The exact Hausdorff dimension for a class of fractal functions
This paper is a supplement to a paper of \textit{K. J. Falconer} [Math. Proc. Comb. Philos. Soc. 103, No. 2, 339-350 (1988; Zbl 0642.28005)]. It deals with some individual fractal functions of the following kind: Let \(A=[0,1]\times[0,1]\) and \(f_ i: A\to A\) for \(i=1,2,\dots,k\), \(k>1\) such that \(f_ i=L_ i+v_ i\), where \(L_ i=\left({\lambda_ i\atop 0}{0\atop\pm\mu}\right)\), \(v_ i=\left({x_ i\atop y_ i}\right)\) and \(0<\lambda_ i<1\), \(\sum \lambda_ i=1\), \(0<\mu<1\), \(x_{i+1}-x_ i\geq \lambda_ i\), \(y_{i+1}-y_ i=n_ i\mu\) for some \(n_ i\in\mathbb{Z}\), \(i=1,\dots,k\). The iterated function system \((f_ 1,\dots,f_ k; A)\) has some attractor \(E\) which is the graph of some function \(f: [0,1]\to[0,1]\). If \(f\) is a continuous function than the fractal dimension of \(E\) is the unique positive solution \(s\) of \(\sum^ k_{i=1}|\mu|\lambda^{t-1}_ i=1\) as was shown in \textit{M. F. Barnsley, J. Elton, D. Hardin} and \textit{P. Massopust} [SIAM J. Math. Anal. 20, No. 5, 1218-1242 (1989; Zbl 0704.26009)]. The authors gives now a sufficient condition such that \(s\) is also the Hausdorff dimension of \(E\). This condition is an inequality in terms of the elements of \(L_ i\) and a value \(M\) which arises from a rather technical matrix construction. It is shown that Kieswetter's and Bedford's fractals satisfy it for \(M=2\) and \(M=3\), respectively. Unfortunately, the paper seems to contain a lot of misprints that make the reading difficult.
scientific article; zbMATH DE number 150584
scientific article