Minimal Cantor Sets: The Combinatorial Construction of Ergodic Families and Semi-Conjugations

Abstract

Combinatorially obtained minimal Cantor sets are acquired as the inverse limit of certain directed topological graphs where specific nonnegative integer matrices, called winding matrices, are used to describe the projection between each graph. Examples of non-uniquely ergodic combinatorially obtained minimal Cantor sets first appeared in the 2006 article Algebraic topology for minimal Cantor sets of Gambaudo and Martens and are constructed using winding matrices whose entries grow “fast enough.” In this work, we will introduce families of minimal Cantor sets which may be combinatorially obtained in such a way that the corresponding winding matrices possess unbounded entries given by explicit sequences of nonnegative integers. For each of these families, the growth rate needed to achieve either unique or non-unique ergodicity will be specifically addressed and the result will be applied to the case of minimal Cantor sets corresponding to Lorenz maps. We will explore the construction of topological semi-conjugations between combinatorially obtained minimal Cantor sets. Theorems guaranteeing the existence of a topological semi-conjugation between specific families of these sets will be proved and utilized to introduce examples possessing additional intriguing properties. We will also show that there exist both finitely and infinitely non- uniquely ergodic minimal Cantor sets semi-conjugated to an adding machine in such a way that the semi-conjugation map is almost-everywhere injective with respect to the unique ergodic invariant probability measure on the adding machine. Furthermore, we will prove that this construction can be realized by the dynamical system (ω(c),q) where q is a logistic unimodal map with critical point c and omega-limit set ω(c).