The Hopf argument
Résumé
Theorem: Let $(X,d)$ be a metric space, $µ$ be a Borel measure on $X$ such that $X$ can be covered by a countable family of open sets with finite $µ$ measure. Let $T\colon X\to X$ be a measurable, $µ$ preserving transformation. Then any bounded, measurable $T$ eigenfunction is $W^{ss}$ invariant. Here $W^{ss}$ is the 'stable foliation' generated by $T$. The foliation $W^{ss}$ is defined by defining its leaf $W^{ss}(x)$ through $x\in X$ by setting $W^{ss}(x) = {y\in X | d(T^nx,T^ny)\to 0 \text {as} n\to \infty}$. There are a number of dynamical systems for which ergodicity is deduced from the triviality of $W^{ss}$ invariant functions. For such systems, this result proves that, in fact, they are weakly mixing. For transformations preserving infinite measure, most of the versions of 'Hopf argument' make use of estimates coming from the underlying geometric structure. Another advantage of the above result is that it avoids these geometric considerations. The proof involves a blend of the ratio approach of E. Hopf and the induction method of V. Kaimanovich.