How big is the intersection of two thick Cantor sets? (English) Zbl 0734.54022

Williams, R. F.

Continuum theory and dynamical systems, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Arcata/CA (USA) 1989, Contemp. Math. 117, 163-175 (1991).

[For the entire collection see Zbl 0723.00027.]
The concept of thickness for Cantor sets was introduced by Newhouse in his thesis. In the simplest case of a regular, or “middle part”, Cantor set the Newhouse thickness is the ratio of the length of any of two remaining intervals to the length of the middle gap. Newhouse showed the two Cantor sets must intersect provided that the ratio of their thicknesses exceeds 1 and none is contained in a complementary domain of the other. This paper goes further in trying to determine how “large” the intersection must be. Surprisingly, for two quite thick Cantor sets (thickness arbitrarily close to \(\sqrt{2}+1)\), the intersection may still be one point. On the other hand, if both thicknesses are greater than this number, the intersection must contain a Cantor set.

Reviewer: G.Swiatek (Stony Brook)

MSC:

54F50

Topological spaces of dimension \(\leq 1\); curves, dendrites

Keywords:

thickness for Cantor sets

Citations:

Zbl 0723.00027