Non-hyperbolic One-dimensional Invariant Sets with a Countably Infinite Collection of Inhomogeneities

In this paper we examine the structure of countable closed invariant sets under a dynamical system on a compact metric space. We are motivated by a desire to understand the possible structures of inhomogeneities in one-dimensional non-hyperbolic sets (inverse limits of finite graphs) particularly when those inhomogeneities form a countable set. All of the previous literature regarding the collection of inhomogeneities of these spaces focuses on the case when the collection of inhomogeneities is either finite, a Cantor set, or the entire space. These are interesting cases; however they do not exhaust all of the possibilities. We address the first case not previously covered: when the collection of inhomogeneities is countable. OR???: We examine the possible structure of In, the set of of inhomogeneities of a one-dimensional non-hyperbolic set (inverse limits of finite graphs). All of the previous literature regarding the inhomogeneities of such spaces focuses on the case when In is either finite, a Cantor set, or the entire space. These are interesting cases; however they by no means exhaust the possibilities. We address the first case not previously covered: when the collection of inhomogeneities is countable. We prove a surprising restriction on the topology of countable In. Conversely, using a novel application of techniques from descriptive set theory to construct various tent map cores, we show that this restriction in fact completely characterizes the structure of countable In.