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Intended for beginners in ergodic theory, this book addresses students as well as researchers in mathematical physics. The main novelty is the systematic treatment of characteristic problems in ergodic theory by a unified method in terms of convergent power series and renormalization group methods, in particular. Basic concepts of ergodicity, like Gibbs states, are developed and applied to, e.g., Asonov systems or KAM Theory. Many examples illustrate the ideas and, in addition, a substantial number of interesting topics are treated in the form of guided problems.

Inhaltsverzeichnis

1. General Qualitative Properties

Abstract
We begin with a few comments on the meaning of the word ergodic.
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

2. Ergodicity and Ergodic Points

Abstract
The problem of determining which sets are visited with defined frequency by the motions of a dynamical system (Ω, S) can be satisfactorily solved in the case of particularly simple systems; for instance in the case in which $$S = {S_{{t_0}}}$$ and (S t )t∈ℝ is a Hamiltonian flow which is analytically integrable on a region W ⊂ ℝ2r and Ω = W, cf. definition 1.3.1. This means looking at motions observed at time intervals t 0. More precisely the following proposition holds.
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

3. Entropy and Complexity

Abstract
Continuing the analysis of general structural properties of motions of an invertible discrete dynamical system (Ω, S) we shall now discuss the foundations of the notion of complexity of motions on Ω and of its theory.
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

4. Markovian Pavements

Abstract
In the previous chapters the problem of studying the statistics of motions of a dynamical system (Ω, S), as seen from a partition P, has been shown to be equivalent to studying probability distributions on the space of sequences of symbols associated with a partition P (cf. proposition 2.3.2).
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

5. Gibbs Distributions

Abstract
The study of the general structure of dynamical systems, begun in the previous sections, could continue and constitutes one of the directions in which ergodic theory can be developed. We shall, however, look in a somewhat different direction dedicating attention to a few concrete problems that do not belong to the general theory. The more concrete studies involve analytic work of “classical” type and are more directly related to the applications.
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

6. General Properties of Gibbs and SRB Distributions

Abstract
The study of Gibbs distributions can be performed in remarkable depth as we shall hint in this section and in the forthcoming ones. We begin with a structure theorem which can be articulated into various propositions.
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

7. Analyticity, Singularity and Phase Transitions

Abstract
There are various instances in which the construction of the Gibbs distributions can be performed in great detail, almost completely explicitly, allowing us to answer satisfactorily questions concerning, for instance, mixing rates of Gibbs states and smoothness of their dependence on the potential.
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

8. Special Ergodic Theory Problems in Nonchaotic Dynamics

Abstract
A very natural and important question that one can ask about stability in Hamiltonian systems is what becomes of the simple foliation of phase space into invariant tori when a perturbing force is switched on.
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

9. Some Special Topics in KAM Theory

Abstract
In Chap. 8 the proof of the KAM theorem was based on diagrams and notions that became visually clear if the diagrammatic interpretation of the various terms building the Lindstedt series was kept in mind The analogy with the diagrams used in perturbation theory in quantum field theory and in statistical mechanics is, we feel, quite striking. Therefore one can wonder whether one could go ahead and apply other techniques widely employed in those fields.
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

10. Special Problems in Chaotic Dynamics

Abstract
A general theorem on Anosov maps allows us to say that in a certain sense Anosov maps that are close enough in C 2 can be considered as derived one from the other by a “change of coordinates”, which, however, is not really smooth. This is the theorem of structural stability of Anosov that can be formulated as follows.
Giovanni Gallavotti, Federico Bonetto, Guido Gentile

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