Combinatorial invariant for Morse-Smale diffeomorphisms on surfaces with orientable heteroclinic

Area of Science:

• Dynamical Systems and Differential Geometry
• Topology

Background:

• Morse-Smale diffeomorphisms on orientable surfaces possess a finite number of heteroclinic orbits.
• Classification of these diffeomorphisms is linked to distinguishing orientable graphs, a problem lacking efficient algorithms.

Purpose of the Study:

• To develop a novel, effective algorithm for classifying orientation-preserving Morse-Smale diffeomorphisms.
• To address the challenge of distinguishing complex orientable graphs associated with these dynamical systems.

Main Methods:

• Associating each diffeomorphism with a specific graph structure.
• Developing an effective algorithm for graph isomorphism testing based on these associated graphs.
• Identifying a class of admissible graphs realizable by surface diffeomorphisms.

Main Results:

• A new approach to classifying Morse-Smale diffeomorphisms is proposed.
• An effective algorithm for determining graph isomorphism is constructed.
• A class of admissible graphs is identified, linked to surface diffeomorphisms with orientable heteroclinics.

Conclusions:

• The findings provide a method for constructing representatives in homotopy classes of homeomorphisms.
• This work offers an approach to solving the open problem of realizing Nielsen's algebraically finite type homeomorphisms.