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

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