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 $\{f_k:[b_k,b_{k-1}]\to [0,1]\}_{k\in\N}$ 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.