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

Consider a compact X 2 . The set  2 is identified with . For P a function S:P is called a rotation, if there exist ρ P , z P with |ρ P |=1 such that Tx=ρ P x+z P for every xP. A map T:XX is called a piecewise rotation, if there exists a finite partition 𝒫 of X such that T| P is a rotation for every P𝒫. If P𝒫 ρ P k P =1 for integers k P 0 implies k P =0 for all P𝒫, 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 ω of r symbols, and define ω T as the set of all xX whose coding equals ω.

Denote by λ the two-dimensional Lebesgue measure. It is proved that lim T ˜T λ(ω T ˜ )=λ(ω T ), if T is an irrational piecewise rotation. Moreover, if T is an irrational piecewise rotation and λ(ω T )>0, then ω T ˜ converges to ω T in the Hausdorff metric, if T ˜T.

Define B T as the set of all xX whose orbit intersects the boundary of an element of 𝒫. The author proves that lim sup T ˜T λ(B T ^ )λ(B T ) for every irrational piecewise rotation. This implies that the map Tλ(B T ) is continuous on a dense G δ subset of the space of piecewise rotations on X.

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