Bowen’s Formula for the Hausdorff Dimension of Self-Similar Sets
Geometric self-similarity of a set J (with a metric) means that the microscopic structure of the set (in any neighborhood of any point) can be magnified to ressemble the macroscopic structure of the set. The Hausdorff dimension t of J is then a natural notion, as stressed by Mandelbrot . Bowen’s formula expresses t in terms of concepts of statistical mechanics, and can be used to show that the dimension t is not an integer (see Bowen ) or that it depends smoothly on parameters (see Ruelle ).
KeywordsHausdorff Dimension Gibbs Measure Admissible Sequence Markov Partition Macroscopic Structure
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- P. Billingsley. Ergodic theory and information. John Wiley New York, 1965.Google Scholar
- R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. N° 470. Springer, Berlin, 1975.Google Scholar
- H. Brolin. Invariant sets under iteration of rational functions. Arkiv för Mat. 6, 106–144 (1965).Google Scholar
- B. Mandelbrot. Fractals: form, chance, and dimension. W.H. Freeman, San Francisco, 1977.Google Scholar
- D. Ruelle. Thermodynamic formalism. Addison-Wesley, Reading, 1978.Google Scholar
- D. Ruelle. Repellers for real analytic maps. Ergodic Theory and dynamical systems. To appear.Google Scholar