Smale diffeomorphisms of surfaces: a classification algorithm (Q1884222)

The object of the paper is to present a finite algorithm to determine whether two Smale diffeomorphisms \(f\) and \(g\) of compact surfaces \(M_f\) and \(M_g\) are topologically conjugate on neighborhoods of basic sets \(K_f\subset M_f\) and \(K_g\subset M_g\). A Smale diffeomorphism is one that is \(C^1\) structurally stable. It is well known that the nonwandering set of such a diffeomorphism is composed of a finite number of pairwise disjoint compact subsets, each of which is invariant and contains a dense orbit; these are the basic sets. Since the restriction of a Smale diffeomorphism to a basic set is hyperbolic, in the case of a surface, a basic set is either trivial (a single periodic orbit), a Cantor set, or it contains a one-dimensional lamination and is either an attractor or a repeller. The conjugacy question is trivial in the first case, and has been answered in the last case [see \textit{C. Bonatti} and \textit{R. Langevin}, Difféomorphismes de Smale des surfaces. Avec la collaboration d'Emmanuelle Jeandenans. (Smale diffeomorphisms of surfaces. With the collab. of Emmanuelle Jeandenans), Astérisque. 250, Paris: Société Mathématiques de France (1998; Zbl 0922.58058)], so the current paper addresses the case where the basic set is totally disconnected, infinite, and of saddle type (and also concentrates on the case of orientation-preserving diffeomorphisms of oriented surfaces). The work of Bonatti and Langevin referenced above provides combinatorial descriptions of such diffeomorphisms on their basic sets, and shows that if two basic sets \(K_f\), \(K_g\) admit the same combinatorial description, then \(f\) and \(g\) are conjugate on neighborhoods of the basic sets. A complication is that this combinatorial description of a basic set is not unique. The paper under review extends this work by showing that (i) the set of combinatorial descriptions of a basic set \(K\) can be give as a countable list of finite sets \(S_j(K)\) of combinatorial descriptions of \(K\), (ii) given two basic sets \(K_f\) and \(K_g\), one can compute an integer \(p\) with the property that the two basic sets are locally conjugate if and only if \(S_p(K_f) = S_p(K_g)\), and (iii) there is a finite algorithm for computing each set \(S_j(K)\).