2.3. Ergodk theory

For any .t06X the sequence of points {*„, Txih T?x0       } is called the discrete trajectory generated by exists for almost all x£X. Furthermore, /* is measurable, absolutely integrable and for almost all x. Also, if the measure p is finite.

x0. The relative period of time the trajectory spends in a set A is given by the average n'1 £"=, I(Txo) where I(y) is the indicator function of an arbitrarily chosen measurable set AG B; I(y) = 1 if y e A and J(y) * 0 if yf£ A. Under certain conditions on T the limit as n -»°° of the sum for the period of time spent in A exists and is denoted by ja(A). This is the ergodic theorem [14] of Birkhoff: Let / : X-» R be a measurable, absolutely integrable function (/e^f1). Then with T measure-preserving hereafter.

 

 

A set is invariant under the map T if T 'A = A and a function, is invariant under T if g(.v) = g(Tx) for all

In order to characterize the behaviour of systems with erratic or chaotic trajectories we need a notion of behaviour that does not contain stable subcomponents which remain invariant under transformation in time. T is said to be ergodic if and only if every invariant set (or invariant function) is almost everywhere equal to e constant. So, if A is invariant then either ju (A) - 0 of /x(X - A) = 0. An ergodic system cannot be decomposed into invariant portions.

If T is ergodic then the ergodic theorem shows that for fEJ£l and almost every v 6 X.

 

 

The effect of ergodicity on the ergodic theorem is to make /* invariant and equal to jYCO M almost everywhere, (a.e). When / is the indicator function of a se' A then

The form (2.17) says that afte; a large number of applications of the mapping T moving A' forwards in time one approaches a state of statistical independence of A on the average. If A is invariant, A = TA,

 

and only sets A with /x(A) = 0 or /x(A) = 1 are admitted in accord with the discussion above. The mapping

 

then setting A' = A in (2.17) we have

is not ergodic when A is rational (= k/m). The set of jco^0 together with its iterates x0 + j/m, where

j = 0     'w-1, is invariant under T but does not have measure zero or unity. Each orbit of T

contains m noints. On the other hand, if A is irrational then T is ergodic with respect to the Lebesgue measure and its orbits are dense on \'he circle.

Equivalents, a transformation T is said to be ergodic if for any two sets A, A € B

 

If A and A' had been independent we would just have had

 

Ergodicity is not a very strong statistical property; it just indicates that a measurable subset of a system is visited by a trajectory with a frequency proportional to its measure. Ergodic systems need not have sensitive dependence on initial conditions like example (1.1). A much stronger notion is that of trajectory mixing [14].

A stationary process T is strongly mixing if for any two sets A, A' 6 B

 

 

In this case the limiting relation holds without averaging and mixing is therefore a stronger property than ergodicity; the map (2.20) with A irrational is not mixing. It is saying that if we hold A fixed and let A' evolve under the map T„ then A' will spread out and 'mix' through the entire phase space of possibilities, eventually intersecting the fixed set A (both A and A' are assumed to have finite measuie). As the mixing becomes more thorough, any part of A will locally resemble the whole space and memory of the initial conditions will eventually be lost. Therefore, the measure of the intersection u (77V n A) will lie in the same ratio to fi(A) as fi(TA') does to the measure of the whole space, which s one. However, since the fiow is measure-preserving fi(TA')= fi(A') and so mixing ensures

Mbringt ensures that non-equilibrium distributions converge on n as n-**. Ergodicity alone is insufficient to guarantee this.

A graphic illustration of these properties, due originally to Gibbs [15] and Halmos [14], envisages a fluid mixture of 10% rum and 90% cola. H one now considers the proportion of rum in any fluid volume then an "ergodic cocktail' ensures that this proportion is 10% on the (time) average. A 'weakly mixed cocktail1 ensures that the proportion is eventually 10% except on occasional, infrequent moments, whilst a 'strongly mixed cocktail' has the property that after some time the proportion of rum is always i0%. Mixing systems tent1 to an equilibrium as

A still more powerful random property which we shall use is the Bernoulli shift. This describes systems which are completely random. Suppose the phase space is partitioned into n sections, each labeled by some kt and having a probability pt of arising during the evolution. If the system evolves at discrete intervals of 'time' then the dynamics are coded by a random sequence of k,. The simplest system would be tossing a coin with two possible outcomes ku k2 and p, = p2 = 0.5. The motion is described by a randc a two-symbol sequence like

 

 

More formally, if Z is a set containing k elements and the j'th element has probability measure p, let Z„ -oo < i < oo, be copies of Z and let X be the product of Zf with the product measure. Each point in X is now a doubly infinite sequence of points. A Bernoulli shift T now acts as a transformation on a set of symbols {yd by shifting {yj}-»{yi} where y,' = yi+t. This is equivalent to spinning a roulette wheel containing k slots of width pt. The set X contains the possible oi-tcomes of an infi nite sequence of spins. We obtain the probability of any finite sequence of outcomes occurring whilst the action of T corresponds to examining the result one spin later.