On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ is connected. We show that a non-trivial portion of $M$ near the imaginary axis is contained in the closure of its interior (it is conjectured that all non-real points of $M$ are in the closure of the set of interior points of $M$). Next we turn to the attractors $A_\lambda$ themselves and to natural measures $\nu_\lambda$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures $\nu_\lambda$. Next we investigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$ in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on the absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greater than $\sqrt{1/2}$.Comment: 22 pages, 5 figure

Schur analysis in an indefinite setting

Schur analysis comprises topics like: the Schur transformation on the class of Schur functions (by definition, the functions which are holomorphic and bounded by 1 on the open unit disk) and the Schur algorithm, Schur parameters and approximation, interpolation problems for Schur functions, factorization of rational 2 × 2 matrix polynomials, which are -unitary on the unit circle, and a related inverse scattering problem. This note contains a survey of indefinite versions of these topics related to the class of scalar generalized Schur functions. These are the meromorphic functions s(z) on the open unit disk for which the kernel has finitely many negative squares. We also review a generalization of the Schur transformation to classes of functions on a general domain one of which is the class of scalar generalized Nevanlinna functions. These are the meromorphic functions n(z) on the open upper half plane for which the kernel has finitely many negative squares.</p

Schur analysis in the quaternionic setting: The fueter regular and the slice regular case

This chapter is a survey on recent developments in quaternionic Schur analysis. The first part is based on functions which are slice hyperholomorphic in the unit ball of the quaternions, and have modulus bounded by 1. These functions, which by analogy to the complex case are called Schur multipliers, are shown to be (as in the complex case) the source of a wide range of problems of general interest. They also suggest new problems in quaternionic operator theory, especially in the setting of indefinite inner product spaces. This chapter gives an overview on rational functions and their realizations, on the Hardy space of the unit ball, on the half-space of quaternions with positive real part, and on Schur multipliers, also discussing related results. For the purpose of comparison this chapter presents also another approach to Schur analysis in the quaternionic setting, in the framework of Fueter series. To ease the presentation most of the chapter is written for the scalar case, but the reader should be aware that the appropriate setting is often that of vector-valued functions