Idempotent measures on semigroups

Murray Rosenblatt's interest in random walks on compact semigroups probably came from his work on representations of stationary processes as shifts of functions of independent random variables described in [11], where products of matrices were studied. His papers [12], [5], [13], [14] generalized the work of Levy [2] on random walks on the circle and the work of Kawada and ltd [4] on random walks on compact groups. In [14], he also completely characterized the structure of the limit measures for the special case of compact semigrougs of n x n stochastic matrices.To describe Rosenblatt's work in this area in more detail, we first introduce some definitions and basic facts about compact topological semigroups (see [15] and [3]).1. Every compact topological semigroup S has a minimal two sided ideal K which is called the kernel of S.2. The kernel if is a compact completely simple semigroup which has a Rees structure 1x6x7, where (a) X is a compact left-zero semigroup, Y is a compact right-zero semigroup and G is a compact group.(b) The topology on X x G x Y is the product topology.

show abstract

“…So the components a and j3 of the product measure are the extra elements in the semigroup case. J. S. Pym also obtained this results in [10].…”

supporting

confidence: 54%

Rosenblatt’s Contributions to Random Walks on Compact Semigroups

Davis

Lii

Politis

2011

Selected Works of Murray Rosenblatt

View full text Add to dashboard Cite

show abstract

“…Also we prove that if ¡x is idempotent, then F is completely simple. This was proved for compact 5 in [3] and [5] and under various compactness conditions in [4] and [6] and conjectured in [6]. As in the compact case, the result that F is completely simple enables us to formulate a complete characterization of idempotent measures on locally compact semigroups as products of a Haar measure and two regular Borel measures (see Theorem 3.1).…”

mentioning

confidence: 86%

“…p is said to be r*-invariant, whenever for every 73£(B and xES, piB) =piBx~1). If p is idempotent, 7 is a (closed) semigroup such that cl(77)=7 (see [5] and [4]). For aEF, we denote by jua=ju(-a-1) the measure paiB)=piBa~l), BE<$>-(Similarly for ap(B) =p(a~lB).)…”

mentioning

confidence: 99%

Idempotent Measures on Locally Compact Semigroups

Mukherjea¹,

Tserpes²

1971

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

Abstract.A conjecture that the support of an /■"'-invariant regular finite measure on a locally compact semigroup is a left group is proven. Moreover we also prove that the support of an idempotent measure on a locally compact semigroup is completely simple, thus extending a well-known result of Pym and HebleRosenblatt on compact semigroups to the locally compact case. These results are also shown to be true in a complete metric semigroup.

show abstract

mentioning

confidence: 86%

Idempotent measures on locally compact semigroups

Mukherjea

Tserpes

1971

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

A conjecture that the support of an /■"'-invariant regular finite measure on a locally compact semigroup is a left group is proven. Moreover we also prove that the support of an idempotent measure on a locally compact semigroup is completely simple, thus extending a well-known result of Pym and Heble-Rosenblatt on compact semigroups to the locally compact case. These results are also shown to be true in a complete metric semi-group.

show abstract

Idempotent measures on semigroups

Cited by 45 publications

References 9 publications

Rosenblatt’s Contributions to Random Walks on Compact Semigroups

Rosenblatt’s Contributions to Random Walks on Compact Semigroups

Idempotent Measures on Locally Compact Semigroups

Idempotent measures on locally compact semigroups

Contact Info

Product

Resources

About