Algebraic Topology for Minimal Cantor Sets

Abstract

It will be shown that every minimal Cantor set can be obtained as a projective limit of directed graphs. This allows to study minimal Cantor sets by algebraic topological means. In particular, homology, homotopy and cohomology are related to the dynamics of minimal Cantor sets. These techniques allow to explicitly illustrate the variety of dynamical behavior possible in minimal Cantor sets.

All minimal Cantor systems are slow

“…In this section, we recall notions from the theory of graph covers that we are going to use in the proof of Theorem . For more on graph covers, and various recent applications see, for example, .…”

On invariant measures of finite affine type tilings

“…This last theorem allows us to exhibit some criteria to bound the number of invariant probabilities by a standard way (see [4]). Proof.…”

Realization of a Choquet simplex as the set of invariant probability measures of a tiling system

“…and (A n ∈ M k n ×k n+1 (N)) n≥0 is the sequence of incidence matrices associated to (P n ) n≥0 [10].…”