Distance expanding random mappings, thermodynamical formalism, Gibbs measures and fractal geometry /
Mayer, Volker, 1964-
Distance expanding random mappings, thermodynamical formalism, Gibbs measures and fractal geometry / Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski. - Heidelberg ; New York : Springer-Verlag Berlin Heidelberg, ©2011. - 1 online resource (x, 112 pages) : illustrations (some color) - Lecture notes in mathematics, 2036 0075-8434 ; . - Lecture notes in mathematics (Springer-Verlag) ; 2036. .
Includes bibliographical references (pages 109-110) and index.
1 Introduction -- 2 Expanding Random Maps -- 3 The RPF-theorem -- 4 Measurability, Pressure and Gibbs Condition -- 5 Fractal Structure of Conformal Expanding Random Repellers -- 6 Multifractal Analysis -- 7 Expanding in the Mean -- 8 Classical Expanding Random Systems -- 9 Real Analyticity of Pressure.
Annotation The theory of random dynamical systems originated from stochasticdifferential equations. It is intended to provide a framework andtechniques to describe and analyze the evolution of dynamicalsystems 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 Bowens 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 manyproperties 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.
English.
9783642236501 3642236502 3642236499 9783642236495
10.1007/978-3-642-23650-1 doi
Springer
015965467 Uk
Random dynamical systems.
Fractals.
Systèmes dynamiques aléatoires.
Fractales.
fractals.
Fractales
Fractals
Random dynamical systems
Mathematics Differentiable dynamical systems Dynamical Systems and Ergodic Theory
QA614.835 / .M39 2011
515/.39
Distance expanding random mappings, thermodynamical formalism, Gibbs measures and fractal geometry / Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski. - Heidelberg ; New York : Springer-Verlag Berlin Heidelberg, ©2011. - 1 online resource (x, 112 pages) : illustrations (some color) - Lecture notes in mathematics, 2036 0075-8434 ; . - Lecture notes in mathematics (Springer-Verlag) ; 2036. .
Includes bibliographical references (pages 109-110) and index.
1 Introduction -- 2 Expanding Random Maps -- 3 The RPF-theorem -- 4 Measurability, Pressure and Gibbs Condition -- 5 Fractal Structure of Conformal Expanding Random Repellers -- 6 Multifractal Analysis -- 7 Expanding in the Mean -- 8 Classical Expanding Random Systems -- 9 Real Analyticity of Pressure.
Annotation The theory of random dynamical systems originated from stochasticdifferential equations. It is intended to provide a framework andtechniques to describe and analyze the evolution of dynamicalsystems 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 Bowens 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 manyproperties 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.
English.
9783642236501 3642236502 3642236499 9783642236495
10.1007/978-3-642-23650-1 doi
Springer
015965467 Uk
Random dynamical systems.
Fractals.
Systèmes dynamiques aléatoires.
Fractales.
fractals.
Fractales
Fractals
Random dynamical systems
Mathematics Differentiable dynamical systems Dynamical Systems and Ergodic Theory
QA614.835 / .M39 2011
515/.39