Solenoids of Split Sequences
Sarasi Jayasekara
TL;DR
This work introduces Solenoids of Split Sequences as inverse limits of fold maps, develops a robust topology and natural singular foliated structure for these solenoids, and establishes a criterion for when such solenoids are minimal and uniquely ergodic. Central to the analysis is the space TM(X) of transverse measures, shown to be isomorphic to the inverse limit of a sequence of convex weight-cones Λ_j(ζ) driven by fold-transition matrices T_j; this yields a finite-dimensional, cone-shaped TM(X) that can be effectively studied via linear-algebraic data. A key contribution is the Semi-Normality criterion, which identifies a wide class of solenoids that are uniquely ergodic and, in turn, fully Mingling and topologically minimal. The framework also provides a detailed atlas of the solenoid using Turn/Extended Turn Tunnels and Star Tunnel Sets, connecting the foliation-like structure to explicit combinatorial and geometric data, with implications for future connections to currents and dynamics on free groups. Overall, the paper blends inverse-limit dynamics, combinatorial graph-folding, and singular foliated geometry to illuminate the ergodic and minimal properties of a broad family of solenoids.
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.