The Dynamics of a Piecewise Linear Map and its Smooth Approximation

The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fe: (x, y) → (1 - y - fe(x), x), where fe(x) is real analytic and close to |x| for small values of e. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fe, provided that e is small.