Geometry of Cantor Systems

Geometry of Cantor Systems

A Cantor system is defined. The geometry of a certain family of Cantor systems is studied. Such a family arises in dynamical systems as hyperbolicity is created. We prove that the bridge geometry of a Cantor system in such a family is uniformly bounded and that the gap geometry is regulated by the size of the leading gap.

Dimension of Cantor Sets with Complicated Geometry

Noncommutative Riemannian Geometry and Diffusion on Ultrametric Cantor Sets

An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree [28]. This tree allows to define a family of spectral triples (CLip(C),H, D) using the l space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here CLip...

Real Coboundaries for Minimal Cantor Systems

In this paper we investigate the role of real-valued coboundaries for classifying of minimal homeomorphisms of the Cantor set. This work follows the work of Giordano, Putnam, and Skau who showed that one can use integer-valued coboundaries to characterize minimal homeomorphisms up to strong orbit equivalence. First, we prove a rigidity result. We show that there is an orbit equivalence between ...

Flow-orbit Equivalence for Minimal Cantor Systems

This paper is about ‡ow-orbit equivalence, a topological analogue of even Kakutani equivalence. In addition to establishing many basic facts about this relation, we characterize the conjugacies of induced systems that can be extended to a ‡ow-orbit equivalence. We also describe the relationship between ‡ow-orbit equivalence and a distortion function of an orbit equivalence. We show that if the ...