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DYNAMICS REPORTED reports on recent developments in dynamical systems. Dynamical systems of course originated from ordinary differential equations. Today, dynamical systems cover a much larger area, including dynamical processes described by functional and integral equations, by partial and stochastic differential equations, etc. Dynamical systems have involved remarkably in recent years. A wealth of new phenomena, new ideas and new techniques are proving to be of considerable interest to scientists in rather different fields. It is not surprising that thousands of publications on the theory itself and on its various applications are appearing DYNAMICS REPORTED presents carefully written articles on major subjects in dy­ namical systems and their applications, addressed not only to specialists but also to a broader range of readers including graduate students. Topics are advanced, while detailed exposition of ideas, restriction to typical results - rather than the most general one- and, last but not least, lucid proofs help to gain the utmost degree of clarity. It is hoped, that DYNAMICS REPORTED will be useful for those entering the field and will stimulate an exchange of ideas among those working in dynamical systems Summer 1991 Christopher K. R. T Jones Drs Kirchgraber Hans-Otto Walther Managing Editors Table of Contents The "Spectral" Decomposition for One-Dimensional Maps Alexander M. Blokh Introduction and Main Results 1. 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 0. 1. 1. Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. 2. A Short Description of the Approach Presented . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. 3. Solenoidal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Basic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 4.

Inhaltsverzeichnis

Frontmatter

The “Spectral” Decomposition for One-Dimensional Maps

Abstract
We construct the “spectral” decomposition of the sets \(\overline {Per\,f}\), ω(f) = ∪ω(x) and Ω(f) for a continuous map f: [0,1] → [0,1]. Several corollaries are obtained; the main ones describe the generic properties of f-invariant measures, the structure of the set Ω(f)\\(\overline {Per\,f}\) and the generic limit behavior of an orbit for maps without wandering intervals. The “spectral” decomposition for piecewise-monotone maps is deduced from the Decomposition Theorem. Finally we explain how to extend the results of the present paper for a continuous map of a one-dimensional branched manifold into itself.
Alexander M. Blokh

A Constructive Theory of Lagrangian Tori and Computer-assisted Applications

Abstract
Perturbative techniques are among the most powerful tools in the theory of conservative dynamical systems. Besides giving finite time predictions (something well known to the astronomers of the eighteenth century), perturbation methods may be used to establish the existence of regular motions. H. Poincaré used thoroughly such methods in his investigation in Celestial Mechanics [Po], obtaining, e.g., his celebrated results on periodic orbits for Hamiltonian systems. A more recent success of perturbation ideas is the so called “KAM (Kolmogorov [Ko]-Arnold [A1]-Moser [Mo1]) theory”, which ensures, under suitable smoothness assumptions, the survival under a small perturbation of “most” of the invariant maximal tori which foliate the phase-space of “integrable” conservative systems (see [B] for review and exhaustive references and [ChG] for recent developments).
Alessandra Celletti, Luigi Chierchia

Ergodicity in Hamiltonian Systems

Abstract
We discuss the Sinai method of proving ergodicity of a discontinuous Hamiltonian system with (nonuniform) hyperbolic behavior.
Carlangelo Liverani, Maciej P. Wojtkowski

Linearization of Random Dynamical Systems

Abstract
At the end of the last century the French mathematician Henri Poincaré laid the foundation for what we call nowadays the qualitative theory of ordinary differential equations. Roughly speaking, this theory is devoted to studying how the qualitative behavior (e.g. the asymptotic behavior) of solutions to certain initial value problems changes as the initial condition is varied. In order to make this more explicit, consider the autonomous differential equation
$$ x = f(x), $$
(1.1)
where f : ℝd → ℝd is a C1-mapping with f (x 0) = 0 for some x 0 ∈ ℝd. Obviously, the point x 0 is a constant solution of (1.1). But what can be said about the behavior of solutions of (1.1) starting in some small neighborhood of this point? Of course one is tempted to first study the linearized equation
$$ x = Df(x_0 )x $$
(1.2)
near the origin, since it may be hoped that the nonlinear behavior of (1.1) near x 0 will be “basically” the same.
Thomas Wanner
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