The box dimension of self-affine graphs and repellers. (English) Zbl 0691.58025

In this important paper the author calculates the box dimension (capacity) of self-affine sets \(E\). He describes a canonical continuous surjection from \(\Sigma = {0,1,...,k-1}^{z^+}\) onto \(E\). Considering the shift transformation on \(\Sigma\) allows to apply methods of dynamical systems to study the structure of the set \(E\). It is proved (easily) that \(E\) is the graph of a continuous function. The main result of the paper says that the box dimension of \(E\) is given by a formula which has a similar form as Bowen’s formula for the Hausdorff dimension of quasi-circles and the formula of McCluskey and Manning for the Hausdorff dimension of plane Smales horseshoes. This formula is expressed in terms of the dynamics of the shift transformation on \(\Sigma\) and involves the topological pressure of a certain Hoelder continuous function. For the proof of this fact a complicated machinery is built which uses some advanced results of thermodynamic formalism in the version of Ruelle. An elementary proof (using only standard properties of Gibbs states for the shift map) can be found in a later paper of Bedford and Urbanski. The paper contains also a discussion of the problem of equality of box and Hausdorff dimensions of self-affine sets. This problem has stimulated an interesting development in the theory of these sets.
Reviewer: M.Denker


37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37A99 Ergodic theory
28A75 Length, area, volume, other geometric measure theory
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