Using the ratio ergodic theorem for a measure preserving transformation in a σ-finite measure space we give a straightforward proof of Derriennic’s reverse maximal inequality for the supremum of ergodic ratios.

We consider models of random Schrr odinger operators which exist thanks to a cohomological theorem in ergodic theory. Examples are ergodic Schrr odinger operators with random magnetic uxes on discrete two-dimensional lattices or non-periodic situations like Penrose lattices.

We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications ensemble require computation long time averages. establish consistency this setting by proving ergodic theorem As part proof, we derive explicit variance formulas could be useful optimizing method.