Invariant curves and explosion of periodic Islands in systems of piecewise rotations

Abstract

Invertible piecewise isometric maps (PWIs) of the plane, in spite of their apparent simplicity, can show a remarkable number of dynamical features analogous to those found in nonlinear smooth area preserving maps. There is a natural partition of the phase space into an exceptional set, ⋶, consisting of the closure of the set of points whose orbits accumulate on discontinuities of the map, and its complement. In this paper we examine a family of noninvertible PWIs on the plane that consist of rotations on each of four atoms, each of which is a quadrant. We show that this family gives examples of global attractors with a variety of geometric structures. On some of these attractors, there appear to be nonsmooth invariant curves within ⋶ that form barriers to ergodicity of any invariant measure supported on ⋶. These invariant curves are observed to appear on perturbations of an “integrable” case where the exceptional set is a union of annuli and it decomposes into a one-dimensional family of interval exchange maps that may be minimal but nonergodic. We have no adequate theoretical explanation for the curves in the nonsmooth case, but they appear to come into existence at the same times as an explosion of periodic islands near where the interval exchanges used to be located. We exhibit another example—a piecewise rotation on the plane with two atoms that also appears to have nonsmooth invariant curves.