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Dynamics of a piecewise rotation. (English) Zbl 0965.37037

This paper deals with the dynamics of systems generalizing interval exchanges to a class of Euclidean two-dimensional piecewise isometries. Due to the lack of hyperbolicity this class exhibits many features of attractors. More precisely, the author investigates noninvertible piecewise isometries in the 2D case with particular interest of the maximal invariant sets and $\omega$-limit sets.

The main result states that for a certain class of noninvertible piecewise isometries, orbits visiting both atoms in finitely often must accumulate on the boundaries of the attractor consisting of two maximal invariant discs ${D}_{0}$ and ${D}_{1}$. The key idea is the proof of a dynamical and geometric observation about the monotonic behaviour of orbits of the first-return map to one of the atoms.

##### MSC:
 37E30 Homeomorphisms and diffeomorphisms of planes and surfaces 37B10 Symbolic dynamics
##### Keywords:
isometry; noninvertible; piecewise map; attractor; hyperbolicity