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18) k=0 This result holds also for the absolute value. While the distance traveled by the particle along the x-direction is alway xn = nα, the distance traveled by the particle along y-direction depend on the initial condition (x, ε) and seems as a random walk. 19) 24 The mean value of Sn over the initial conditions is zero and the problem is to estimate the asymptotic behavior of the variance of Sn . We shall come back to this question in the next section. Second example: G = S 1 A particle moves on an infinite plane among periodically distributed obstacles with spatial period equal to 1 along both (q1 , q2 )-directions.

Space with a distance d. 1 1 Given > 0, a set Y ⊆ X is -separated iff for any different x, y ∈ Y one has d(x, y) ≥ . 2 The number C (X, d) = C := max{|Y |, Y is an -separated set}, where | · | denotes the cardinality of a set, is called the -complexity of X. 3 An -separated set Y is optimal iff |Y | = C . Let us show the following natural inequality. 1 Given D1 , D2 ⊆ X and ε > 0 one has Cε (D1 ∪ D2 ) ≤ Cε (D1 ) + Cε (D2 ). 1 Let Y ⊆ D1 ∪ D2 be an optimal ε-separated set in D1 ∪ D2 . Then Yi = Y ∩ Di is an ε-separated set in Di and |Y | ≤ |Y1 | + |Y2 | ≤ Cε (D1 ) + Cε (D2 ).

1) Notes on Spectral Theory, Mixing and Transport. 17 In order to stress on the dependence of U with respect to T, we should use the notation UT , but we shall omit it and use it only when necessary. The spectral properties of this operator are called the spectral properties of the DS. The first important spectral property of a DS is a characterization of its ergodicity. e. T A = A, eventually modulo a subset of zero measure) has either zero measure or full measure. 2 (X, T, µ) is ergodic if and only if the only eigenfunction of U for the eigenvalue 1 is the constant function.