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

TL;DR

This paper classifies Morse-Smale diffeomorphisms on high-dimensional spheres using colored graphs and automorphisms, providing a topological conjugacy criterion and an efficient algorithm for graph isomorphism detection.

Contribution

It introduces a graph-based classification method for Morse-Smale diffeomorphisms on spheres and proves a conjugacy criterion based on graph isomorphism, with a linear-time algorithm for comparison.

Findings

Topological conjugacy corresponds to graph isomorphism.

A linear-time algorithm exists for distinguishing diffeomorphisms via their graphs.

The classification applies to Morse-Smale diffeomorphisms with non-intersecting invariant manifolds.

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 $f, f'\in G(S^n)$ are topologically conjugated iff the graphs $\Gamma_f$, $\Gamma_f'$ are isomorphic. Moreover we establish the existence of a linear-time algorithm for distinguishing two colored graphs of diffeomorphisms from the class $G(S^n)$.