Moduli of stability of two-dimensional diffeomorphisms

Abstract

INTRODUCTION IN THIS paper we consider diffeomorphisms on a compact two-dimensional manifold having a pair of hyperbolic periodic points such that the unstable manifold of one of them is tangent to the stable manifold of the other along an orbit. They occur naturally in Bifurcation Theory: there is an open set of arcs, starting at a MorseSmale diffeomorphism, whose first bifurcation point is such a diffeomorphism[3]. It is well known that a diffeomorphism f which exhibits an orbit of tangency between stable and unstable manifolds of hyperbolic periodic points cannot be structurally stable[7]. In fact, this situation gives rise to interesting invariants of topological equivalence, as pointed out in [4], which implies the existence of an uncountable number of different topological equivalence classes in any small neighborhood of f. However, it might be possible to parametrize all these equivalence classes with finitely many real parameters which would provide a pretty good description of the diffeomorphisms near f. In this case we say that the modulus of stability of f is finite and equal to the minimum number of parameters. Here we will prove that in some cases the modulus of stability is finite and in other cases it is infinite and that both situations occur for diffeomorphisms which are first bifurcation points for an open set of arcs starting at Morse-Smale diffeomorphisms. The existence of those tangencies is not a persistent phenomenum in the space of all diffeomorphisms. However, if we restrict our perturbations to a specific subspace it may become persistent. This is the case when we consider the set of G-equivariant diffeomorphisms, where G is a finite group acting on the manifold. The action of G induces a partition of the manifold in submanifolds whose points have the same orbit type and this partition is let invariant by any equivariant diffeomorphism f. If f has two hyperbolic periodic points pl and p2 such that the stable manifold of pI intersects the unstable manifold of p2 and both are contained in the same one-dimensional submanifold of the partition then the same happens for any equivariant diffeomorphism near f. We prove here that, given any integer k, there is an open set of equivariant diffeomorphisms whose modulus of stability is exactly k.