Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets

Suppose that f is a surface diffeomorphism with a hyperbolic fixed point O and this fixed point has a transversal homoclinic orbit. It is well known t hat in a vicinity of this type of homoclinic there are hyperbolic invariant s sets. We introduce smooth invariants for the homoclinic orbit which we ca rl the multipliers. As an application, we study the influence of the multip liers on numerical invariants of the hyperbolic invariant sets as the vicin ity becomes small.