A Simple Family of Exceptional Maps with Chaotic Behavior

Abstract

A simple family of maps in ${\mathbb {T}}^2$ is considered in this note. It displays chaos in the sense that the dynamics has sensitive dependence to initial conditions and topological transitivity. Furthermore the set of points displaying chaotic behavior has full Lebesgue measure in ${\mathbb {T}}^2$. However the maps have neither homoclinic nor heteroclinic orbits and have a single fixed point which is parabolic, with an unstable branch and a stable one. The role of returning infinitely many times near the fixed point is taken by quasi-periodicity. The maximal Lyapunov exponent is zero. This family was presented as a one-page example in Garrido and Simó (Some ideas about strange attractors. Dynamical systems and chaos (Sitges/Barcelona, 1982). Lecture notes in physics, Springer, Berlin, 1983) (section 2.8). Later we present generalizations and variants.