My title - Departamento de Matemática da Universidade do Minho

16 ERGODICITY AND CONVERGENCE OF TIME MEANS 105
statement that Lebesgue almost any real number is normal in every base m ≥ 2, is due to Emile
Borel (1909).
It is not so easy to give examples of normal numbers, actually of series whose sum is a normal
number. Much more difficult is to show that a “given” number, such as π, √ 2 or e ..., is normal.
Here we quote Mark Kac: 35
“As is often the case, it is much easier to prove that an overhelming majority of objects
possess a certain property that to exhibit even one such object. The present case is
no exception. It is quite difficult to exhibit a ‘normal’ number! The simplest example
is the number (written in decimal notation) x = .1234567891011 . . . where after the
decimal point we write the positive integers in succession. The proof that this number
is normal is by no means trivial.”
Continued fractions and Gauss map.
to be done
16.4 Unique ergodicity
Unique ergodicity. A homeomorphism f : X → X of a compact metric space (X, d) is uniquely
ergodic if it admits one, and only one, invariant Borel probability measure µ. The above discussion
implies that this unique invariant measure is ergodic.
This notion is the probabilistic counterpart of minimality, and indeed both minimality and
unique ergodicity are often observed simultaneously (this means that, although equivalence of the
two is false, it is not easy to think at a couterexample!). Observe that we defined unique ergodicity
in the context of continuous transformations. The reason is that this notion is interesting due to
the following
Oxtoby’s theorem. Let f : X → X be a homeomorphism of a compact metric space X. The
following statements are equivalent:
i) f is a uniquely ergodic,
ii) there exists an invariant Borel probability measure µ such that, for any continuous observable
ϕ, the time averages ϕ (x) exist and are equal to ∫ ϕdµ for any initial condition x ∈ X,
X
iii) there exists an invariant Borel probability measure µ such that, for any continuous observable
ϕ, the convergence
1
n∑
ϕ ( f k (x) ) ∫
→ ϕdµ
n + 1
k=0
as n → ∞ holds and is uniform in x ∈ X.
X
Kronecker-Weyl equidistribution theorem.
ergodic.
An irrational rotation of the circle is uniquely
Proof. Indeed, let +α : x + Z ↦→ x + α + Z be an irrational rotation. We must check that time
means of continuous observables ϕ converge uniformly to the average ∫ ϕdl, where l is Lebesgue
probability measure on the circle. Since, according to Weierstrass theorem, trigonometric polinomials
are dense in the space of continuous functions of the circle, it suffices to check that the above
holds for any of the functions x ↦→ ϕ k (x) = e i2πkx with k ∈ Z. A computation gives, for k ≠ 0,
1
n∑
ϕ j
((+α) (x)) ∣ ∣ ∣∣∣∣ j =
1
n∑ ∣∣∣∣∣
∣n + 1
e i2πkjα ≤ 2
∣n + 1
n + 1 · 1
|1 − e i2πkα | → 0
j=0
j=0
uniformly in x, while the time averages of ϕ 0 are constant and equal to 1. Hence, the time means
of each ϕ k converge uniformly to their space means as times goes to infinity, and we are done. The
35 Mark Kac, Statistical independence in probability, analysis, and number theory, Carus Math. Monographs, 12,
New York 1959 (pag. 18).