A multifractal (Benzi, Paladin, Parisi and Vulpiani 1984; Frisch and Parisi 1985; Halsey et al. 1986; Feigenbaum 1987) is a fractal for which a probability measure on the fractal support is given. For example, if the fractal is the attractor of a map in a numerical experiment, the probability measure is given by the relative frequencies of the iterates, which are interpreted as probabilities, as already explained in chapter 2. It is then useful to introduce more general dimensions D(β) that also contain information about the probability distribution on the fractal.

The grid of boxes of equal size

Suppose an arbitrary (possibly fractal) probability distribution is given (for example, the natural invariant measure of an ergodic map). For the following it is important that we divide the phase space into boxes of equal size, not of variable size. The different possibilities of partitioning the phase space have already been mentioned in section 3.1. In a d-dimensional space the boxes are d-dimensional cubes with side length ɛ. Again let us denote the number of boxes with nonzero probability by r. We label these boxes by i = 1, 2, …, r. The number r should be distinguished from the total number of boxes R ∼ ɛ−d. For a given value ɛ the probability attributed to a box i centred at some point x will be called pi. As Pi is attributed to a single box, it is a ‘local’ quantity.

Suppose we have an arbitrary, possibly fractal, probability distribution. For example, this might be the natural invariant density of a chaotic map. We wish to analyse the most important properties of this complicated distribution in a quantitative way. A fundamental idea, which has turned out to be very useful in nonlinear dynamics, is the following. To a given probability distribution a set of further probability distributions is attributed, which, in a way, have the ability to scan the structure of the original distribution. These distributions have the same form as thermodynamic equilibrium distributions and yield the key for various analogies between chaos theory and thermodynamics. We shall call them ‘escort distributions’. The considerations of this chapter are valid for arbitrary probability distributions, no matter how they are generated.

Temperature in chaos theory

Whereas in statistical thermodynamics probability distributions are sought on the basis of incomplete knowledge (usually only a few macroscopic variables are given, which are interpreted as statistical mean values), in chaos theory the central question can sometimes be reversed in a sense: here the distributions p may be given in the form of observed relative frequencies in a computer experiment. Then a characterization of the system is sought using relatively few global quantities that describe the relevant features. In thermodynamics we were led to the generalized canonical distribution by the unbiased guess.

In recent years methods borrowed from thermodynamics and statistical mechanics have turned out to be very successful for the quantitative analysis and description of chaotic dynamical systems. These methods, originating from the pioneering work of Sinai, Ruelle, and Bowen in the early seventies on the thermodynamic formalism of dynamical systems, have been further developed in the mean time and have attracted strong interest among physicists. The quantitative characterization of chaotic motion by thermodynamic means and the thermodynamic analysis of multifractal sets are now an important and rapidly evolving branch of nonlinear science, with applications in many different areas.

The present book aims at an elucidation of the various thermodynamic concepts used for the analysis of nonlinear dynamical systems. It is intended to be an elementary introduction. We felt the need to write an easily readable book, because so far there exist only a few classical monographs on the subject that are written for mathematicians rather than physicists. Consequently, we have tried to write in the physicist's language. We have striven for a form that is readable for anybody with the knowledge of mathematics a student of physics or chemistry at the early graduate level would have. No advanced mathematical preknowledge on the part of the reader is required. On the other hand, we also tried to avoid serious loss of rigour in our presentation.

Fractals are complex geometrical objects that possess nontrivial structure on arbitrary scales. In this chapter we first describe a few examples of fractals with a simple recurrent structure. Starting from these simple examples we explain the concept of a ‘fractal dimension’ and ‘Hausdorff dimension’. Finally, we consider some more complicated examples of fractals that are of utmost interest in nonlinear dynamics, yielding a glimpse of the beauty inherent in ‘self-similar’ structures: these are the Mandelbrot set, Julia sets, and fractals generated by iterated function systems.

Simple examples of fractals

The Koch curve A standard example of a fractal is the so called ‘Koch curve’. It is constructed as follows. We start with an equilateral triangle with sides of unit length and divide each side into three equal parts. Then, as illustrated in fig. 10.1, we put onto the middle part of each side a smaller equilateral triangle with a third of the side length. This step is then repeated for each of the new sides that were generated in the preceding step. The figure that arises after an infinite number of steps is the famous ‘Koch island’. Its border is called the ‘Koch curve’. It does not possess a finite length nor a tangent at any point. In contrast to the smooth lines and curves of Euclidean geometry such a geometric creation is called a ‘fractal’.

Let us use the following procedure to measure the length of the Koch curve or of an irregularly shaped coastline of an island.

Most of the considerations in this chapter are only valid for special classes of chaotic maps f, namely for either expanding or hyperbolic maps (for the definitions see section 15.6). We shall first derive a very important variational principle for the topological pressure. The notion of Gibbs measures and SRB measures (Sinai–Ruelle–Bowen measures) will be introduced. We shall then show that for one-dimensional expanding systems one type of free energy is sufficient: all interesting quantities such as the (dynamical) Rényi entropies, the generalized Liapunov exponents, and the Rényi dimensions can be derived from the topological pressure. A disadvantage is that, due to the expansion condition, we consider quite a restricted class of systems.

A variational principle for the topological pressure

Similar to the principle of minimum free energy in conventional statistical mechanics (see section 6.3) there is also a variational principle for the topological pressure. This variational principle allows us to distinguish the natural invariant measure of an expanding (or hyperbolic) map from other, less important invariant measures. In fact, the ‘physical meaning’ of the variational principle could be formulated as follows. Among all possible invariant measures of a map one is distinguished in the sense that it is the smoothest one along the unstable manifold.