Chaos, inverse limit spaces, and recurrence

In 1975, Li and Yorke gave a definition of chaos on a compact metric space. On the interval, if a continuous map has positive topological entropy, then it is chaotic. But the converse is not true. In this thesis, we define a concept called {dollar}omega{dollar}-chaos. The definition of {dollar}omega{dollar}-chaos parallels the definition of chaos given by Li and Yorke. But a continuous map is {dollar}omega{dollar}-chaotic if, and only if, it has positive topological entropy. Moreover, we prove that a continuous map is {dollar}omega{dollar}-chaotic if, and only if, it is chaotic in the sense of Devaney.; If a continuous map f from a compact metric space to itself is {dollar}omega{dollar}-chaotic, then the shift map {dollar}sigmasb f{dollar} on the inverse limit space induced by it is also {dollar}omega{dollar}-chaotic. With some modification of the definition, the converse is true. {dollar}sigmasb f{dollar} is chaotic in the sense of Devaney if, and only if, f is chaotic in the sense of Devaney. An expanding map under which every point is nonwandering on a 1-branched manifold is both {dollar}omega{dollar}-chaotic and chaotic in the sense of Devaney. We give a sufficient condition for a transitive continuous map to have sensitive dependence on initial conditions. A new method of constructing chaotic maps on the pseudoarc is given. On any compact metric space, we prove that the chain recurrent set of the shift map equals the inverse limit space of the chain recurrent set of the sole bonding map. Similar results are proved for the nonwandering set, {dollar}omega{dollar}-limit set, recurrent set, and almost periodic set.; On the interval, we also prove that a nonwandering point, which is not an {dollar}omega{dollar}-limit point, is in the orbit of a turning point. We say a point is a {dollar}gamma{dollar}-limit point if it is both an {dollar}omega{dollar}-limit point and an {dollar}alpha{dollar}-limit point of a point. A nonwandering point which is not a {dollar}gamma{dollar}-limit is in the closure of the orbit of turning points.; Last, we study Type 2{dollar}spinfty{dollar} maps on the interval and the structure of Type 2{dollar}spinfty{dollar} unimodal maps.