Fabrizio Polo

We prove that for a minimal rotationTon a two-step nilmanifold and any measureμ, the push-forwardTⁿ_⋆μofμunderTⁿtends toward Haar measure if and only ifμprojects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density one. These results strengthen Parry’s result [Ergodic properties of affine transformations and flows on nilmanifolds.Amer. J. Math.91(1968), 757–771] that such systems are uniquely ergodic. Extending the work of Furstenberg [Strict ergodicity and transformations of the torus.Amer. J. Math.83(1961), 573–601], we get the same result for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally we characterize limits ofTⁿ_⋆μfor some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties that strengthen unique ergodicity in a way analogous to that in which mixing and weak mixing strengthen ergodicity for measure-preserving systems.

A continuous action of a group G G on a compact metric space has sensitive dependence on initial conditions if there is a number ε > 0 \varepsilon > 0 such that for any open set U U we can find g ∈ G g \in G such that g . U g.U has diameter greater than ε . \varepsilon . We prove that if a countable G G acts transitively on a compact metric space, preserving a probability measure of full support, then the system either is minimal and equicontinuous or has sensitive dependence on initial conditions. Assuming ergodicity, we get the same conclusion without countability. These theorems extend the invertible case of a theorem of Glasner and Weiss. We prove that when a finitely generated, solvable group acts transitively and certain cyclic subactions have dense sets of minimal points, the system has sensitive dependence on initial conditions. Additionally, we show how to construct examples of non-compact monothetic groups and transitive, non-minimal, almost equicontinuous, recurrent G G -actions.