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Classical Expanding Random Systems

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2036)

Abstract

Having treated a very general situation up to here, we now focus on more concrete random repellers and, in the next section, random maps that have been considered by Denker and Gordin. The Cantor example of Chap. 5.3 and random perturbations of hyperbolic rational functions like the examples considered by Brück and Büger are typical random maps that we consider now. We classify them into quasi-deterministic and essential systems and analyze then their fractal geometric properties. Here as a consequence of the techniques we have developed, we positively answer the question of Brück and Büger (see [9] and Question 5.4 in [8]) of whether the Hausdorff dimension of almost all (most) naturally defined random Julia sets is strictly larger than 1. We also show that in this same setting the Hausdorff dimension of almost all Julia sets is strictly less than 2.

Keywords

  • Hausdorff Dimension
  • Polynomial System
  • Compact Riemannian Manifold
  • Random System
  • Classical Conformal

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Volker Mayer .

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© 2011 Springer-Verlag Berlin Heidelberg

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Mayer, V., Skorulski, B., Urbanski, M. (2011). Classical Expanding Random Systems. In: Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry. Lecture Notes in Mathematics(), vol 2036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23650-1_8

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