Fixed Point Composition and Toeplitz-Composition C*-algebras

Let $\varphi$ be a linear-fractional, non-automorphism self-map of $\mathbb{D}$ that fixes $\zeta \in \mathbb{T}$ and satisfies $\varphi^{\prime}(\zeta) \neq 1$ and consider the composition operator $C_{\varphi}$ acting on the Hardy space $H^2(\mathbb{D}).$ We determine which linear-fractionally-induced composition operators are contained in the unital C$^*$-algebra generated by $C_{\varphi}$ and the ideal $\mathcal{K}$ of compact operators. We apply these results to show that $C^*(C_{\varphi}, \mathcal{K})$ and $C^*(\mathcal{F}_{\zeta})$, the unital C$^*$-algebra generated by all composition operators induced by linear-fractional, non-automorphism self-maps of $\mathbb{D}$ that fix $\zeta$, are each isomorphic, modulo the ideal of compact operators, to a unitization of a crossed product of $C_0([0,1])$. We compute the K-theory of $C^*(C_{\varphi}, \mathcal{K})$ and calculate the essential spectra of a class of operators in this C$^*$-algebra. We also obtain a full description of the structures, modulo the ideal of compact operators, of the C$^*$-algebras generated by the unilateral shift $T_z$ and a single linear-fractionally-induced composition operator.