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Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets

Kathryn E. Hare, Franklin Mendivil

TL;DR

This work extends the theory of Φ-dimensions for random 1-variable measures from homogeneous Moran sets to non-homogeneous Moran constructions with unequal scaling, under a uniform separation condition. By introducing θ-dependent random variables and the function G(θ), it establishes a threshold-driven dichotomy: in the large-Φ regime, almost-sure Φ-dimensions are dictated by whether G(ψ) lies below or above ψ, often yielding a unique crossing α where G(α)=α; in the small-Φ regime, the almost-sure dimensions are given by explicit essential-supremum/infimum formulas involving the mass split p_n and scales a_n,b_n. The paper provides concrete deterministic and probabilistic examples, extends the analysis to higher dimensions and more general branching, and clarifies when the Φ-dimensions of the random measure align with those of the supporting random set. Collectively, the results offer precise, computable descriptions of local fractal geometry for a broad class of non-homogeneous random Moran sets and measures and illuminate the relationship between measure- and set-dimensions in this stochastic setting.

Abstract

In this paper, we determine the almost sure values of the $Φ$-dimensions of random measures $μ$ supported on random Moran sets in $\R^d$ that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets \cite{HM} to the case of unequal scaling factors. The $Φ$-dimensions are intermediate Assouad-like dimensions with the (quasi-)Assouad dimensions and the $θ$-Assouad spectrum being special cases. The almost sure value of $\dim_Φμ$ exhibits a threshold phenomena, with one value for ``large'' $Φ$ (with the quasi-Assouad dimension as an example of a ``large'' dimension) and another for ``small'' $Φ$ (with the Assouad dimension as an example of a ``small'' dimension). We give many applications, including where the scaling factors are fixed and the probabilities are uniformly distributed. The almost sure $Φ$ dimension of the underlying random set is also a consequence of our results.

Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets

TL;DR

This work extends the theory of Φ-dimensions for random 1-variable measures from homogeneous Moran sets to non-homogeneous Moran constructions with unequal scaling, under a uniform separation condition. By introducing θ-dependent random variables and the function G(θ), it establishes a threshold-driven dichotomy: in the large-Φ regime, almost-sure Φ-dimensions are dictated by whether G(ψ) lies below or above ψ, often yielding a unique crossing α where G(α)=α; in the small-Φ regime, the almost-sure dimensions are given by explicit essential-supremum/infimum formulas involving the mass split p_n and scales a_n,b_n. The paper provides concrete deterministic and probabilistic examples, extends the analysis to higher dimensions and more general branching, and clarifies when the Φ-dimensions of the random measure align with those of the supporting random set. Collectively, the results offer precise, computable descriptions of local fractal geometry for a broad class of non-homogeneous random Moran sets and measures and illuminate the relationship between measure- and set-dimensions in this stochastic setting.

Abstract

In this paper, we determine the almost sure values of the -dimensions of random measures supported on random Moran sets in that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets \cite{HM} to the case of unequal scaling factors. The -dimensions are intermediate Assouad-like dimensions with the (quasi-)Assouad dimensions and the -Assouad spectrum being special cases. The almost sure value of exhibits a threshold phenomena, with one value for ``large'' (with the quasi-Assouad dimension as an example of a ``large'' dimension) and another for ``small'' (with the Assouad dimension as an example of a ``small'' dimension). We give many applications, including where the scaling factors are fixed and the probabilities are uniformly distributed. The almost sure dimension of the underlying random set is also a consequence of our results.
Paper Structure (18 sections, 17 theorems, 176 equations, 2 figures)

This paper contains 18 sections, 17 theorems, 176 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Assouad-like dimensions
  3. Random Moran sets and Measures
  4. Definition of random Moran sets $\mathcal{C}_{\omega }$ and random measures $\mu_\omega$
  5. Dimension results for large $\Phi$
  6. Statement of the dimension theorem for large $\Phi$ and preliminary results
  7. Proof of the Theorem
  8. Consequences of the Theorem
  9. Example: The deterministic Moran set $\mathcal{C}_{ab}$
  10. $p_n$ chosen uniformly over $[0,1]$
  11. $p_n$ chosen from the two-element set $\{ p, 1 -p \}$ for fixed $0 < p < 1/2$
  12. $p_n$ deterministic and $(a,b)$ random
  13. Further remarks on $G(\theta )$
  14. Relating $\dim_{\Phi} \mu$ to $\dim_{\Phi} {\mathcal{C}}_\omega$
  15. Comments on a more general construction
  16. ...and 3 more sections

Key Result

Proposition 1

Let $\Phi ,\Psi$ be dimension functions and $\mu$ be a measure. (i) If $\Phi (t)\leq \Psi (t)$ for all $t>0$, then $\overline{\dim }_{\Psi }\mu$$\leq \overline{\dim }_{\Phi }\mu$ and $\underline{\dim }_{\Phi }\mu \leq \underline{\dim }_{\Psi }\mu$. (ii) We have that and $\dim _{L}\mu \leq \underline{\dim }_{\Phi }\mu$. If $\mu$ is doubling, then $\underline{\dim }_{\Phi }\mu \leq \underline{\dim

Figures (2)

  • Figure 1: $\overline{\dim }_{\Phi }\mu$ as a function of $(a,b)$ for $\mu$ on $C_{ab}$ with $p\sim U[0,1]$.
  • Figure 2: Four different examples of $G(\theta)$, described in the appendix.

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 36 more