A class of self-affine sets and self-affine measures
Let \({\mathcal I}=\{\phi_j\}_{j=1}^m\) be an iterated function system (IFS) consisting of a family of contractive affine maps on \(\mathbb R^d\), then there exists a unique compact set \(K=K({\mathcal I})\), called the self-affine set, or attractor of the IFS \({\mathcal I}\), such that \(K=\bigcup_{j=1}^m\phi_j(K)\). Moreover, for any given probability vector \({\mathbf p}=(p_1,\ldots ,p_m)\), i.e., \(p_j>0\) for all \(j\) and \(\sum_{j=1}^m p_j=1\),there exists a unique compactly supported probability measure \(\nu =\nu _{{\mathcal I},{\mathbf p}}\) such that \[ \nu =\sum_{j=1}^m{p_j\nu \circ \phi_j^{-1}}, \] the measure is called self-affine measure. When the maps are self-similar, and the open set condition is fulfilled, then the structure (containing the dimensions) of the self-similar sets and measures are rather clear, but for the general self-affine maps, this is a very difficult problem, even determining the Hausdorff dimension and box dimension under the open set condition. The following are partial results concerning the problem. McMullen and Bedford independently considered the type for \({\mathcal I}=\{\phi_j\}_{j=1}^m\) in which all \(\phi_j\) have the form \[ \phi_j(x)= \left[\begin{matrix} n^{-1}&0\\ 0&k^{-1}\\ \end{matrix}\right]x+ \left[\begin{matrix} a_j/h\\b_j/k\\ \end{matrix}\right] \] where all \(a_j,b_j\) are integers, \(0\leq a_j<n\) and \(0\leq b_j <m\). They determined the Hausdorff dimension and the box dimension which do not coincide in general. Lallay and Gatzouras, in a highly technical article along the same spirit of McMullen, obtained the dimension results for a broader class of IFS \({\mathcal I}=\{\phi_j \}_{j=1}^m\), in which all rectangles \(\phi_j((0,1)^2)\) are parallel to the axes and have longer sides parallel to the \(x\)-axis. Aside from a few other special cases such as the graph-directed McMullen class studied by Kenyon and Peres, the Lalley-Gatzouras class (which includes the McMullen class) remains the only deterministic class of self-affine sets whose Hausdorff dimension are known. Hu obtained the box dimension of a class of nonlinear self-affine sets in terms of the topological pressure. More precisely, he considered expanding maps on \(\mathbb R^2\) having an invariant ``strong unstable foliation'' and obtained a formula for the box-dimension of its closed invariant sets involving topological pressures. Falconer gave a variational formula for the Hausdorff and box dimensions for ``almost all'' self-affine sets under some assumptions. Later Hueter and Lally, and Solomyak proved that Falconer's formula remains true under some weaker condition. The article under review, by a completely different method, studies fundamental properties of a class of self-affine sets and measures, such as the \(L^q\) spectrum, the Hausdorff dimension and the entropy dimension. These quantities are important basic ingredients in the study of fractal geometry, particularly in the study of multifractal phenomena. The authors calculate first the \(L^q\) spectrum and entropy dimension for a class of self-affine measures in \(\mathbb R^2\). This class of self-affine measures \(\nu_{\mathcal I,\mathbf p}\) requires only that the underlying IFS's \({\mathcal I}=\{\phi_j\}_{j=1}^m\) satisfy the rectangular open set condition (ROSC). It is a much larger class than the McMullen class and the Lally-Gatzouras class. Simply speaking, \({\mathcal I} =\{\phi_j\}_{j=1}^m\) in \(\mathbb R^2\) satisfies the ROSC, if there is an open rectangle \(T\) such that the maps \(\phi_j\) map \(T\) into disjoint rectangles parallel to the axes inside \(T\). As an application the authors obtain the formula for the box dimension of \(K(\mathcal I)\) under the ROSC as well as the Hausdorff dimension of \(\nu_{\mathcal I,\mathbf p}\) under some additional assumptions. The results on box dimension can be viewed as an extension of the box dimension results by Lally and Gatzouras and Hu.
scientific article; zbMATH DE number 2171866