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This monograph grew out of the authors' efforts to provide a natural geometric description for the class of maps invariant under parabolic renormalization and for the Inou-Shishikura fixed point itself as well as to carry out a computer-assisted study of the parabolic renormalization operator. It introduces a renormalization-invariant class of analytic maps with a maximal domain of analyticity and rigid covering properties and presents a numerical scheme for computing parabolic renormalization of a germ, which is used to compute the Inou-Shishikura renormalization fixed point.   Inside, readers will find a detailed introduction into the theory of parabolic bifurcation,  Fatou coordinates, Écalle-Voronin conjugacy invariants of parabolic germs, and the definition and basic properties of parabolic renormalization.   The systematic view of parabolic renormalization developed in the book and the numerical approach to its study will be interesting to both experts in the field as well as graduate students wishing to explore one of the frontiers of modern complex dynamics.

Differential equations, Parabolic. --- Differential equations, Parabolic --- Renormalization (Physics) --- Numerical solutions. --- Charge and mass renormalization --- Mass and charge renormalization --- Electric charge and distribution --- Mass (Physics) --- Physical measurements --- Quantum field theory --- Numerical analysis --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Differentiable dynamical systems. --- Functions of complex variables. --- Numerical analysis. --- Dynamical Systems and Ergodic Theory. --- Functions of a Complex Variable. --- Numerical Analysis. --- Mathematical analysis --- Complex variables --- Elliptic functions --- Functions of real variables --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Dynamics. --- Ergodic theory. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics