  In this work, we examine spectral properties of Markov transition operators
corresponding to Gaussian perturbations of discrete time dynamical systems on
the circle. We develop a method for calculating asymptotic expressions for
eigenvalues (in the zero noise limit) and show that changes to the number or
period of stable orbits for the deterministic system correspond to changes in
the number of limiting modulus 1 eigenvalues of the transition operator for the
perturbed process. We call this phenomenon a $\lambda$-bifurcation. Asymptotic
expressions for the corresponding eigenfunctions and eigenmeasures are also
derived and are related to Hermite functions.
