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Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry
Αγγλική
146 b
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In this book we introduce measurable expanding random systems, develop the thermodynamical formalism and establish, in particular, the exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen s formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we arrive at a natural division of the systems into two classes: quasi-deterministic systems which share many properties of deterministic ones, and essential random systems which are rather generic and never bilipschitz equivalent to deterministic systems. We show in the essential case that the Hausdorff measure vanishes, refuting a conjecture by Bogenschutz and Ochs. Lastly, we present applications of our results to various specific conformal random systems and positively answer a question from Bruck and Buger concerning the Hausdorff dimension of random Julia sets.
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