Dimension Of Inhomogeneous Sub-Self-Similar Sets

TL;DR

This paper introduces inhomogeneous sub-self-similar sets, explores their properties, and investigates their dimensions, extending prior work on self-similar and sub-self-similar fractals.

Contribution

It defines ISSS sets, provides construction methods, and analyzes their box and Hausdorff dimensions, advancing understanding of inhomogeneous fractal structures.

Findings

ISSS sets can be constructed using the proposed method.

Upper and lower box dimensions of ISSS sets are characterized.

Continuity of the Hausdorff dimension for ISSS sets is discussed.

Abstract

In this paper, we introduce the concept of Inhomogeneous sub-self-similar (ISSS) sets, building upon the foundations laid by Falconer (Trans. Amer. Math. Soc. 347 (1995) 3121-3129) in the study of sub-self-similar sets and drawing inspiration from Barnsley's work on inhomogeneous self-similar sets (Proc. Roy. Soc. London Ser. A 399 (1985), no. 1817, 24). We explore a range of examples of ISSS sets and elucidate a method to construct ISSS sets. We also investigate the upper and lower box dimensions of ISSS sets and discuss the continuity of the Hausdorff dimension.