Minkowskis question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in the sense o
f Ullman--Stahl--Totik and moreover it belongs to a Nevai class: we provide numerical evidence of the validity of these conjectures. In addition, we study the zeros of its orthogonal polynomials and the associated Christoffel functions, for which asymptotic formulae are derived. Rigorous results and numerical techniques are based upon Iterated Function Systems composed of Mobius maps.
We study the one-dimensional projection of the extremal Gibbs measures of the two-dimensional Ising model, the Schonmann projection. These measures are known to be non-Gibbsian at low temperatures, since their conditional probabilities as a function
of the two-sided boundary conditions are not continuous. We prove that they are g-measures, which means that their conditional probabilities have a continuous dependence on one-sided boundary condition.
There are many tests for determining the convergence or divergence of series. The test of Raabe and the test of Betrand are relatively unknown and do not appear in most classical courses of analysis. Also, the link between these tests and regular var
iation is seldomly made. In this paper we offer a unified approach to some of the classical tests from a point of view of regular varying sequences.
A space of pseudoquotients $mathcal P (X,S)$ is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) sim (y,g)
$ if $gx=fy$. In this note we assume that $(X,Sigma,mu)$ is a measure space and that $S$ is a commutative semigroup of measurable injections acting on $X$ and investigate under what conditions there is an extension of $mu$ to $mathcal P (X,S)$.