When the weak separation condition implies the generalize finite type in $\mathbb{R}^d$

TL;DR

The paper addresses overlap controls for self-similar IFSs on $\4$ with full support and restricted rotations by proving that the weak separation condition (WSC) is equivalent to the generalized finite-type condition (GFTC) under these hypotheses. It advances the net-interval framework to higher dimensions, introduces the finite neighbor condition (FNC), and uses these tools to relate WSC to GFTCco while enabling practical computation of local dimensions of self-similar measures via a directed graph with transition matrices. By extending the Hare–Hare–Matthews approach to $\4$, it provides a concrete, algorithmic method to estimate local dimensions from net-interval data in higher dimensions. The work also highlights open questions about equivalences of overlap conditions in $\4$ and the role of rotations, suggesting directions for generalizing the theory beyond current technical constraints.

Abstract

Let $\mathcal{S}$ be an iterated function system in $\mathbb{R}^d$, with full support and some restrictions on the allowable rotations. We show that $\mathcal{S}$ satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. With this in mind, we extend the notion of net intervals from $\mathbb{R}$ to $\mathbb{R}^d$. We also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support.

Figures (7)

• Figure 1: Example \ref{['exnetinterval']}.
• Figure 2: Net intervals. Example \ref{['exampenetintervaldif']}
• Figure 3: Example in $\mathbb{R}^2$ of notation in Theorem \ref{['thm4.4']}.
• Figure 4: First iteration of the IFS $\mathcal{S}$. Example \ref{['Netintervalslookweird']}.
• Figure 5: Here we have iteration 2 and 3 of the IFS $\mathcal{S}$. Example \ref{['Netintervalslookweird']}.

Theorems & Definitions (45)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark
• Definition 2.5
• Example 2.6
• Example 2.7
• Definition 2.8
• Definition 2.9