Suppose that $f$ is a surface diffeomorphism with a
hyperbolic fixed point $\mathcal O$
and this fixed point has a transversal homoclinic orbit.
It is well known that in a
vicinity of this type of homoclinic there are hyperbolic
invariants sets. We introduce
smooth invariants for the homoclinic orbit which we call
the multipliers. As an
application, we study the influence of the multipliers
on numerical invariants of the
hyperbolic invariant sets as the vicinity becomes small.