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Multifractal decompositions of digraph recursive fractals. (English) Zbl 0764.28007

Let \((V,E)\) be a directed multigraph which is strongly connected. A ratio \(r(e)\in]0,1[\) is assigned to each edge \(e\in E\). For each path \(\gamma=e_ 1e_ 2\dots e_ k\) of length \(k\) and consisting of edges \(e_ 1,\dots,e_ k\) such that \(e_ i\) and \(e_{i+1}\) have at least a common node let \(r(\gamma)=r(e_ 1)\cdot\dots\cdot r(e_ k)\). Furthermore, for each node \(u\in V\) a non-empty compact set \(J_ u\) in \(\mathbb{R}^ n\) is given such that it is equal to the closure of its interior. If \(E_ u^{(k)}\) denotes the set of all paths \(\gamma\) with initial node \(u\) and length \(k\) then the associated Mauldin-Williams fractal is the set \[ K_ u=\bigcap^ \infty_{k=0}\bigcup_{\gamma\in E_ u^{(k)}}J(\gamma), \] where the \(J(\gamma)\) are chosen recursively such that the \(J(\gamma)\) are geometrically similar to \(J_ u\) with reduction ratio \(r(\gamma)\) and different \(J(\gamma)\), \(J(\gamma')\) do not overlap. Furthermore, \(K_ u\) can be represented as the image of a certain model \(E_ u^{(\omega)}\) of infinite strings using the symbols of \(E\) and directed by \((V,E)\) under a map \(h_ u\). If, furthermore, admissible transition probabilities \(p(e)\) are given for each edge, they give rise to some unique Markov measure \(\hat\mu_ u\) on \(E_ u^{(\omega)}\) and an image measure \(\mu_ u\) on \(K_ u\). If \(\sigma\in E_ u^{(\omega)}\) is an infinite string then \(\gamma=\sigma| k\) is the restriction to the \(k\) first letters of \(\sigma\). Analogously to \(r(\gamma)\) let \(p(\gamma)=p(e_ 1)\cdot\dots\cdot p(e_ k)\). For given real numbers \(\alpha>0\) the multifractal components \(K_ u^{(\alpha)}\) of \(K_ u\) are defined as follows \[ \hat K_ u^{(\alpha)}=\left\{\sigma\in E_ u^{(\omega)}:\lim_{k\to\infty}{\log p(\sigma| k)\over\log r(\sigma| k)}=\alpha\right\},\quad K_ u^{(\alpha)}=h_ u(\hat K_ u^{(\alpha)}). \] Let \(A(q,\beta)\) be a square matrix of type \(| V|\times| V|\) with the entries \(A_{uv}(q,\beta)=\sum_{e\in E_{uv}}p(e)^ qr(e)^ \beta\). For any real \(q\) there is a unique \(\beta\) such that \(A(q,\beta)\) has spectral radius 1. \(\beta\) is an analytic function of \(q\). The function \(f\) is defined as \(f=q\alpha+\beta\), where \(\alpha=-d\beta/dq\). The main result of this excellently written paper says that packing and Hausdorff dimension of the multifractal components \(K_ u^{(\alpha)}\) are equal to \(f(\alpha)\). The authors also study the behavior of the auxiliary functions \(\beta\) and \(\alpha\).

MSC:

28A80 Fractals
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