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.
Table of contents:
Theorems and examples; Cantor sets and thickness; Proof of Theorem 1.2; Third kind of overlapped point; Second kind of overlapped point; First kind of overlapped point; The dynamics of *V *t *t' ; Results about the geometric process; Proof of Theorem 6.2(1); The boundary between *L1 and *L2; Proof of Theorem 1.1.
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.
Table of contents:
Theorems and examples; Cantor sets and thickness; Proof of Theorem 1.2; Third kind of overlapped point; Second kind of overlapped point; First kind of overlapped point; The dynamics of *V *t *t' ; Results about the geometric process; Proof of Theorem 6.2(1); The boundary between *L1 and *L2; Proof of Theorem 1.1.
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.