Interseções estáveis de conjuntos de cantor

Abstract

In this work, we study stable intersections and arithmetic differences of regular Cantor sets. We present techniques that help us to detect the existence of stable intersections or extremal stable intersections between pairs or Cantor sets. We begin with the Newhouse Gap Lemma, which makes use of the thickness of Cantor sets to its end, and the Generalized Thickness Test, which analizes the intersections of Markov domains and to conclude, the existence of a recurrent compact set in the space of relative configurations of these Cantor sets. We also present some examples which shows how such techniques are applied. In certain examples, we emphasize that one of the techniques works, tough the other may fail, in this way showing the importance of each of them. We conclude with a brief study of the topology of the arithmetic difference of Affine Cantor sets, presenting a family of Cantor sets whose arithmetic difference is almost always a R-Cantorval.