For a tuple A = (A₁, A₂, ... , A_n) 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 + ... + z_nA_n is not invertible. We call the complement of P(A) the projective resolvent set, P^c(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 Ω_A = – ω_A^* ∧ ω_A where ω_A is the holomorphic Maurer-Cartan type B-valued (1,0)-form ω_A(z) = A^–1(z)dA(z), we define the Douglas-Yang metric on P^c((I, λ(a), λ(t))). We demonstrate that P^c(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).