On local integrability and transience of convolution semigroups of measures

The analytic proof due to M. Itô of the Kesten-Ornstein transience criterion for continuous convolution semigroups of nonnegative contraction measures on a compactly generated Abelian locally compact group has been reworked and given a self-contained form. The new proof still relies on the existence of the equilibrium measure but dispenses with the complete maximum principle.     

On the continuity of convolution semigroups

Communicated by Karl H. Hofmann     

On the factor sets of measures and local tightness of convolution semigroups over Lie groups

It is shown that for a large class of Lie groups (called weakly algebraic groups) including all connected semisimple Lie groups the following holds: for any probability measure on the Lie group the set of all two-sided convolution factors is compact if and only if the centralizer of the support of inG is compact. This is applied to prove that for any connected Lie groupG, any homomorphism of any real directed (submonogeneous) semigroup into the topological semigroup of all probability measures onG is locally tight.     

On summability of pairs of convolution sequences of probability measures on locally compact semigroups

In this paper we study the problem of convergence in the weak and the vague topology of the sequence

On summability of pairs of convolution sequences of probability measures on locally compact semigroups

In this paper we study the problem of convergence in the weak and the vague topology of the sequence

$$\left( {\mathop \Sigma \limits_{i = 1}^\infty \mathop \Sigma \limits_{j = 1}^\infty a_{ij}^{(n)} \mu ^i *v^j ,n \in \mathbb{N}} \right)$$     

Measurable centres in convolution semigroups

In certain convolution semigroups over locally compact groups, the only measurable translations are those defined by Radon measures. In other words, the measurable centre of every such convolution semigroup consists of Radon measures.     

The structure of limit measures and their supports on topological semigroups

Given a sequence of measures (μ _n ) on a topological semigroupS, a measure λ onS is called a tail limit of (μ _n ) if for some subsequence of integers (n _i ), converges weakly to ν _k for allk and λ is a weak limit point of the sequence (ν _k ). The main theorem of this paper characterizes the supports of the tail limits on compact completely simple semigroups. Some applications of the theorem and various open problems are discussed.     

On free semigroups of automaton transformations

It is established that the subset of freek-generated subsemigroups of the semigroup of all automaton transformations over a finite alphabet is a second category set (in the sense of the Baire category approach) in the set of allk-generated subsemigroups. A continuum series of pairs of automaton transformations each of which generates a free semigroup of rank two is indicated. A criterion is established for this semigroup to be a finite-automaton group. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 248–259, February, 1998. The author wishes to express his deep gratitude to Professor V. I. Sushchans'kii for permanent help and attention to the research. This research was partially supported by the ISSEP under grant No. GSU 051341.     

On semi-invariant probability measures on semigroups

Summary Let a regular Borel measure m on a locally compact semigroup S be upper semi-invariant i.e., m(C x)m(C) and m(x C)m(C) for every compact C and x in S. It is shown: (i) Every subsemigroup of S of positive measure contains an idempotent. (ii) S admits an upper semi-invariant probability measure iff S has a kernel K which is a compact group.We should like to thank the referee for pointing out certain redundancies in the theorems. Also we thank Dr. Tze-Chien Sun for some helpful observations.     

On equations in free semigroups and groups

The unsolvability of some algorithmic problems is proved for equations in free groups and semigroups, namely, some simple properties of the solutions of the equations are determined and the absence of an algorithm permitting the determination of whether an arbitrary equation in a free group or semigroup has a solution with the properties introduced is proved.Translated from Matematicheskie Zametki, Vol. 16, No. 5, pp. 717–724, November, 1974.     

On optimal factorization of free semigroups into free subsemigroups

Optimal factorizations of a subsemigroup into very pure and weakly prefix subsemigroups are considered. It is shown that the intersection of the family of very pure (weakly prefix) subsemigroups containing a given subset A of X⁺ is itself a very pure (weakly prefix) subsemigroup and procedures for constructing the basis of such minimal subsemigroups are provided.