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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.

Mathematics --- Physical Sciences & Mathematics --- Geometry --- Calculus --- Functions, Meromorphic. --- Gibbs' equation. --- Fractals. --- Expanding universe. --- Universe, Expanding --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Equation, Gibbs' --- Meromorphic functions --- Mathematics. --- Dynamics. --- Ergodic theory. --- Dynamical Systems and Ergodic Theory. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Math --- Science --- Astrophysics --- Big bang theory --- Cosmology --- Red shift --- Dimension theory (Topology) --- Differential equations --- Phase rule and equilibrium --- Thermodynamics --- Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics

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Systems driven far from thermodynamic equilibrium can create dissipative structures through the spontaneous breaking of symmetries. A particularly fascinating feature of these pattern-forming systems is their tendency to produce spatially confined states. These localized wave packets can exist as propagating entities through space and/or time. Various examples of such systems will be dealt with in this book, including localized states in fluids, chemical reactions on surfaces, neural networks, optical systems, granular systems, population models, and Bose-Einstein condensates.This book should appeal to all physicists, mathematicians and electrical engineers interested in localization in far-from-equilibrium systems. The authors - all recognized experts in their fields - strive to achieve a balance between theoretical and experimental considerations thereby giving an overview of fascinating physical principles, their manifestations in diverse systems, and the novel technical applications on the horizon.

Chemical engineering -- Tables. --- Chemical engineering. --- Chemical reactions. --- Solitons. --- Thermodynamics -- Tables. --- Thermodynamics. --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Solitons --- Fractals. --- Research. --- Mathematical models. --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Pulses, Solitary wave --- Solitary wave pulses --- Wave pulses, Solitary --- Physics. --- System theory. --- Quantum optics. --- Amorphous substances. --- Complex fluids. --- Phase transitions (Statistical physics). --- Complex Systems. --- Quantum Optics. --- Phase Transitions and Multiphase Systems. --- Soft and Granular Matter, Complex Fluids and Microfluidics. --- Statistical Physics and Dynamical Systems. --- Dimension theory (Topology) --- Connections (Mathematics) --- Nonlinear theories --- Wave-motion, Theory of --- Statistical physics. --- Mathematical statistics --- Statistical methods --- Dynamical systems. --- Complex liquids --- Fluids, Complex --- Amorphous substances --- Liquids --- Soft condensed matter --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics --- Optics --- Photons --- Quantum theory --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics