zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. (English) Zbl 1080.37003

Consider G is a nilpotent Lie group, X a compact homogeneous space of G and X=G/Γ the nilmanifold on which G acts by left translations. A sequence {g(n)} n in G of the form g(n)=a 1 p 1 (n) a m p m (n) where the a i G and p i are polynomials taking on integer values on the integers is called a polynomial sequence. The author establishes the following pointwise convergence result for continuous functions on such a nilmanifold:

For any xX, for any continuous function fC(X) and for any Folner sequence Φ N in

lim N 1 Φ N nΦ N (g(n)x)

exists. The proof is done by studying carefully the distribution of the orbit of a point on a compact nilmanifold.

This pointwise result and its multidimensional extension obtained by the author [ibid. 25, 215–225 (2005; Zbl 1080.37004)] are useful tools as they allow one to reduce the study of the norm and pointwise convergence of several nonconventional averages to orthocomplements of characteristic factors.

37A15General groups of measure-preserving transformation
28D15General groups of measure-preserving transformations
22F30Homogeneous spaces