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.