Surface diffeomorphisms with connected but not path-connected minimal sets containing arcs

摘要

In 1955, Gottschalk and Hedlund introduced in their book that Jones constructed a minimal homeomorphism whose minimal set is con-nectd but not path-connected and contains infinitely many arcs. However the homeomorphism is defined only on this set. In 1991, Walker first constructed a homeomorphism of S~1 × R with such a minimal set. In this paper, we will show that Walker's homeomorphism cannot be a diffeomorphism (Theorem 2). Furthermore, we will construct a C~∞ diffeomorphism of S~1 × R. with a compact connected but not path-connected minimal set containing arcs (Theorem 1) by using the approximation by conjugation method.