Doeblin's condition, $\rho$-mixing and spectra of convolution operators on the circle

We study the asymptotic behavior of Markov operators $P_{\mu }$ defined by convolution with a probability measure $\mu$ on the unit circle $\mathbb{T}$. We prove that when $\mu$ is adapted, $P_{\mu }$ satisfies Doeblin’s condition if and only if some power ${\mu }^k$ is non-singular. We give an example of a symmetric probability measure $\mu$ on $\mathbb{T}$, such that the reversible stationary chain induced by $P_{\mu }$ is $\rho$-mixing, but $P_{\mu }$ does not satisfy Doeblin’s condition. We look at the spectra of $P_{\mu }$ in the different $L_p$ spaces when $P_{\mu }$ is, or is not, $\rho$-mixing.