Characterizations of $A_\infty$ Weights in Ergodic Theory

We establish a discrete weighted version of Calder\'{o}n-Zygmund decomposition from the perspective of dyadic grid in ergodic theory. Based on the decomposition, we study discrete $A_\infty$ weights. First, characterizations of the reverse H\"{o}lder's inequality and their extensions are obtained. Second, the properties of $A_\infty$ are given, specifically $A_\infty$ implies the reverse H\"{o}lder's inequality. Finally, under a doubling condition on weights, $A_\infty$ follows from the reverse H\"{o}lder's inequality. This means that we obtain equivalent characterizations of $A_{\infty}$. Because $A_{\infty}$ implies the doubling condition, it seems reasonable to assume the condition.