Spectra of Composition Operators on the Unit Ball in Two Complex Variables

Let φ be a self-map of B₂, the unit ball in C². We investigate the equation C_φf=λf where we define C_φf : -f◦φ, with f a function in the Drury Arves on Space. After imposing conditions to keep C_φ bounded and well-behaved, we solve the equation C_φf - λf and determine the spectrum σ(C_φ) in the case where there is no interior fixed point and boundary fixed point without multiplicity. We then investigate the existence of one-parameter semigroups for such maps and discuss some generalizations.