My title - Departamento de Matemática da Universidade do Minho

16 ERGODICITY AND CONVERGENCE OF TIME MEANS 106
theorem owes its name to the fact that
1
n + 1
n∑
∫
ϕ (x + jα) →
j=0
ϕdl
uniformly for any continuous function ϕ on the circle, and this is interpreted as saying that the
sequence of points {x, x + α, x + 2α, x + 3α, ...} is “equidistributed” w.r.t. Lebesgue measure.
Linear flows on tori. Now, consider the torus X = R n /Z n of dimension n ≥ 2, and the linear
flow φ t : x + Z n ↦→ x + tα + Z n defined by the differential equation
ẋ = α
where α ∈ R n . The “frequency vector” α = (α 1 , α 2 , ..., α n ) is said non resonant if the scalar
product 〈α, k〉 = ∑ n
j=1 α jk j ≠ 0 for any k ∈ Z n \ {0}. As above, one can approximate any
continuous function on the torus with trigonometric functions. One then checks that
∫
1 T
(
)
e i2π〈k,x+tα〉 dt = ei2π〈k,x〉
e i2πT 〈k,x〉 − 1 → 0
T 0
iT 〈k, x〉
as T → ∞, for any k ∈ Z n \ {0}, while the time mean of the observable 1 is constant and equal
to one. There follows that a non resonant linear flow on the torus is uniquely ergodic w.r.t. to
Lebesgue measure.
Dyadic adding machine.
to be done