By Hakan Eliasson, Sergei Kuksin, Stefano Marmi, Jean-Christophe Yoccoz, Stefano Marmi, Jean-Christophe Yoccoz

Authors illustrate the newest advancements of the dynamical structures thought either in finite and limitless dimensions. Softcover.

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**Extra resources for Dynamical systems and small divisors: Lectures CIME school**

**Example text**

H. Eliasson Theorem 12. Let D ∈ N F(α, . . 1-2), and assume that D is truncated at distance ν from the diagonal. Let F be a covariant matrix, smooth on P and | Fab |C k ≤ εe−α|b−a| γ k ∀k ≥ 0. Then there exists a constant C – C depends only on dim L, κ, τ, s, α, β, γ, λ, µ, ν, ρ, σ, #P – such if ε ≤ C then there exists a matrix U such that U (x)−1 (D(x) + F (x))U (x) = D∞ (x), ∀x ∈ X, and D∞ (x) is a norm limit of normal form matrices Dj (x). e. x. The limit limj→∞ Ej (x) = E∞ (x) is uniform and satisﬁes, for all y ∈ / 2πZ, Lebesgue{x : E∞ (x + y) − E∞ (x) = 0} = 0.

Eliasson For v i (0) in Λj (0), let 1 Pˇj (D(x), x)v i (0), i ˇ Pj (E (0), 0) v i (x) = x ∈ U. This gives a basis for Λj (x) for x near 0 – how near? We have m m | v i |C k ≤ ( )m−1 (const)m ((const )m γ)k ∀k ≥ 0. r r Hence, for x ∈ W , v i (x) will be close to v i (0) and we get a basis deﬁned in W . 4 gives an ON-basis for each Λj (x) on W which fulﬁlls the estimate. 5, using that m2 ≥ (m + 1) m 2. In order to estimate the inverse of Q, consider the D∗ -invariant decomposition k Cm = ˜ j (x), Λ i=1 such that the eigenvalues of D∗ (x)Λ˜ j (x) are complex conjugates of the eigen˜ j satisﬁes the same estimate as Λj and Λ ˜j = values of D(x)Λj (x) .

E)&T (σ, s) have almost-multiplicity ≤ µµ if, for all x and for all t ≤ ρ, the inequality | E(x + y) − E(x) |> t( is fulﬁlled outside at most µ µ µ ) µ many intervals of length less than 1 t 1 2(4βµ ) s ( ) sµ2 . σ Lemma 9. Let the eigenvalues of D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω, P)&T (σ, s) have almost-multiplicity ≤ µ µ and assume ρ ≤ ( κ 1 2τ sµ2 ) σ. 3) Then D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω, P)&C(λ , µ , ν ) for any ν = 1 µ κ 1 σ τ sµ ( )τ ( ) 2 , 2µ 4βµ ρ λ = µ (ν + 2λ). 4) Proof. For each x, | E(x + y) − E(x) |> ρ( is fulﬁlled outside at most µ µ µ ) µ many intervals of length less than 1 1 ρ L = 2(4βµ ) s ( ) sµ2 .