Israel Journal of Mathematics

, Volume 90, Issue 1–3, pp 115–123 | Cite as

On convolution powers on semidirect products

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Abstract

LetK be a compact group of linear operators of thed-dimensional spaceR d andG K,d denote the semidirect productK byR d . It is shown that if an adapted probability measureμ onG K,d is not scattered (i.e. for some compactF we havex 0 ∈ R d (gF)>0), then there exists a nonzero vectorx 0 ∈R d such thatk 1(x 0)=k 2(x 0) holds for all (k 1,x 1) and (k 2,x 2) belonging to the topological supportS(μ) of the measureμ. As a result we obtain that every adapted and strictly aperiodic probability measure on the group of all rigid motions of thed-dimensional Euclidian space is scattered.

Keywords

Probability Measure Compact Group Haar Measure Semidirect Product Rigid Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. [B]
    W. Bartoszek,On concentrated probabilities, submitted.Google Scholar
  2. [BO]
    P. Bougerol,Une majoration universelle des fonctions de concentration sur les groupes localement compacts non compacts, Springer Lecture Notes in Mathematics706 (1979), 36–40.MathSciNetCrossRefGoogle Scholar
  3. [C]
    I. Csiszar,On infinite products of random elements and infinite convolutions of probability distributions on locally compact groups, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete5 (1966), 279–299.MATHCrossRefMathSciNetGoogle Scholar
  4. [D]
    Y. Derriennic,Lois « zéro ou deux » pour les processus de Markov. Applications aux marches aléatoires, Annales de l’Institut Henri Poincaré. Probabilités et Statistiques12 (1976), 111–129.MathSciNetMATHGoogle Scholar
  5. [DL]
    Y. Derriennic and M. Lin,Convergence of iterates of averages of certain operator representations and convolution powers, Journal of Functional Analysis85 (1989), 86–102.MATHCrossRefMathSciNetGoogle Scholar
  6. [E]
    P. Eisele,On shifted convolution powers of a probability measure, Mathematische Zeitschrift211 (1992), 557–574.MATHCrossRefMathSciNetGoogle Scholar
  7. [EG]
    W. R. Emerson and F. P. Greenleaf,Covering properties and Følner Condition for locally compact groups, Mathematische Zeitschrift102 (1967), 370–384.MATHCrossRefMathSciNetGoogle Scholar
  8. [HM]
    K. H. Hofmann and A. Mukherjea,Concentration function on a class of noncompact groups, Mathematische Annalen256 (1981), 535–548.MATHCrossRefMathSciNetGoogle Scholar
  9. [HR]
    E. Hewitt and K.A. Ross,Abstract Harmonic Analysis I, Springer-Verlag, Berlin, 1963.MATHGoogle Scholar
  10. [W]
    G. Willis,The structure of totally disconnected, locally compact group, 1993.Google Scholar

Copyright information

© Hebrew University 1995

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of South AfricaPretoriaSouth Africa

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