My title - Departamento de Matemática da Universidade do Minho

4
NÚMEROS E DINÂMICA 24
Exercícios.
• Describe periodic and pre-preriodic points of σ : Σ + → Σ + . Show that they are dense in Σ + .
• Consider the alphabet A = {0, 1, 2, . . . , 9}. Define a map h : A N → [0, 1] as
(x 1 , x 2 , x 3 , . . . ) ↦→ 0.x 1 x 2 x 3 . . . .
Show that h is onto, and describe the map sending h(x) into h(σ(x)).
4.3 Rotações do círculo/toro
Arithmetic progressions and exponential sums.
The dynamics of an arithmetic progression
a a + α a + 2α a + 3α . . . a + nα . . . ,
obtained from the initial condition x 0 = a using the recursion x n+1 = x n + α, is quite trivial. All
trajectories x n = a + nα diverge, provided α ≠ 0.
Nevertheless, if we average the observable ϕ(x) = e 2πix , something interesting appears: apart
from a constant factor e 2πia and the normalization 1/n, the Birkhoff averages read
n−1
∑
S α (n) = e 2πikα .
n=0
Sums as E(N) = ∑ n
k=1 e2πix k
are called exponential sums, and contain “spectral information”
about the distribution of the numbers x k modulo 1. Triangular inequality gives the trivial bound
|E(N)| ≤ N, i.e. E(N) = O(N). If the different exponentials e 2πix k
were “uncorrelated”, as
successive positions of a random walk in the plane, we should expect E(N) = O( √ N). This, of
course, does not happen with “deterministic” generic sequences. The best we can hope is some
bound as E(N) = o(N) (which, in our case, would mean that the Birkhoff averages ϕ n → 0).
Exercícios.
• Show that the sum of the first n terms of an arithmetic progression x k = a + kα is
n−1
∑
x k = n 2 (x 0 + x n−1 ) = na +
k=0
n(n − 1)
α
2
• Observe that, for q ∈ N, the complex number z = e 2πi/q is a non-trivial q-th root of unity.
Hence,
1 + z + z 2 + · · · + z q−1 = 0 .
Deduce that if α = p/q ∈ Q with p ∈ Z, then
∑q−1
S α (q) = e 2πikp/q = 1
so that the exponential sum S p/q (n) is periodic, and in particular is o(N).
n=0
O círculo. O círculo (o esfera de dimensão 1) é o espaço quociente S 1 = R/Z do grupo comutativo
R pelo subgrupo Z, munido da topologia quociente herdada da topologia euclidiana da reta.
A métrica euclidiana da reta induz uma métrica invariante d no círculo, definida por
d (x + Z, x ′ + Z) = min |y − y ′ |
y∈π −1 {x+Z}, y ′ ∈π −1 {x ′ +Z}
onde π : R → R/Z denota a projeção x ↦→ x + Z.
= min
n∈Z |x − x′ + n|