zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Fully generic sequences and a multiple-term return-times theorem. (English) Zbl 0910.28013

Let (X,,μ,T) be a dynamical systems consisting of a measure-preserving transformation defined on a standard probability space. A bounded sequence {a i } of complex numbers are called universal weights (for the pointwise ergodic theorem) if for any dynamical system (Y,𝒢,ν,S) and gL 1 (ν), 1 N j=0 N-1 a i g(S j y) converges.

The following theorem was proved by J. Bourgain, H. Furstenberg, Y. Katznelson and D. S. Ornstein [Publ. Math., Inst. Haut. √Čtud. Sci. 69, 5-45 (1989; Zbl 0705.28008)] (and a joinings proof was given by D. J. Rudolph [Ergodic Theory Dyn. Syst. 14, No. 1, 197-203 (1994; Zbl 0799.28010)]):

For any dynamical system (X,,μ,T) and fL (μ) for μ a.e. x, the sequence of values f(T i x) are universal weights.

The author proves a multi-term version of the above result first proposed by I. Assani, i.e., involving averages of the form

1 N i=0 N-1 f 1 (T 1 i x 1 )f 2 (T 2 i x 2 )f k (T k i x k ),

where all the f i are bounded and where for each j<k, the points x 1 ,x 2 ,,x j guaranteeing convergence can be chosen universally, without knowledge of the transformations and functions to follow. The author is able to avoid the L 2 orthogonality arguments in the proof of Bourgain et al., which requires the splitting of the L 2 space into Kronecker functions and their orthocomplement, by using the notion of pointwise genericity. In particular, a disjointness result on joinings avoids the need to identify distinguished factor algebras for the higher term averages. A disadvantage of this approach is that delicate information about characteristic factors is not available (as in, for example, the work of D. J. Rudolph [Lond. Math. Soc. Lect. Note Ser. 205, 369-432 (1995; Zbl 0877.28012)], concerning the Conze-Lesigne algebra).

MSC:
28D05Measure-preserving transformations