The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions

TL;DR

This paper studies the Hausdorff dimension of a family of conformal iterated function systems related to generalized complex continued fractions, showing its continuity, real-analyticity, and subharmonicity in parameter space.

Contribution

It introduces a new example of infinite conformal iterated function systems with a complex parameter space and analyzes the properties of their Hausdorff dimension functions.

Findings

Hausdorff dimension function is continuous in the parameter space.

The dimension function is real-analytic and subharmonic inside the parameter space.

The maximum of the Hausdorff dimension occurs on the boundary of the parameter space.

Abstract

We consider the family of CIFSs of generalized complex continued fractions with a complex parameter space. This is a new interesting example to which we can apply a general theory of infinite CIFSs and analytic families of infinite CIFSs. We show that the Hausdorff dimension function of the family of the CIFSs of generalized complex continued fractions is continuous in the parameter space and is real-analytic and subharmonic in the interior of the parameter space. As a corollary of these results, we also show that the Hausdorff dimension function has a maximum point and the maximum point belongs to the boundary of the parameter space.