Raines, Brian Edward, 1975-Sherman, Casey L.2012-11-292017-04-072012-11-292017-04-072012-082012-11-29Good, C., Greenwood, S., Raines, B. E., & Sherman, C. L. "A compact metric space that is universal for orbit spectra of homeomorphisms." Advances in Mathematics 229, #5 (2012): 2670-2685.Sherman, Casey. "A Lebesgue-like measure for inverse limit spaces of piecewise strictly monotone maps of an interval." Topology and its Applications 159, 8 (2012): 2062-2070.http://hdl.handle.net/2104/8516In this dissertation we answer the following question: If X is a Cantor set and T: X → to X is a homeomorphism, what possible orbit structures can T have? The answer is given in terms of the orbit spectrum of T. If X is a Cantor set, then there is a homeomorphism T : X → to X with σ(T) = (0, ζ, σ₁, σ₂, σ₃, …) if and only if one of the following holds: 1) ζ = 0, there exists k ∈ N and a set {n₁ … ,nk} with σ _{n_i} > 0 for each 1 ≤ i ≤ k such that if σ _j > 0 then there exists i ∈ {1, 2, …, k} with n_i|j and there is an m ∈ N with σ _{mj} = c. 2) 1 ≤ ζ < c, {n: σ_ n= c} is infinite, and ∑ σ_ n : σ_ {mn} < c { for all m∈N} ≤ ζ, or 3) ζ = c.en-USBaylor University theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact librarywebmaster@baylor.edu for inquiries about permission.Cantor set.Homeomorphism.Orbit structure.Inverse limit space.Dynamical systems.Universal compact metric space.ThesisWorldwide access