2,897 research outputs found
A nilpotent IP polynomial multiple recurrence theorem
We generalize the IP-polynomial Szemer\'edi theorem due to Bergelson and
McCutcheon and the nilpotent Szemer\'edi theorem due to Leibman. Important
tools in our proof include a generalization of Leibman's result that polynomial
mappings into a nilpotent group form a group and a multiparameter version of
the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.Comment: v4: switch to TeXlive 2016 and biblate
Z^d-actions with prescribed topological and ergodic properties
We extend constructions of Hahn-Katznelson and Pavlov to Z^d-actions on
symbolic dynamical spaces with prescribed topological and ergodic properties.
More specifically, we describe a method to build Z^d-actions which are
(totally) minimal, (totally) strictly ergodic and have positive topological
entropy.Comment: 19 pages, 1 figure, referee suggestions incorporated, to appear in
Ergodic Theory and Dynamical System
Set-polynomials and polynomial extension of the Hales-Jewett Theorem
An abstract, Hales-Jewett type extension of the polynomial van der Waerden
Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established:
Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring
of the set of subsets of V={1,...,N}^{d} x {1,...,q} there exist a set a
\subset V and a nonempty set \gamma \subseteq {1,...,N} such that a \cap
(\gamma^{d} x {1,...,q}) = \emptyset, and the subsets a, a \cup (\gamma^{d} x
{1}), a \cup (\gamma^{d} x {2}), ..., a \cup (\gamma^{d} x {q}) are all of the
same color.
This ``polynomial'' Hales-Jewett theorem contains refinements of many
combinatorial facts as special cases. The proof is achieved by introducing and
developing the apparatus of set-polynomials (polynomials whose coefficients are
finite sets) and applying the methods of topological dynamics.Comment: 43 pages, published versio
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