UNSTABLE DIMENSION VARIABILITY - A SOURCE OF NONHYPERBOLICITY IN CHAOTIC SYSTEMS

The hyperbolicity or nonhyperbolicity of a chaotic set has profound im plications for the dynamics on the set. A familiar mechanism causing n onhyperbolicity is the tangency of the stable and unstable manifolds a t points on the chaotic set. Here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible (respectively noninvertible) maps of dimension 3 (respectively 2) and higher. In pa rticular, we investigate a situation (first considered by Abraham and Smale in 1970 for different purposes) in which the dimension of the un stable (and stable) tangent spaces are not constant over the chaotic s et; we call this unstable dimension variability. A simple two-dimensio nal map that displays behavior typical of this phenomenon is presented and analyzed.