Nonsmoothable, unstable group actions

Abstract

For k > 1 k > 1 there is a nonempty open set of C 1 {C^1} actions of Z k {{\mathbf {Z}}^k} on S 1 {S^1} , no element of which is either topologically conjugate to a C 2 {C^2} action or structurally stable. The C 1 {C^1} closure of this set contains all C 2 {C^2} actions which have compact orbits, so no such action is structurally stable in the space of C 1 {C^1} actions.