Solenoids of Split Sequences

Abstract

Solenoids induced by split sequences are introduced, as the inverse limit object of a sequence of fold maps. The topology of a solenoid is explored, and it is established that solenoids have naturally arising singular foliated structures. Our main goal is to answer the question: ``When is a solenoid minimal, both in a topological sense, and a measure theoretic sense?" To aid this, we introduce the notions of leaves, partial leaves and transversals of a solenoid and explore their properties. A combinatorial criterion for topological minimality of a solenoid, is introduced. The primary tool we construct to study dynamics of solenoids is contained in the following theorem: When a given solenoid $X$ doesn't contain finite partial leaves, the space of transverse measures of $X$, denoted $TM(X)$, is equal to the inverse limit of a certain sequence of linear maps on convex cones. We use this machinery to show that $TM(X)$ is a finite dimensional cone, and then to provide a combinatorial criterion called ``Semi-Normality" that allows us to recognize a wide class of uniquely ergodic solenoids.