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Moran-Type Iterated Function Systems and Dimensions of Moran Self-Similar Sets

Yong-Shen Cao, Qi-Rong Deng, Ming-Tian Li

TL;DR

The paper extends fractal dimension theory to Moran-type iterative systems, introducing Moran-type attractors $K_n$ and Moran-type invariant measures ${\mu_n}$ for sequences of bi-Lipschitz IFS. It develops a comprehensive separation framework with MOSC, MWSC, MSSC and auxiliary properties MWHP/MBDP, connecting them to classical OSC/WSC/SSC in the constant-IFS case. It then derives dimension results for Moran-type self-similar sets, giving explicit formulas and thresholds for Hausdorff, box, and packing dimensions via products $\prod_{i=1}^k\sum_{j=1}^{N_i} r_{i,j}^s$, and showing when $K_n$ form uniform $s$-sets. The work is complemented by illustrative examples and counterexamples clarifying the necessity of assumptions and the rich behavior of Moran-type fractals across self-similar, self-conformal, and self-affine regimes.

Abstract

Moran-type iterated function systems (Moran-type IFS or MIFS) are defined by a sequence of iterated function systems, and their basic theoretical framework is established. We define Moran-type attractors and invariant probability measures associated with a sequence of probability weight vectors. Furthermore, separation conditions for MIFS are introduced, and the dimension theory of Moran-type self-similar sets is investigated. Appropriate examples are provided to illustrate and support the definitions and results.

Moran-Type Iterated Function Systems and Dimensions of Moran Self-Similar Sets

TL;DR

The paper extends fractal dimension theory to Moran-type iterative systems, introducing Moran-type attractors and Moran-type invariant measures for sequences of bi-Lipschitz IFS. It develops a comprehensive separation framework with MOSC, MWSC, MSSC and auxiliary properties MWHP/MBDP, connecting them to classical OSC/WSC/SSC in the constant-IFS case. It then derives dimension results for Moran-type self-similar sets, giving explicit formulas and thresholds for Hausdorff, box, and packing dimensions via products , and showing when form uniform -sets. The work is complemented by illustrative examples and counterexamples clarifying the necessity of assumptions and the rich behavior of Moran-type fractals across self-similar, self-conformal, and self-affine regimes.

Abstract

Moran-type iterated function systems (Moran-type IFS or MIFS) are defined by a sequence of iterated function systems, and their basic theoretical framework is established. We define Moran-type attractors and invariant probability measures associated with a sequence of probability weight vectors. Furthermore, separation conditions for MIFS are introduced, and the dimension theory of Moran-type self-similar sets is investigated. Appropriate examples are provided to illustrate and support the definitions and results.
Paper Structure (5 sections, 10 theorems, 174 equations)

This paper contains 5 sections, 10 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Definitions of attractors and invariant measures
  3. Definitions of separation conditions
  4. Dimensions of Moran-type self-similar sets
  5. Remarks and Examples

Key Result

Theorem 2.1

Let $\left\{\Phi_n\right\}_{n=1}^\infty$ be an MIFS on $(X,\ \rho)$ as in Definition def1.1. We have the following statements. (i) The following sets are well defined compact subsets of $X$ and are independent of $a\in X$. Moreover, the above $\left\{K_n\right\}_{n=1}^\infty$ is the unique sequence of compact subsets of $X$ such that (ii) For any given sequence of probability weights ${\mathcal{

Theorems & Definitions (28)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Remark 3.2
  • ...and 18 more