈\omega 〉}_{T}$ in the Hausdorff metric, if $\stackrel{˜}{T}\to T$.

Define ${B}_{T}$ as the set of all $x\in X$ whose orbit intersects the boundary of an element of $𝒫$. The author proves that ${lim sup}_{\stackrel{˜}{T}\to T}\lambda \left({B}_{\stackrel{^}{T}}\right)\le \lambda \left({B}_{T}\right)$ for every irrational piecewise rotation. This implies that the map $T↦\lambda \left({B}_{T}\right)$ is continuous on a dense ${G}_{\delta }$ subset of the space of piecewise rotations on $X$.

##### MSC:
 37E99 Low-dimensional dynamical systems 37B10 Symbolic dynamics 37C75 Stability theory 37B05 Transformations and group actions with special properties