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

TL;DR

This paper studies Markov operators on the unit circle given by convolution with a probability measure $\mu$, focusing on Doeblin's condition, $\rho$-mixing, and spectral properties across $L_p$ spaces. It proves that for adapted $\mu$, Doeblin's condition for $P_\mu$ is equivalent to the existence of a power $k$ with $\mu^{*k}$ non-singular, and it characterizes quasi-compactness and $L_p$-convergence of $P_\mu^n$ to the projection $E$ in various norms. The authors analyze the spectrum: $\sigma(P,L_2)$ equals the closure of $\{\hat{\mu}(n): n\in\mathbb{Z}\}$, with Rajchman measures yielding equal $L_p$ spectra for $1<p<\infty$, while $L_1$ can differ; they also exhibit symmetric counterexamples where $\rho$-mixing occurs without Doeblin. Additionally, the work establishes almost everywhere convergence results for convolution powers under different regularity assumptions, linking uniform ergodicity, spectral properties, and ergodic transforms, and discusses several sharp counterexamples that delineate the boundaries of these phenomena.

Abstract

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.

Theorems & Definitions (37)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Proposition 2.4
• proof
• Corollary 2.5
• Proposition 2.6