On the conditions for the continuity of the Hausdorff measure

Abstract

Let $(b_k{)}_{k=0}^{\infty }$ be strictly decreasing sequence of real numbers such that $b_0=1$ and $\{f_k:[b_k,b_{k-1}]\to [0,1]{\}}_{k\in \mathbb{N}}$ 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)~$\underset{n\to \infty }{\lim }\left(1-h_n\right)\ln n=0$ where $h_n$ is the Hausdorff dimension of $J_n$, and (2)~$\underset{k\in \mathbb{N}}{\sup }\left\{\frac{b_k-b_{k+1}}{b_{k+1}}\right\}<\infty$, then $\underset{n\to \infty }{\lim }{H}_{h_n}(J_n)=1=H_1(J)$, 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 $\underset{n\to \infty }{\lim }{H}_{h_n}(J_n)=1=H_1(J)$, 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.

28 pages, 3 figures