Hausdorff dimension of the limit sets of Tree Iterated Function Systems

TL;DR

This paper introduces Tree Iterated Function Systems (TIFSs), defines their limit sets, and derives a Hausdorff dimension formula generalizing Bowen's formula, highlighting differences from non-autonomous IFSs.

Contribution

The paper defines TIFSs, establishes a Hausdorff dimension formula for their limit sets, and provides an example distinguishing TIFSs from non-autonomous IFSs.

Findings

Derived a Hausdorff dimension formula for TIFSs

Established a generalization of Bowen's formula

Presented an example showing differences from non-autonomous IFSs

Abstract

We investigate Tree Iterated Function Systems (TIFSs), which we introduce in this paper. TIFSs are the generalizations of Iterated Function Systems in which we take the maps independently at each step and each block. In this paper, we give the definition of TIFSs and the limit sets of them. We show a formula for the Hausdorff dimension of the limit sets of TIFSs, which is a generalization of Bowen's formula. Moreover, we give an example which emphasizes the difference between TIFSs and non-autonomous IFSs.