V. Ergodic theory of differentiable dynamical systems: Axiom A systems

As we have just seen, it is important for the physics of chaos to understand the structure of invariant probability measures of differentiable time evolutions. This topic also has great mathematical interest; it is a rich but difficult area of research called the ergodic theory of differentiable dynamical systems.

The first systems that were studied from this viewpoint are Smale's Axiom A systems [5]. We need not specify here what these systems are. Let it just be said that they are accessible to study because of the existence of Markov partitions. Markov partitions, introduced by Sinai and Bowen, provide a coding of points of a manifold by a sequence of symbols (from a finite alphabet):

x®(..., x_-1, x₀, x₁, ...)

and the time evolution xrightarrow fx (in the discrete time case) corresponds to the shift operation:

(...,x_-1, x₀, x₁, ...)®(..., x₀, x₁, x₂, ...).

(The situation is slightly more complicated in the continuous time case.) If we interpret the x_i as spin values, the sequence

(..., x_-1, x₀, x₁, ...)

may be viewed as a configuration of a one-dimensional classical spin system. Time invariant probability measures for an Axiom A dynamical system correspond thus to translationally invariant states of a one-dimensional spin system, in the sense of statistical mechanics (time invariance has been replaced by space invariance). As it happens, our knowledge of the Gibbs states of equilibrium statistical mechanics is just what was needed to understand the time-invariant probability measures for Axiom A dynamical systems. One may view this as a little scientific miracle. Or one may reflect that equilibrium statistical mechanics is the only area of science where a really deep analysis of invariant measures had been made; it was therefore natural that this analysis would yield useful results here as well.

Specifically, the result obtained by Sinai [14], Ruelle [15], Bowen and Ruelle [16] is as follows. If (f^t}) is an Axiom A time evolution on a (compact) manifold M, then for almost every xÎM with respect to a smooth measure on M ("Lebesgue measure"), the time averages of observables A (continuous functions A:M

R), i.e.,

where r is a probability measure on the attractor to which f^tx tends when t®¥ (there are finitely many choices). The measure r on an attractor (SRB measure) is a particular Gibbs measure specified in terms of the time evolution (f^t).

The ideas leading to SRB measures come therefore in part from mathematics and in part from physics. The physical ideas come, interestingly, from two completely different sources: interest for chaos on one hand provides motivation, the tools of equilibrium statistical mechanics on the other hand provide the solution. It should be noted that statistical mechanics intervenes here only at the mathematical level: there is no relation between the physics of one-dimensional lattice spin systems and the physics of chaos of Axiom A dynamical systems.