Doeblin's condition, $\ddot{I}$-mixing and spectra of convolution operators on the circle

Abstract

We study the asymptotic behavior of Markov operators $P_{\hat{I}}¼$ defined by convolution with a probability measure $\hat{I}¼$ on the unit circle $\mathbb{T}$. We prove that when $\hat{I}¼$ is adapted, $P_{\hat{I}}¼$ satisfies Doeblin's condition if and only if some power $\hat{I}{¼}^k$ is non-singular. We give an example of a symmetric probability measure $\hat{I}¼$ on $\mathbb{T}$, such that the reversible stationary chain induced by $P_{\hat{I}}¼$ is $\ddot{I}$-mixing, but $P_{\hat{I}}¼$ does not satisfy Doeblin's condition. We look at the spectra of $P_{\hat{I}}¼$ in the different $L_p$ spaces when $P_{\hat{I}}¼$ is, or is not, $\ddot{I}$-mixing.