A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1?p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operators

     

A mean ergodic theorem for resolvent operators

Let {R(t)} _t≥0 be a uniformly bounded strongly continuous resolvent operator for the Volterra equation of convolution typeu=g+k*Au, whereA is a closed and densely defined operator on a Banach spaceX andk is a scalar kernel. We show that whenX is reflexive and that the average given by {R(t)} _t≥0 andk converges on the closed subspace to a bounded projection. This work was partially supported by DICYT 92-33LY and FONDECYT 91-0471     

Uniform distribution and the mean ergodic theorem

Generalizing a result of Veech, [14] Theorem 1.1.5 and answering a question, [14], 1.4 we prove the existence of certain well distributed sequences in topological semigroups having a countable dense subset.     

On the multiplicative ergodic theorem for uniquely ergodic systems

We consider the question of uniform convergence in the multiplicative ergodic theorem

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A mean ergodic theorem on vector spaces

A mean ergodic theorem for a matrix is first proved from which a mean ergodic theorem for affine operators on a vector space without any topological structure is obtained.     

On ergodic averages for affine lattice actions on tori

We prove that for a ?^ν-action on a torus as a group of affine transformations all the ergodic averages exist if and only if all the eigenvalues of the automorphism parts of all the affine transformations are roots of unity.     

A complex Tauberian theorem and mean ergodic semigroups

Communicated by Rainer Nagel     

Convergence of polynomial ergodic averages

We prove theL ² convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilsystems, that controls the limit behavior of the average. For a general system, we prove the convergence for certain families of polynomials. Dedicated to Hillel Furstenberg upon his retirement The second author was partially supported by NSF grant DMS-0244994.