where (Fk) ∞ k=1 is a sequence of measurable sets Fk ∈ B with positive measure μ(Fk) > 0; specifically, we consider questions of when this limit exists, and when it exists, what that limit is for f ∈ L(X, μ). Before proceeding, we pause to distinguish these problems from two other kinds of differentiation problems which we will call temporal and spatial differentiation problems. A temporal differentiation problem might look like limk→∞ 1 k ∑ k−1 i=0 ∫ Ti f dμ, and a spatial differentiation… Expand

A bstract . We introduce a non-autonomous generalization of spatial-temporal differentiations, and prove results about probabilistically and topologically generic behaviors of certain… Expand

Let f be a real-valued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic f -average is as… Expand

LetXbe a Cantor set,S a minimal self-homeomorphism ofX, and Μ anS-invariant Borel probability. LetT be an ergodic automorphism of a non-atomic Lebesgue probability space(Y,Ν). Then there is a minimal… Expand

Ergodic Theory of Numbers is an introduction to the ergodic theory behind common number expansion, like decimal expansions, continued fractions, and many others. However, its aim does not stop there.… Expand

Let T be an ergodic measure-preserving transformation of a Lebesgue measure space with entropy h(T). We prove that T has a generator of size k where eh(T) _ k < eh(T)+ 1

Using a combinatorial result of N. Hindman one can extend Jewett’s method for proving that a weakly mixing measure preserving transformation has a uniquely ergodic model to the general ergodic case.… Expand