On Topological Classification of Morse-Smale Diffeomorphisms on the Sphere $S^n$

Abstract

We consider a class $G(S^n)$ of orientation preserving Morse-Smale diffeomorphisms of the sphere $S^n$ of dimension $n>3$ in assumption that invariant manifolds of different saddle periodic points have no intersection. We put in a correspondence for every diffeomorphism $f\in G(S^n)$ a colored graph ${\Gamma }_f$ enriched by an automorphism $P_f$. Then we define the notion of isomorphism between two colored graphs and prove that two diffeomorphisms $f,f^{\prime }\in G(S^n)$ are topologically conjugated iff the graphs ${\Gamma }_f$, ${\Gamma }_f^{\prime }$ are isomorphic. Moreover we establish the existence of a linear-time algorithm for distinguishing two colored graphs of diffeomorphisms from the class $G(S^n)$.