We show that effective 2ℓ-multiple correlations imply quantitative ℓ-multiple pointwise ergodic theorems. The result has a wide class of applications which include subgroup actions on homogeneous spaces, nilmanifold automorphisms, subshifts finite type and Young towers.

A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence struct...

We show that for every ergodic system (X, μ,T1, . . . ,Td) with commuting transformations, the average 1 Nd+1 ∑ 0≤n1,...,nd≤N−1 ∑ 0≤n≤N−1 f1(T n 1 d ∏ j=1 T n j j x) f2(T n 2 d ∏ j=1 T n j j x) · · · fd(T n d d ∏ j=1 T n j j x). converges for μ-a.e. x ∈ X as N → ∞. If X is distal, we prove that the average 1 N N ∑ i=0 f1(T n 1 x) f2(T n 2 x) · · · fd(T n d x) converges for μ-a.e. x ∈ X as N → ∞...

We show that if f is locally in L log logL then the lacunary spherical means converge almost everywhere. The argument given here is a model case for more general results on singular maximal functions and Hilbert transforms along plane curves [6].

We study pointwise convergence of the solutions to Schrödinger equations with initial datum f ∈ H(R). The conjecture is that the solution ef converges to f almost everywhere for all f ∈ H(R) if and only if s ≥ 1/4. The conjecture is known true for one spatial dimension and the convergence when s > 1/2 was verified for n ≥ 2. Recently, concrete progresses have been made in R for some s < 1/2. Ho...

Let P be a compact n-dimensional convex polyhedron in R n containing the origin in its interior and let e H(t) = Z 1 0 Z vP e 2it ddv, t2 R n ;where vP is the characteristic function of the dilated polyhedron vP. Let H N (t) = X m2Z n e H 1 N+1 (t+m), t 2 T n , where e H " (t) = " ?n e H(t="). We prove that (e H " f)(t) ! f(t) a.e., as " ! 0, for any f 2 L 1 (R n), and that (H N f)(t) ! f(t) a....

This work concerns controlled Markov chains with denumerable state space, (possibly) unbounded cost function, and an expected average cost criterion. Under a Lyapunov function condition, together with mild continuity-compactness assumptions, a simple necessary and sufficient criterion is given so that the relative value functions and differential costs produced by the value iteration scheme con...