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

By Vítor Araújo, Maria José Pacifico (auth.)

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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. Indeed, if σ ∈ Λ is a singularity then it is hyperbolic but the dimension of the central sub-bundle is zero since the flow is zero at σ . Therefore the dimensions of either the stable or the unstable direction at σ and those of a transitive regular orbit in Λ do not match. In other words an invariant subset Λ containing a singularity accumulated by regular orbits cannot be uniformly hyperbolic.

This result provides sensitive dependence on initial conditions for singularhyperbolic attractors, and shows that this class of attractors are both past and future chaotic, once we have shown that these attractors have a dense subset of periodic orbits, in Chap. 6. Proof Let Λ have a dense regular orbit and a dense subset of periodic orbits. Claim There exists δ0 > 0 such that for each x ∈ Λ we can find p ∈ Per(X) ∩ Λ such that dist(OX (p), x) > δ0 /2. Indeed there are at least two distinct periodic orbits OX (q1 ), OX (q2 ) in Λ and we can set δ0 = dist(OX (q1 ), OX (q2 )) > 0.

We say that a vector field X ∈ Xr (M), r ≥ 1, is C r -structurally stable if there exists a neighborhood V of X in Xr (M) such that every Y ∈ V is topologically equivalent to X. Roughly speaking, a vector field is structurally stable if its qualitative features are robust under small perturbations. 1 One-Dimensional Flows The only connected one-dimensional compact boundaryless manifold M is the circle S1 , which we represent by R/Z or by the unit interval I = [0, 1] with its endpoints identified 0 ∼ 1.

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