# Gaussian perturbations of circle maps: A spectral approach - Mathematics > Probability

Abstract: In this work, we examine spectral properties of Markov transition operatorscorresponding to Gaussian perturbations of discrete time dynamical systems onthe circle. We develop a method for calculating asymptotic expressions foreigenvalues in the zero noise limit and show that changes to the number orperiod of stable orbits for the deterministic system correspond to changes inthe number of limiting modulus 1 eigenvalues of the transition operator for theperturbed process. We call this phenomenon a $\lambda$-bifurcation. Asymptoticexpressions for the corresponding eigenfunctions and eigenmeasures are alsoderived and are related to Hermite functions.

Author: John Mayberry

Source: https://arxiv.org/