Doeblin’s Condition, ρ-Mixing and Spectra of Convolution Operators on the Circle

We study the asymptotic behavior of Markov operators Pμ defined by convolution with a probability measure μ on the unit circle T. We prove that when μ is adapted, Pμ satisfies Doeblin’s condition if and only if some power μk is non-singular. We give an example of a symmetric probability measure μ on T, such that the reversible stationary chain induced by Pμ is ρ-mixing, but Pμ does not satisfy Doeblin’s condition. We look at the spectra of Pμ in the different Lp spaces when Pμ is, or is not, ρ-mixing.