On loosely self-similar sets (Q1910408)

The author introduces the notion of loosely self-similar sets \(K\). The main difference from ordinary self-similar sets in the sense of Hutchinson is that the similitudes \(\varphi_{i_1,i_2,\dots, i_k}\) in each step of the construction may change up to a common contraction rate \(0<r_{i_k}<1\), where \(\omega=(i_1,i_2,\dots, i_k,\dots)\in\{1,2, \dots, m\} ^\mathbb{N}\) is some address for \(K\). For a given probability vector \((P_1, \dots, P_m)\) of positive numbers, the set \(K(P_1,\dots, P_m)=\{\varphi (\omega): N_i(\omega, n)/n\to P_i\) as \(n\to \infty\) for \(i=1,\dots,m\}\) is associated to \(K\), where \(\varphi\) is the address to a point map of \(K\) and \(N_i (\omega,n)\) counts the frequency of numbers \(i_k=i\) for \(k=1,\dots,n\) in \(\omega\). The author compares the \(\alpha\)-dimensional Hausdorff measure \(H^\alpha\), where \(\sum^m_{i=1}r^\alpha_i=1\), and the Bernoulli measure \(\nu_{(P_1,\dots, P_m)}\). For example, it is shown that they are absolutely continuous to each other on \(K\) for \(P_i=r^\alpha_i\). If \(P_i\neq r^\alpha_i\) for some \(i\) then the Bernoulli measure is not absolutely continuous with respect to \(H^\beta\) with \(\beta\) the Hausdorff dimension of \(K(P_1,\dots,P_m)\). The number \(\alpha\) is the maximum of Hausdorff dimensions for the sets \(K(P_1,\dots,P_m)\) and is always equal to the lower and upper box dimension.