Title:

Diffeomorphisms on surfaces with a finite number of moduli

One of the purposes of the theory of Dynamical Systems is to understand the orbit structure of diffeomorphisms. Here we say that two diffeomorphisms f and g have the same orbit structure if they are conjugate, that is, if there is a homeomorphism h of the ambient manifold such that hf = gh. Basically the only kind of diffeomorphisms whose dynamics are well studied, are the class of structurally stable diffeomorphisms. These are diffeomorphisms f such that all nearby diffeomorphisms are conjugate to f. The well known Structural stability Theorem says that a diffeomorphism is structurally stable if it is Axiom A and if all invariant manifolds are transversal to each other. If these transversality conditions are not satisfied then the diffeomorphism not only fails to be stable, but also this gives rise to the appearence of moduli. That is, one needs several real parameters to parameterise all conjugacy classes of nearby diffeomorphisms. (The minimum number of parameters needed is called the number of moduli.) Here we deal with diffeomorphisms on two dimensional manifolds whose asymptotic dynamics are well understood, (the class of Axiom A diffeomorphisms). The Main Result characterises those Axiom A diffeomorphisms which have a finite number of moduli. This result can be regarded as a generalisation of the Structural Stability Theorem. From the proofs it follows that the dynamics of these diffeomorphisms can also be well understood. In the proof of our Main Theorem we need certain invariant foliations to be quite smooth. In an Appendix we prove a differentiable version of the Lambda Lemma.
