Consider a compact . The set is identified with . For a function is called a rotation, if there exist , with such that for every . A map is called a piecewise rotation, if there exists a finite partition of such that is a rotation for every . If for integers implies for all , then the piecewise rotation is called irrational. The author defines a very natural topology on the set of all piecewise rotations on . In a standard way is semi-conjugate to a subshift of the one-sided shift on symbols via a coding, if consists of elements. Consider a one-sided sequence of symbols, and define as the set of all whose coding equals .
Denote by the two-dimensional Lebesgue measure. It is proved that , if is an irrational piecewise rotation. Moreover, if is an irrational piecewise rotation and , then converges to in the Hausdorff metric, if .
Define as the set of all whose orbit intersects the boundary of an element of . The author proves that for every irrational piecewise rotation. This implies that the map is continuous on a dense subset of the space of piecewise rotations on .