Morse-Smale Diffeomorphisms and the Standard Family

Abstract

Every Morse-Smale diffeomorphism of the circle is conjugate to a diffeomorphism belonging to the set defined by \[ f_{\omega,\varepsilon,k}(x)=x+\omega+\frac{\varepsilon}{2\pi}\sin(2k\pi x) \quad (0<\omega<1, 0<\varepsilon<1, k \text{ with } 0<\varepsilon k<1) \] and Morse-Smale diffeomorphisms in the set is $C^1$ open and dense, with respect to the relative topology, in Amol'd tongue.