0. Introduction. M. M. Day has shown that an abelian group has only one invariant mean if, and only if, the group is finite [3, Theorem 4, p. 534]. Knowing this we now ask to what extent the invariant means on a given abelian group are alike. In this paper we shall show that they all have the same "measure-theoretic character." An invariant mean on an abelian group G is a normalized, positive, ...

In [17] the author and A. Vershik have shown that for β = 1 2 (1 + √ 5) and the alphabet {0, 1} the infinite Bernoulli convolution (= the Erdös measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type II1 under the β-shift, and the natural extension of the β-shift provided with the measure equivalent to the Erdös measure, is Bernoulli. In this note we exten...

Statistical mechanics of three-dimensional flows of an ideal incompressible fluid is considered. An ideal fluid differs from the usual Hamiltonian systems of statistical mechanics by possessing an infinite number of integrals of motion that are circulations of velocity over closed fluid contours. To reduce the problem to a standard one, the governing equations should be written in a Hamiltonian...

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss [7, 8], we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist...

Explicit invariant measures are derived for a family of finite-toone, ergodic transformations of the unit interval having indifferent periodic orbits. Examples of interesting, non-trivial maps of [0, 1] for which one can readily compute an invariant measure absolutely continuous to Lebesgue measure are not easy to come by. The familiar examples are the Gauss map, the backward continued fraction...

We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and exampl...

We present a novel proof of de Finetti's Theorem characterizing permutation-invariant probability measures infinite sequences variables, so-called exchangeable measures. The is phrased in the language Markov categories, which provide an abstract categorical framework for and information flow. diagrammatic nature arguments makes intuitive easy to follow. also show how usual measure-theoretic ver...

The topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. The vertex PI polynomial is defined as PIv (G)  euv nu (e)  nv (e). Then Omega polynomial (G,x) for counting qoc strips in G is defined as (G,x) = cm(G,c)xc with m(G,c) being the number of strips of length c. In this paper, a new infinite class of fullerenes is constructed. ...