For a tuple A = (A1, A2, ... , An) of elements in a unital Banach algebra B its projective (joint) spectrum P(A) is the collection of z ∈ Cn such that A(z) = z1A1 + z2A2 + ... + znAn is not invertible. We call the complement of P(A) the projective resolvent set, Pc(A) = Cn \ P(A). In this dissertation the primary focus will be on the infinite dihedral group D∞ = ⟨ a,t | a2 = t2 = 1 ⟩ and the left regular representation λ acting on l2(D∞) giving the tuple (I, λ(a), λ(t)). First, using the fundamental form ΩA = – ωA* ∧ ωA where ωA is the holomorphic Maurer-Cartan type B-valued (1,0)-form ωA(z) = A–1(z)dA(z), we define the Douglas-Yang metric on Pc((I, λ(a), λ(t))). We demonstrate that Pc(A) with respect to this metric is not complete. Next we turn our attention to the mapping defined by Grigorchuk and Yang (2017) from the self-similarity of D∞, F(z) = (z0(z02 – z12 – z22), z12 z2, z2(z02 – z22)) and examine Fatou-Julia theory of the mapping. Utilizing projective space to examine F : P2 → P2 allows us to define the Julia set of F(z). Then we demonstrate a clear connection between the projective spectrum of (I, λ(a), λ(t)) and the Julia set of F(z).
|Commitee:||Beceanu, Marius, Radulescu, Anca, Stessin, Michael, Tchernev, Alexandre|
|School:||State University of New York at Albany|
|Department:||Mathematics and Statistics|
|School Location:||United States -- New York|
|Source:||DAI-B 80/09(E), Dissertation Abstracts International|
|Keywords:||Douglas-Yang metric, Dynamics, Grigorchuk group, Infinite dihedral group, Julia set, Projective spectrum|
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