this book gives a comprehensive and up-to-date survey on dynamics and related topics, such as fatou-julia type theory, the ergodic theorem and invariant sets, hyperbolicity in differentiable or complex dynamics, iterant ion theory on pm, complex dynamics in cm and the foundations of differentiable and complex dynamics. the main aims of this volume are, firstly, to advance the study of the above-named topics and to establish the corresponding fatou-julia results for complex manifolds, and, secondly, to provide some advanced account of dynamical systems within the framework of geometry and analysis, presented from a unified approach applicable to both real and complex manifolds. audience: this work will be of interest to graduate students and researchers involved in the fields of global analysis, analysis on manifolds, several complex variables and analytic spaces, partial differential equations, differential geometry, measure and integration.