144 problems of the Austrian-Polish Mathematics Competition, by Kuczma M.E. (ed.)

By Kuczma M.E. (ed.)

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The symplectic maps f : M → M and F/K : K → K are conjugated by π ˜ . For every point ρ ∈ K, Tρ K is a linear symplectic subspace of the symplectic linear space Tρ (T ∗ M ) and therefore admits a unique symplectic ⊥ orthogonal (Tρ K) . 3) is invariant under the map F because the trapped set K is invariant and because F preserves the symplectic form Ω. In the next proposition, we introduce convenient local coordinates, called normal coordinates or Darboux coordinates. We will use them later in the proof.

Then the coordinates Φ (and the local trivialization) constructed in the proof have the required properties. The differential of the Anosov map Dfx : Tx M → Tf (x) M splits according to the invariant decomposition Tx M = Eu (x) ⊕ Es (x) as Dfx = ax ⊕ bx with ax := Df |Eu (x) : Eu (x) → Eu (f (x)) and bx := Df |Es (x) : Es (x) → Es (f (x)) . But since Eu (x) and Es (x) are Lagrangian subspaces, ω provides an isomorphism(2) ∗ : Es (x) → Eu (x) defined by the relation ( (S))(U ) = ω (S, U ) for S ∈ Es (x) and U ∈ Eu (x).

If a -dependent family of functions u = (u ) >0 is micro-localized at point x ∈ Rn as → 0 and its -Fourier transform is micro-localized at point ξ ∈ Tx∗ Rn , which means that these functions decay fast outside these points as → 0, then the operator Fˆ transforms these functions u to functions Fˆ u micro-localized in another point (x , ξ ) = F (x, ξ) ∈ T ∗ Rn where F is the associated canonical map. , globally, the phase space is not the symplectic space (M, ω) but its cotangent space T ∗ M . 5.

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