A Very Small Remark on Smooth Denjoy Maps of the Circle.

Maps of a circle into itself, when considered as discrete dynamical systems, have been observed to contain some of the qualitative behavior of complicated experimental situations. Much of this structure can be understood for homeomorphisms of the circle by considering the invariant called the rotation number. In fact, the homeomorphisms of the circle with irrational rotation number can be classified up to topological conjugacy, provided the map and its inverse are sufficiently smooth. Counter examples to this classification scheme exist which are C sub 1 diffeomorphisms or C infinity homeomorphisms with non differentiable inverse. Whether C infinity can be replaced with analytic in the second examples is an open question. In this report the authors show that if certain assumptions are made on the rotation number and on the derivative of a homeomorphism from the circle onto itself, then even if the map does not fit into the classification scheme we can still obtain estimates on the behavior of its orbits under iteration.