Self-similar constructions in smooth dynamics: Rigidity, smoothness and dimension

In the first part of this paper, for each d≥2, we construct diffeomorphisms of the d-dimensional ball which have zero entropy, one periodic orbit with period 2n for each n≥0, no other periodic orbits, and a single invariant Cantor set which has a continuum of possible but, in any case, simple geometric structures. These diffeomorphisms are Cr(d)-smooth, where r(d) is a strictly increasing function of d, which goes to infinity with d. The second part contains a more general result about smooth maps obtained by an infinite sequence of surgeries, and further particular cases. © 1992 Springer-Verlag.