Intersections of thick Cantor sets

Problems in dynamical systems involving homoclinic tangencies, homoclinic bifurcations, and the creation of horseshoes have led to the problem of analysing the difference sets ( t *G. ( *G' + t ) = 0) of Cantor sets *G and *G' embedded in the real line. In this work, the author proves two theorems about difference sets of Cantor sets, both of which involve the concept of the thickness of a Cantor set. The first gives conditions on the thicknesses of two Cantor sets that determine if the intersection of the two Cantor sets must contain a Cantor set or if the intersection may, in a nontrivial way, be as small as one point. The second theorem states that if the product of the thicknesses of two Cantor sets is strictly greater than one, then for a generic point t in their difference set, *G. ( *G' + t ) contains a Cantor set.