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## Über dieses Buch

This book, along with its companion volume, Nonlinear Dynamics New Directions: Models and Applications, covers topics ranging from fractal analysis to very specific applications of the theory of dynamical systems to biology. This first volume is devoted to fundamental aspects and includes a number of important new contributions as well as some review articles that emphasize new development prospects. The second volume contains mostly new applications of the theory of dynamical systems to both engineering and biology. The topics addressed in the two volumes include a rigorous treatment of fluctuations in dynamical systems, topics in fractal analysis, studies of the transient dynamics in biological networks, synchronization in lasers, and control of chaotic systems, among others.

This book also:

· Presents a rigorous treatment of fluctuations in dynamical systems and explores a range of topics in fractal analysis, among other fundamental topics

· Features recent developments on large deviations for higher-dimensional maps, a study of measures resisting multifractal analysis and a overview of complex Kleninan groups

· Includes thorough review of recent findings that emphasize new development prospects

## Inhaltsverzeichnis

### A Note on the Large Deviations for Piecewise Expanding Multidimensional Maps

We present here the large deviation principle for some systems admitting a spectral gap, by using the functional approach of Hennion and Hervé, with slight modification. Our main application concerns multidimensional expanding maps introduced by Saussol.
R. Aimino, S. Vaienti

### Directional Metric Entropy and Lyapunov Exponents for Dynamical Systems Generated by Cellular Automata

The deterministic dynamics of a spatially extended physical or chemical or biological system may be complex, as in the case of turbulent flows in contrast with the simple motion of laminar fluids. The complexity of extended dynamical systems has been described in many ways using several characteristics and several models. An important role, In investigating the complexity of dynamical systems, entropy and quantities connected with it plays an important role. In the smooth dynamics the Lyapunov exponents are quantities of this type.
Maurice Courbage, Brunon Kamiński

### On the Complexity of Some Geometrical Objects

We recall the definition of the ϵ-distortion complexity of a set defined in Bonanno et al. (Nonlinearity 24:459–479, 2011) and the results obtained in this chapter for Cantor sets of the interval defined by iterated function systems. We state an analogous definition for measures which may be more useful when dealing with dynamical systems. We prove a new lower bound in the case of Cantor sets of the interval defined by analytic iterated function systems. We also give an upper bound for the ϵ-distortion complexity of invariant sets of uniformly hyperbolic dynamical systems.
P. Collet

### Fluctuations of Observables in Dynamical Systems: From Limit Theorems to Concentration Inequalities

We start by reviewing recent probabilistic results on ergodic sums in a large class of (nonuniformly) hyperbolic dynamical systems. Namely, we describe the central limit theorem, the almost-sure convergence to the Gaussian and other stable laws, and large deviations.
Next, we describe a new branch in the study of probabilistic properties of dynamical systems, namely concentration inequalities. They allow to describe the fluctuations of very general observables and to get bounds rather than limit laws. We end up with two sections: one gathering various open problems, notably on random dynamical systems, coupled map lattices, and the so-called nonconventional ergodic averages; and another one giving pointers to the literature about moderate deviations, almost-sure invariance principle, etc.
Jean-René Chazottes

### On Flow Barriers in Discontinuous Dynamical Systems

In this chapter, the theory and concept of flow barriers in discontinuous dynamical systems is presented. The coming and leaving flow barriers are discussed first, followed by the boundary flow barriers relative to the corresponding domains on the boundary. This chapter is dedicated to the 65th birthday of Prof. Valentin Afraimovich.
Albert C. J. Luo

### Nonstandard Analysis of the Behavior of Ergodic Means of Dynamical Systems on Very Big Finite Probability Spaces

In this chapter we discuss the behavior of ergodic means of discrete time dynamical systems on a very big finite probability space Y (discrete dynamical systems below). The G. Birkhoff Ergodic Theorem states the eventual stabilization of ergodic means of integrable functions for almost all points of the probability space. The trivial proof of this theorem for the case of finite probability spaces shows that this stabilization happens for those time intervals, whose length n exceeds significantly the cardinality $$|Y|$$ of Y, i.e., $$\frac{n}{|Y|}$$ is a very big number. For the case of very big number $${|Y|}$$ we introduce the class of S-integrable functions and we prove that the ergodic means of these functions exhibit a regular behavior even for intervals whose length is comparable with $${|Y|}$$.
E. I. Gordon, L. Yu. Glebsky, C. W. Henson

### On Measures Resisting Multifractal Analysis

Any ergodic measure of a smooth map on a compact manifold has a multifractal spectrum with one point - the dimension of the measure itself - at the diagonal. We will construct examples where this fails in the most drastic way for invariant measures invariant under linear maps of the circle.
Jörg Schmeling, Stéphane Seuret

### An Overview of Complex Kleinian Groups

Classical Kleinian groups are discrete subgroups of $$PSL(2,{\mathbb{C}})$$ acting on the complex projective line $$\mathbb{P}_\mathbb{C}^1$$ (which coincides with the Riemann sphere) with nonempty region of discontinuity. These can also be regarded as the monodromy groups of certain differential equations. These groups have played a major role in many aspects of mathematics for decades, and also in physics. It is thus natural to study discrete subgroups of the projective group $$PSL(n,{\mathbb{C}})$$, $$n> 2$$. Surprisingly, this is a branch of mathematics which is in its childhood, and in this chapter we give an overview of it.

### Semigroups of Mappings and Correspondences: Characters and Representations in Holomorphic Dynamical Systems

We use the theory of representation of semigroups to get algebraic characterizations of conjugacy of semigroups of endomorphisms. This text is a short version of the chapter of a monograph in preparation. Several results are already available on the paper Semigroup representations in holomorphic dynamics published in (Cabrera et al. Discrete Contin. Dyn. Syst. 33 (2013), no. 4, 1333–1349).Complimentary part is On Decomposable Rational Maps which is published in (Cabrera and Makienko, Conform Geom Dyn 15:21–218, 2011) and was prepared after these expository notes.
Carlos Cabrera, Peter Makienko, Peter Plaumann
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