Dimension spectra of self-affine sets
A function \(H(\delta)\) -- the dimension spectrum -- monitoring the dimensions of sections of a self-affine carpet \(X\) is considered. Specifically, \(H(\delta)\) is defined as the Hausdorff dimension of the set of vertical coordinates that correspond to horizontal sections of \(X\) with box-counting (or Hausdorff) dimension \(\delta\). If \(X\) is irreducible, a version of the multifractal formalism is obtained in which \(H(\delta)\) coincides with the (modified) Legendre transform of a suitable free energy associated with \(H(\delta)\). The proof uses the fact that the multifractal formalism holds for irreducible sofic measures, which is proved by the author elsewhere. A variational formula is then derived for the Hausdorff dimension of \(X\), namely \(\dim_{\text{H}} X=\max_{\delta}\{\delta +H(\delta)\}\), which is also expressed in terms of the free energy.
scientific article; zbMATH DE number 1786371