An exact formula for the measure dimensions associated with a class of piecewise linear maps
Let \(w_ 1,...,w_ N\) be similitudes contracting \(R^ n\) by the factors \(0<s_ i<1\), respectively. For given probabilities \(p_ 1,...,p_ N\) there exists a unique (``p-balanced'') measure \(\mu\) (supported on the unique compact set A satisfying \(A=\cup w_ iA)\) having the invariance property \(\mu =\sum p_ i\mu \circ w_ i^{-1}.\) Theorem 2. If \(w_ iA\cap w_ jA=\emptyset\) for \(i=j\), then both the Hausdorff-Besicovitch and the box dimensions of \(\mu\) equal \(\sum p_ i \log p_ i/\sum p_ i \log s_ i.\) For general affine maps \(w_ i\) the equality of both dimensions is no longer true (section 4).
scientific article; zbMATH DE number 4089838
scientific article