2.2. Invariant measures

Suppose we have constructed a return mapping for the system (2.1) and it is one-dimensional

 

 

If the assuiated phase flow behaves in a chaotic manner the sequence of numbers generated b\ iterations of (2.2) which code the phase orbit will have an apparently random behaviour. They will not in general converge towards some limit point which would correspond to the presence of a limit c\ele in the phase flow. The relevant question to ask of such 'random' sequences is: Is there some station.» \ behaviour after a large number of iterates, n->®, and is the frequency of occurrence of a particular number in the infinite sequence {jc0, T{x0). T™(x0)..., Tir\.r„),...} described by a stationan (i e independent of n) probability distribution (measure)? For example, suppose in (2.2) that.

so for 8x+ = 0

and let fi(y) 8y be the relative number of iterates {*,} that He in the interval {y,y + 3y}. If the probability density fi exists then it must be preserved under transformation by T because this action can be viewed as merely altering the initial state Xo to a new one, T(xo). Any stationary statistical properties of T must be independent of this change. The graph of y{r) has two branches and so two disjoint x intervals map into a single y interval

 

 

 

 

 

Taking differentials of (2,4) and substituting in (2.5) we see that fi(y), if it exists, must satisfy tue

 

where

 

There exists a simple linear solution for f(y) so

Since ,u(y) 2=0 and (i is normalized to unity, ax must vanish and

functional equation

 

 

 

 

 

This is the stationary asymptotic probability distribution for the sequence of iterates {*<} generated by the non-linear difference equation (2.3). There may, in principle, be other solutions to (2.6), (2.7) besides (2.8); in some cases there may exist no solutions for the corresponding fi or ncne that can be normalized [14].

This simple example illustrates the determination of invariant measures for discrete non-linear transformations More generally, suppose (X, B, /i) is a probability space of outcomes with a a--algebrat B and an addiuve probability measure fi. If T is a measurable transformation from X into itself then for any A 6 B, we have T !A £ B where T 1 A = T*j/ E A}. The map T is measure-preserving if T is measurable and for every measurable set AG B. By a measure-preserving transformation we therefore mean a map which assigns to each point in the space another point so that each measurable set is transformed onto another set of the same measure. For T given by (2.3) the form of (2.10) is just (2.5).