Knot as a Complete Invariant of the Diffeomorphism of Surfaces with Three Periodic Orbits

It is known thatMorse-Smale diffeomorphisms with two hyperbolic periodic orbitsexist only on the sphereand they are all topologically conjugate to each other.However,if we allow three orbits to existthen the range of manifolds admitting themwidens considerably.In particular,the surfaces of arbitrary genus admitsuch orientation-preserving diffeomorphisms.In this article we find a complete invariant for the topological conjugacy ofMorse-Smale diffeomorphismswith three periodic orbits.The invariant is completely determined by the homotopy type(a pair of coprime numbers)of the torus knotwhich is the space of orbits of an unstable saddle separatrixin the space of orbits of the sink basin.We use the result to calculatethe exact number of the topological conjugacy classesof diffeomorphisms under considerationon a given surfaceas well as to relate the genus of the surfaceto the homotopy type of the knot.