Degenerate perturbations of infinite graph-directed iterated function systems

TL;DR

This paper investigates the Hausdorff dimension of limit sets in infinite graph-directed iterated function systems, providing estimates, formulas, and convergence results for perturbed systems, including applications to nonconformal mappings and complex continued fractions.

Contribution

It extends Hausdorff dimension analysis and convergence results from finite to infinite graph-directed IFS, introducing perturbation concepts and applications.

Findings

Hausdorff dimension estimates for infinite GIFS

Convergence of dimension under perturbations

Applications to nonconformal mappings and continued fractions

Abstract

We study infinite graph-directed iterated function systems (GIFS) whose underlying graph is not strongly connected and has countably many vertices and edges. In addition to a summability condition for the physical potential, we provide lower and upper estimates of the Hausdorff dimension of the limit set of such GIFS. Bowen type formula is also given under conformal condition and suitable separate conditions. We also introduce perturbed GIFS in which the images of arbitrarily chosen contraction mappings shrink to a single point. In other words, the graph of the perturbed GIFS differs from that of unperturbed GIFS. Assuming suitable continuity condition on contraction mappings, we prove that the Hausdorff dimension of the limit set of the perturbed GIFS converges to that of the unperturbed GIFS, This result generalizes for finite graphs in [T.2019, T.2016] to the infinite graph setting. As applications, we consider a perturbed nonconformal mapping, as well as convergence and non-convergence in the Hausdorff dimension for perturbed complex continued fractions with degeneration.