My title - Departamento de Matemática da Universidade do Minho

16 ERGODICITY AND CONVERGENCE OF TIME MEANS 102
16 Ergodicity and convergence of time means
16.1 Ergodicity
Ergodic maps. Let f : X → X be an endomorphism of the measurable space (X, E). The
invariant probability measure µ is said ergodic if any of the following equivalent conditions is
satisfied:
i) for any observable ϕ ∈ L 1 (µ), the time average
1
ϕ (x) = lim
n→∞ n + 1
n∑
ϕ ( f k (x) )
exists and is equal to the mean value ∫ ϕdµ for µ-almost any x ∈ X,
X
ii) any invariant event A ∈ E has probability µ (A) = 0 or 1, namely the invariant σ-algebra
E f is equal to the trivial σ-algebra N generated by events of zero measure,
iii) any invariant (measurable) observable ϕ is constant µ-a.e.
If this happens, one also says that f is an ergodic endomorphism of the probability space
(X, E, µ).
Condition i) is the physical meaning of ergodicity, as it says that “time averages are almost
everywhere constant and equal to space averages”. In particular, taking ϕ equal to the characteristic
function of any event A, almost any trajectory spend in A a fraction of time asymptotically
proportional to µ (A), as dreamed by Boltzmann in his ergodic hypothesis.
Condition ii) is what one usually check in order to prove ergodicity of a probability measure.
To see that i) ⇒ ii), let A be an invariant event, and ϕ its characteristic function. Invariance of A
implies that ϕ is invariant, hence that ϕ = ϕ. There follows fom i) that µ (A) = ∫ ϕdµ = ϕ (x)
X
for some x ∈ X, hence that µ (A) = 0 or 1, the only values of characteristic functions.
Conditions ii) and iii) are clearly equivalent, since any invariant event defines an invariant
function (its characteristic function), and conversly level sets of invariant functions are invariant
events.
Finally, in order to show that iii) ⇒ i), let ϕ ∈ L 1 (µ) be an integrable observable. According
to the Birkhoff-Khinchin ergodic theorem, the time average ϕ (x) exists for µ-almost any x ∈ X
and ∫ X ϕdµ = ∫ ϕdµ. Since ϕ is invariant mod 0, by iii) it is constant with probability one. This
X
implies that ϕ (x) = ∫ ϕdµ for µ-almost any x ∈ X.
X
k=0
Warning. Ergodic dynamical systems exist, and some are listed below. On the other side, to
show that a physically interesting system is ergodic turns out to be extremely difficult, and very
few examples are known. The most famous are some “billards”, systems of hard spheres inside a
billard table interacting via elastic collisions, studied by Yakov Sinai in the sixties...
Ergodic measures as extremal measures. We already saw that the space Prob f of invariant
probability measure is a convex and closed subset of the compact space Prob. Here, we observe
that ergodic measures are the “indecomposable” elements of this set.
Proposition. Ergodic invariant measures are the extremals of Prob f . Namely, an invariant
measure µ is ergodic iff it cannot be written as
µ = tµ 1 + (1 − t) µ 0
where t ∈ ]0, 1[ and µ 0 and µ 1 are distinct invariant measures.
proof. First, observe that if ν is an invariant measure which is absolutely continuous w.r.t. the
ergodic measure µ, then ν = µ. Indeed, one easily verifies that the Radon-Nykodim derivative
ρ = dν/dµ is an invariant function, and ergodicity of µ implies that it is constant and equal to one
µ-a.e. Now, let µ be an ergodic measure, and assume that µ = tµ 1 + (1 − t) µ 0 for some t ∈ ]0, 1[.
Since both µ 0 and µ 1 are absolutely continuous w.r.t. µ, they coincide with µ, hence, are not