4. The last step is to calculate ρ∗ by normalisation: It is ρ∗ = cρτ
(6.49)
max
with 1=
2 X i=1
∗i
ρ · diam(Ii ) = c
2 X i=1
ρiτmax · diam(Ii ) = c
3 4
from which follows c = 4/3. Our final result thus reads 4 ∗ 3 ρ = . 2
(6.50)
(6.51)
3
It is not a bad idea to cross-check whether our solution indeed fulfills the Frobenius-Perron matrix equation Eq. (6.38), 4 4 2 2 1 1 2 +3 3 3 3 = = . X (6.52) 2 2 2 2 1 0 3 3 3
7
Measure-theoretic description of dynamics
So far we have analyzed the dynamics of statistical ensembles by using probability densities, which represents rather a physicist’s approach towards a probabilistic theory of dynamical systems. Mathematicians prefer to deal with measures, which are often more well-behaved than probability densities. In this chapter we introduce these quantities, learn about basic properties of them and eventually briefly elaborate on some ergodic properties of dynamical systems. The first section particularly draws on Ref. [Las94], the second one on Refs. [Dor99, Arn68]. If you have problems to associate any meaning with measures you may wish to take a look into Ref. [All97].
7.1
Probability measures
Let ρ(x), x ∈ I, be a probability density, see Definition 29. If ρ(x) exists and is integrable on a subinterval A ⊆ I then Z µ(A) := dxρ(x) (7.1) A
is the probability of finding a point in A. µ(A) is called a measure of A, in the sense that we ‘assign a positive number to the set A’. Let us demonstrate this idea for a simple example: