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This is a one-stop introduction to the methods of ergodic theory applied to holomorphic iteration. The authors begin with introductory chapters presenting the necessary tools from ergodic theory thermodynamical formalism, and then focus on recent developments in the field of 1-dimensional holomorphic iterations and underlying fractal sets, from the point of view of geometric measure theory and rigidity. Detailed proofs are included. Developed from university courses taught by the authors, this book is ideal for graduate students. Researchers will also find it a valuable source of reference to a large and rapidly expanding field. It eases the reader into the subject and provides a vital springboard for those beginning their own research. Many helpful exercises are also included to aid understanding of the material presented and the authors provide links to further reading and related areas of research.

Conformal geometry. --- Fractals. --- Ergodic theory. --- Iterative methods (Mathematics) --- Iteration (Mathematics) --- Numerical analysis --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Geometry

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"This primer uses straightforward language to give the reader step-by-step instructions for identifying and analyzing fractal patterns and the social process that create them. By making fractals accessible to the social science students, this book has a significant impact on the understanding of human behavior."--Pub. desc.

Quantitative methods in social research --- Social sciences --- Fractals --- Sciences sociales --- Fractales --- Mathematical models --- Statistical methods --- Modèles mathématiques --- Méthodes statistiques --- #SBIB:303H10 --- #SBIB:303H520 --- Methoden en technieken: algemene handboeken en reeksen --- Methoden sociale wetenschappen: techniek van de analyse, algemeen --- Fractals. --- Mathematical models. --- Statistical methods. --- Modèles mathématiques --- Méthodes statistiques --- Fractal analysis --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Fractal geometric analysis --- Geometric analysis --- Social sciences - Mathematical models --- Social sciences - Statistical methods

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This book—an outgrowth of an international conference held in honor of Jacques Peyrière—provides readers with an overview of recent developments in the mathematical fields related to fractals. Included are original research contributions as well as surveys written by experts in their respective fields. The chapters are thematically organized into five major sections: • Geometric Measure Theory and Multifractals; • Harmonic and Functional Analysis and Signal Processing; • Dynamical Systems and Analysis on Fractals; • Stochastic Processes and Random Fractals; • Combinatorics on Words. Recent Developments in Fractals and Related Fields is aimed at pure and applied mathematicians working in the above-mentioned areas as well as other researchers interested in discovering the fractal domain. Throughout the volume, readers will find interesting and motivating results as well as new avenues for further research. Contributors: J.-P. Allouche, A.M. Atto, J.-M. Aubry, F. Axel, A. Ayache, C. Bandt, F. Bastin, P.R. Bertrand, A. Bonami, G. Bourdaud, Z. Buczolich, M. Clausel, Y. Demichel, X.-H. Dong, A. Durand, A.H. Fan, U.R. Freiberg, S. Jaffard, E. Järvenpää, A. Käenmäki, A. Karoui, H. Kempka, T. Langlet, K.-S. Lau, B. Li, M.M. France, S. Nicolay, E. Olivier, D. Pastor, H. Rao, S.Gy. Révész, J. Schmeling, A. Sebbar, P. Shmerkin, E.C. Waymire, Z.-X. Wen, Z.-Y. Wen, S.C. Williams, S. Winter.

Astronomy -- Mathematics. --- Dimension theory (Topology). --- Fractals. --- Mathematics in nature. --- Fractals --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Mathematics. --- Harmonic analysis. --- Dynamics. --- Ergodic theory. --- Functional analysis. --- Functions of complex variables. --- Geometry. --- Probabilities. --- Dynamical Systems and Ergodic Theory. --- Functions of a Complex Variable. --- Probability Theory and Stochastic Processes. --- Abstract Harmonic Analysis. --- Functional Analysis. --- Dimension theory (Topology) --- Differentiable dynamical systems. --- Distribution (Probability theory. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Euclid's Elements --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Complex variables --- Elliptic functions --- Functions of real variables --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics