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

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