Proving ergodicity via divergence of time averages
Volume 248 / 2019
Abstract
A classical fact in ergodic theory is that ergodicity is equivalent to almost everywhere divergence of ergodic sums of all nonnegative integrable functions which are not identically zero. We give two methods, one in the measure preserving case and one in the nonsingular case, which enable one to prove this criterion by checking it on a dense collection of functions and then extending it to all nonnegative functions. The first method (Theorem 1.1) is then used in a new proof of a folklore criterion for ergodicity of Poisson suspensions which does not make any reference to Fock spaces. The second method (Theorem 1.2), which involves the double tail relation, is used to show that a large class of nonsingular Bernoulli and inhomogeneous Markov shifts are ergodic if and only if they are conservative. In the last section we discuss an extension of the Bernoulli shift result to other countable groups including $\mathbb {Z}^{d}$, $d\geq 2$, and discrete Heisenberg groups.
