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$L^q$-spectrum of graph-directed self-similar measures that have overlaps and are essentially of finite type

Yuanyuan Xie

Abstract

For self-similar measures with overlaps, closed formulas of the $L^q$-spectrum have been obtained by Ngai and the author for measures that are essentially of finite type in [J. Aust. Math. Soc. \textbf{106} (2019), 56--103]. We extend the results of Ngai and the author \cite{Ngai-Xie_2019} to the graph-directed self-similar measures. For graph-directed self-similar measures satisfying the graph open set condition, the $L^q$-spectrum has been studied by Edgar and Mauldin \cite{Edgar-Mauldin_1992}. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition. For graph-directed self-similar measures $μ$ on $\R^d$ ($d\ge1$), which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the $L^q$-spectrum of $μ$ for $q\ge 0$, and prove the differentiability of the $L^q$-spectrum. This framework allows us to include graph-directed self-similar measures that are strongly connected and not strongly connected and those in higher dimension.

$L^q$-spectrum of graph-directed self-similar measures that have overlaps and are essentially of finite type

Abstract

For self-similar measures with overlaps, closed formulas of the -spectrum have been obtained by Ngai and the author for measures that are essentially of finite type in [J. Aust. Math. Soc. \textbf{106} (2019), 56--103]. We extend the results of Ngai and the author \cite{Ngai-Xie_2019} to the graph-directed self-similar measures. For graph-directed self-similar measures satisfying the graph open set condition, the -spectrum has been studied by Edgar and Mauldin \cite{Edgar-Mauldin_1992}. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition. For graph-directed self-similar measures on (), which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the -spectrum of for , and prove the differentiability of the -spectrum. This framework allows us to include graph-directed self-similar measures that are strongly connected and not strongly connected and those in higher dimension.

Paper Structure

This paper contains 12 sections, 43 theorems, 225 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Graph-directed self-similar systems and measures that are essentially of finite type
  3. Graph-directed iterated function systems
  4. The essentially finite type condition for graph-directed self-similar measures
  5. Renewal equation and proofs of Theorem \ref{['T:main_thm_str_con']} and Theorem \ref{['T:main_thm_not_str_con']}.
  6. Strongly connected GIFSs on ${\mathbb R}$ and ${\mathbb R}^2$
  7. A strongly connected GIFS on ${\mathbb R}$
  8. A strongly connected GIFS on ${\mathbb R}^2$
  9. GIFSs that are not strongly connected on ${\mathbb R}$ and ${\mathbb R}^2$
  10. A GIFS that is not strongly connected on ${\mathbb R}$ and has a unique basic class
  11. A GIFS that is not strongly connected on ${\mathbb R}$ and has basic classes of height greater than 1
  12. A GIFS that is not strongly connected on ${\mathbb R}^2$

Key Result

Theorem 1.1

Let $\mu=\sum_{i=1}^{M}\mu_i$ be a graph-directed self-similar measure defined by a strongly connected GIFS $G=(V, E)$ on ${\mathbb R}^d$. Assume that $\mu$ satisfies (EFT) with $\{\Omega_i\}_{i=1}^{M}$ being an EFT-family and $({\bf B}, {\bf P}):=(\{B_{1,\ell}\}, \{{\bf P}_{k,\ell}\}_{k\ge1})_{\ell

Figures (20)

  • Figure 1: The first iteration of the GIFS defined in \ref{['e:exam_str_con_R_similitudes']}, where $\Omega_i=(0,1)$ for $i=1,2$. The figure is illustrated using $\rho=1/3$ and $r=2/7$.
  • Figure 2: The first iteration of the GIFS defined in \ref{['e:exam_str_con_R2_similitudes']} with $\Omega_1=\bigcup_{x\in(0,1)}(0,1)\times(x,1-x)$ and $\Omega_2=(0,1)\times(0,1)$.
  • Figure 3: First iteration of the GIFS defined in \ref{['E:exam_str_con_R_1_similitudes']}, where $\Omega_i=(0,1)$ for $i=1,2$. The figure is illustrated using $\rho=1/3$ and $r=2/7$.
  • Figure 4: First iteration of the GIFS defined in \ref{['E:exam_str_con_R_2_similitudes']}, where $\Omega_i=(0,1)$ for $i=1,\ldots, 6$. The figure is drawn with $\rho=1/3$ and $r=2/7$.
  • Figure 5: The first iteration of the GIFS defined in Example \ref{['E:exam_GIFS_not_str_con_R2']}, with $\Omega_1=\cup_{x\in(0,1)}(0,1)\times(x,1-x)$, and $\Omega_2=(2,3)\times(0,1)$.
  • ...and 15 more figures

Theorems & Definitions (60)

  • Theorem 1.1
  • Example 1.2
  • Corollary 1.3
  • Example 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Example 1.7
  • Corollary 1.8
  • Example 1.9
  • Corollary 1.10
  • ...and 50 more