Bizarre Topology Is Natural in Dynamical Systems

We describe an example of a C°° diffeomorphism on a 7-manifold which has a compact invariant set such that uncountably many of its connected components are pseudocircles. (Any 7-manifold will suffice.) Furthermore, any diffeomorphism which is sufficiently close (in the C1 metric) to the constructed map has a similar invariant set, and the dynamics of the map on the invariant set are chaotic. Compact invariant sets of typical diffeomorphisms are often pictured as being points or manifolds or as sets which are locally the product of a manifold and a Cantor set. Such views would mislead one to believe more exotic topological structures are either exceptional or absent. This is not the case. We report here on an example of a C°° diffeomorphism F with the following features: (1) There is a compact invariant set B (that'is, F(B) = B) which is maximal in the sense that there is a neighborhood U of B such that if Fn(x) is in U for all integers n , then x is in B. (2) There is a point in B whose trajectory is dense in B under F . (3) The number of connected components of B is uncountable. Most of the components in the sense of category are, from most perspectives, topologically bizarre objects known as pseudocircles. (4) For each diffeomorphism G in some C1 neighborhood of F , the maximal invariant set BG of some neighborhood Ug has a similar topological structure, in that it consists of an uncountable collection of connected components, and most of the components of the invariant set are pseudocircles. What then is a pseudocircle, and what is bizarre about its topology? Pseudocircles have topological dimension 1, but they are arcwise totally disconnected: every continuous map of an interval into a pseudocircle is constant. A continuum Received by the editors August 24, 1993, and, in revised form, November 23, 1994. 1991 Mathematics Subject Classification. Primary 54F20, 54F50, 54F12.