My title - Departamento de Matemática da Universidade do Minho

10 STATISTICAL DESCRIPTION OF ORBITS 71
As for the ”physical” meaning of this measure, you may imagine that X represents the possible
outcomes when tossing a coin with z sides, and p k is the probability of obtaining the k-th side.
Then points in Σ + represent the outcomes of an infinite sequence of tossings, and the very definition
of µ says that each trial is described by the probability p, and each trial is ”independent” from
any finite collection of different trials.
It is not surprising that µ is indeed an invariant probability measure. This comes from the fact
that the inverse image σ −1 (A) of any A ∈ A is the disjoint union of z elements B 1 , B 2 , ..., B z of
the algebra (obtained from A chosing the first letter in z different ways) with measures µ (B k ) =
p k · µ (A), so that
µ ( σ −1 (A) ) z∑
= p k · µ (A) = µ (A)
k=1
Absolutely continuous invariant measures for maps and flows. Let U be a domain in
some euclidean R n , and let vol denote the Lebesgue measure on U, given locally as dvol = dx =
dx 1 dx 2 ...dx n . A local diffeomorphism f : U → U of class C 1 preserves the measure vol iff
∑
x∈f −1 {x ′ }
1
|det f ′ (x)| = 1
for any point x ′ ∈ U, as one can check using the change of coordinates formula. Also interesting is
to see wheather f preserves an absolutely continuous measure µ = ρvol, and this happens iff the
”density” ρ satisfies the equation
∑
x∈f −1 {x ′ }
ρ (x)
|det f ′ (x)| = ρ (x′ )
for any point x ′ ∈ U.
Now, let φ be the flow of a vector field ξ = ∑ n
considering the Jacobian of the diffeomorphisms φ t . Since
k=1 ξ k ∂
∂x k
on U. The above obviously applies,
det φ ′ t =
∫ t
0
divξ ◦ φ s ds
we get the result that Lebesgue measure vol is invariant under the flow of ξ iff
divξ =
n∑
k=1
∂ξ k
∂x k
= 0
In general, the absolutely continuous measure µ = ρvol is invariant under the flow of ξ iff its density
satisfies div (ρξ) = 0.
Hamiltonian flows. Consider a symplectic manifold (X, ω). Liouville measure dvol = ω n is
invariant under the Hamiltonian flow of any Hamiltonian function H. If X has finite volume, it
can be normalized to give an invariant probability measure.
Geodesic flows. Consider a geodesic flow on the unit tangent bundle π : SM → M of the
Riemannian manifold (M, g). Let dvol = √ gdx denote the Riemannian volume form on M, and
let dσ m denotes the Lebesgue probability measure on the sphere S m M = π −1 {m}. The Liouville
measure µ, defined locally as dvol (m) × dσ m , is invariant under the geodesic flow.