On Topological Classification of Morse-Smale Diffeomorphisms on the Sphere $S^n$
Vyachesval Grines, Elena Gurevich, Olga Pochinka, Dmitrii Malyshev

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 of orientation preserving Morse-Smale diffeomorphisms of the sphere of dimension in assumption that invariant manifolds of different saddle periodic points have no intersection. We put in a correspondence for every diffeomorphism a colored graph enriched by an automorphism . Then we define the notion of isomorphism between two colored graphs and prove that two diffeomorphisms are topologically conjugated iff the graphs , are isomorphic. Moreover we establish the existence of a linear-time algorithm for distinguishing two colored graphs of diffeomorphisms from the class .
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