Orbit structures of homeomorphisms.




Sherman, Casey L.

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In 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.



Cantor set., Homeomorphism., Orbit structure., Inverse limit space., Dynamical systems., Universal compact metric space.


Good, 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.