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By Vítor Araújo, Maria José Pacifico (auth.)

ISBN-10: 364211413X

ISBN-13: 9783642114137


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Admits a continuous DX-invariant tangent bundle decomposition TΛ M = EΛ ⊕ u X EΛ ⊕ EΛ , that is we can write the tangent space T xM as a direct sum Exs ⊕ ExX ⊕ Exu , where ExX is the subspace in Tx M generated by X(x), satisfying i • DXt (x) · Exi = EX t (x) for all t ∈ R, x ∈ Λ and i = s, X, u; 2. e. e. for all x ∈ Λ and every t ≥ 0 • EΛ m(DX t | E u ) ≥ Keλt . 3 Hyperbolic Flows 17 Fig. 3 A saddle singularity σ for bi-dimensional flow By the Invariant Manifold Theory [110] it follows that for every p ∈ Λ the sets WXss (p) = {q ∈ M : dist(Xt (q), X t (p)) −−−→ 0} t→∞ and WXuu (p) = {q ∈ M : dist(X t (q), Xt (p)) −−−−→ 0} t→−∞ Cr are invariant immersed manifolds tangent to and Epu respectively at p.

Ik } of {Λ1 , . . , Λn } such that i0 = ik and WXu (Λij ) ∩ WXs (Λij +1 ) = ∅, ∀0 ≤ j ≤ k − 1. 18 2 Preliminary Definitions and Results Fig. g. [190]). ,dim E u (xn ) of E u (xn ) and X(xn ) of E X (xn ), then these vectors converge to a basis of E s (x), E u (x) and E X (x) respectively. In particular the dimension of the subspaces in the hyperbolic splitting is constant if Λ is transitive. This shows that a uniformly hyperbolic basic set Λ cannot contain singularities, except if Λ is itself a singularity.

If we have such sensitive dependence both for the past and for the future, we say that Λ is chaotic. Note that in this language sensitive dependence on initial conditions is weaker than chaotic, future chaotic or past chaotic conditions. An easy consequence of chaotic behavior is that it prevents the existence of sources or sinks, either attracting or repelling singularities or periodic orbits, inside the invariant set Λ. Indeed, if Λ is future chaotic (for some constant r > 0) then, were it to contain some attracting periodic orbit or singularity, any point of such orbit (or singularity) would admit no point in a neighborhood whose orbit will move away in the future.

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Three-Dimensional Flows by Vítor Araújo, Maria José Pacifico (auth.)

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