TY - BOOK ID - 8062612 TI - Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry AU - Mayer, Volker. AU - Skorulski, Bartlomiej. AU - Urbanski, Mariusz. PY - 2011 SN - 3642236499 3642236502 PB - Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, DB - UniCat KW - Mathematics KW - Physical Sciences & Mathematics KW - Geometry KW - Calculus KW - Functions, Meromorphic. KW - Gibbs' equation. KW - Fractals. KW - Expanding universe. KW - Universe, Expanding KW - Fractal geometry KW - Fractal sets KW - Geometry, Fractal KW - Sets, Fractal KW - Sets of fractional dimension KW - Equation, Gibbs' KW - Meromorphic functions KW - Mathematics. KW - Dynamics. KW - Ergodic theory. KW - Dynamical Systems and Ergodic Theory. KW - Ergodic transformations KW - Continuous groups KW - Mathematical physics KW - Measure theory KW - Transformations (Mathematics) KW - Dynamical systems KW - Kinetics KW - Mechanics, Analytic KW - Force and energy KW - Mechanics KW - Physics KW - Statics KW - Math KW - Science KW - Astrophysics KW - Big bang theory KW - Cosmology KW - Red shift KW - Dimension theory (Topology) KW - Differential equations KW - Phase rule and equilibrium KW - Thermodynamics KW - Differentiable dynamical systems. KW - Differential dynamical systems KW - Dynamical systems, Differentiable KW - Dynamics, Differentiable KW - Global analysis (Mathematics) KW - Topological dynamics UR - https://www.unicat.be/uniCat?func=search&query=sysid:8062612 AB - The theory of random dynamical systems originated from stochastic differential equations. It is intended to provide a framework and techniques to describe and analyze the evolution of dynamical systems when the input and output data are known only approximately, according to some probability distribution. The development of this field, in both the theory and applications, has gone in many directions. In this manuscript we introduce measurable expanding random dynamical 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 classification of the systems into two classes: quasi-deterministic systems, which share many properties of deterministic ones; and essentially random systems, which are rather generic and never bi-Lipschitz equivalent to deterministic systems. We show that in the essentially random case the Hausdorff measure vanishes, which refutes a conjecture by Bogenschutz and Ochs. Lastly, we present applications of our results to various specific conformal random systems and positively answer a question posed by Bruck and Buger concerning the Hausdorff dimension of quadratic random Julia sets. ER -