Zemer Kosloff

Explicit coupling argument for nonuniformly hyperbolic transformations

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. Here it is verified that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For nonuniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated to the inducing time. Finally, for nonuniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

On a type III 1 Bernoulli shift

We provide a construction of a product measure under which the shift on the two point product space is a type III_1 transformation.

Symmetric Birkhoff sums in infinite ergodic theory

We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity.

The zero-type property and mixing of Bernoulli shifts

We prove that every non-singular Bernoulli shift is either zero-type or there is an equivalent invariant stationary product probability. We also give examples of a type Bernoulli shift and a Markovian flow which are power weakly mixing and zero type.