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1. CJM Online first

Handelman, David
Nearly approximate transitivity (AT) for circulant matrices
By previous work of Giordano and the author, ergodic actions of $\mathbf Z$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in C*-algebras and topological dynamics. Here we investigate how far from AT (approximately transitive) can actions be which derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, ATness arises; in addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided.

Keywords:approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrix-valued random walk
Categories:37A05, 06F25, 28D05, 46B40, 60G50

2. CJM 2009 (vol 61 pp. 656)

McCutcheon, Randall; Quas, Anthony
Generalized Polynomials and Mild Mixing
An unsettled conjecture of V. Bergelson and I. H\aa land proposes that if $(X,\alg,\mu,T)$ is an invertible weak mixing measure preserving system, where $\mu(X)<\infty$, and if $p_1,p_2,\dots ,p_k$ are generalized polynomials (functions built out of regular polynomials via iterated use of the greatest integer or floor function) having the property that no $p_i$, nor any $p_i-p_j$, $i\neq j$, is constant on a set of positive density, then for any measurable sets $A_0,A_1,\dots ,A_k$, there exists a zero-density set $E\subset \z$ such that \[\lim_{\substack{n\to\infty\\ n\not\in E}} \,\mu(A_0\cap T^{p_1(n)}A_1\cap \cdots \cap T^{p_k(n)}A_k)=\prod_{i=0}^k \mu(A_i).\] We formulate and prove a faithful version of this conjecture for mildly mixing systems and partially characterize, in the degree two case, the set of families $\{ p_1,p_2, \dots ,p_k\}$ satisfying the hypotheses of this theorem.

Categories:37A25, 28D05

3. CJM 1998 (vol 50 pp. 1163)

Chen, Jingyi; Hsu, Elton P.
Gradient estimates for harmonic Functions on manifolds with Lipschitz metrics
We introduce a distributional Ricci curvature on complete smooth manifolds with Lipschitz continuous metrics. Under an assumption on the volume growth of geodesics balls, we obtain a gradient estimate for weakly harmonic functions if the distributional Ricci curvature is bounded below.

Categories:60D58, 28D05

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