Preprint A06/2001
Homoclinic tangencies and fractal invariants in arbitrary dimension

Marcelo Viana | Moreira, Carlos Gustavo | Palis, Jacob

**Keywords: **
homoclinic tangencies | unfolding bifurcations | codimension one | hyperbolic set | Hausdorff dimension

We consider the unfolding of a homoclinic tangency in higher dimensions, starting from a hyperbolic system. This implies that the homoclinic tangency is associated to a periodic point whose unstable (stable) manifold is of dimension (codimension-one, homoclinic tangency). In general the periodic point is part of a hyperbolic set. We prove that the following paradigm holds for such homoclinic bifurcations in any dimension:
hyperbolicity prevails if and only if the Hausdorff dimension of the associated hyperbolic set is smaller than 1.