On the conditions for the continuity of the Hausdorff measure

TL;DR

This paper investigates the conditions under which the Hausdorff measure of limit sets of iterated function systems (IFS) converges to 1, focusing on nonlinear functions and specific geometric conditions.

Contribution

It establishes new criteria involving Hausdorff dimension and geometric bounds that ensure the Hausdorff measure of IFS limit sets approaches 1.

Findings

If certain limit and boundedness conditions hold, the Hausdorff measure converges to 1.

Four conditions are provided that guarantee measure convergence for nonlinear IFS.

Examples of IFS families satisfying these conditions are presented.

Abstract

Let $(b_k)_{k = 0}^\infty$ be strictly decreasing sequence of real numbers such that $b_0 = 1$ and $\{f_k:[b_k,b_{k-1}]\to [0,1]\}_{k\in\N}$ be decreasing functions such that $f_k(b_k) = 1$ and $f_k(b_{k-1}) = 0$, $k = 1, 2, \dots$. By $g_k: [0,1] \to [b_k, b_{k-1}]$ we denote the inverse of $f_k$ for $k = 1,2 \dots$. First, we define iterated function system (IFS) $S_n$ by limiting the collection of functions $g_k$ to first n, meaning $S_n = \{g_k \}_{\{k=1, \dots n\}}$. Let $J_n$ denote the limit set of $S_n$. In the first part, we show that if $S_n$ fulfills the following two conditions: (1)~$\lim\limits_{n \to \infty} \left(1-h_n\right) \ln{n} = 0 $ where $h_n$ is the Hausdorff dimension of $J_n$, and (2)~$\sup \limits_{k\in \mathbb{N}} \left \{\frac{b_k-b_{k+1}}{b_{k+1}} \right \} < \infty $, then $\lim\limits_{n\to \infty} H_{h_n}(J_n) = 1 = H_1(J)$, where $h_n$ is the Hausdorff dimension of $J_n$ and $H_{h_n}$ is the corresponding Hausdorff measure. In the second part, we provide four conditions for IFS consisting of nonlinear functions $f_k$ which guarantee that $\lim\limits_{n\to \infty} H_{h_n}(J_n) = 1 = H_1(J)$, where $h_n$ is the Hausdorff dimension of $J_n$ and $H_{h_n}$ is the corresponding Hausdorff measure. We also provide a wide collection of examples of families of IFSes fulfilling those assumptions.