By D. M. Rowe, C.M. Bhandari

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Also any finite collection of hyperbolic critical elements is a hyperbolic set. We refer to these sets as trivial hyperbolic sets. The first examples of a non-trivial (different from a singularity or a periodic orbit) hyperbolic basic set (on the whole manifold) was the geodesic flow on any Riemannian manifold with negative curvature, studied by Anosov [15], whose name is attached to this type of systems today, and the Smale Horseshoe, presented in [252] in the setting of diffeomorphisms. We use a global construction of a (linear) Anosov diffeomorphism (hyperbolic with dense orbit) on the 2-torus and then consider its suspension on the solid (3-)torus to obtain an example of a transitive Axiom A flow.

5. The consequences of the sensitiveness to initial conditions on a (albeit simplified) model of the atmosphere were far-reaching: assuming that the weather behaves according to this model, then long-range weather forecasting is impossible. Indeed the unavoidable errors in determining the present state of the weather system are mag- 16 2 Preliminary Definitions and Results nified as time goes by casting off any reliability of the values obtained by numerical integration within a small time period.

Let x ∈ Λ and a neighborhood U of x be given. The denseness of periodic orbits in Λ ensures we can find p ∈ Per(X)Λ such that p ∈ U ∩ B(x, r). Let τ > 0 be the period of OX (p). By the previous claim we can also find a periodic orbit OX (q) ⊂ Per(X) ∩ Λ, which we can assume is distinct from OX (p) and with smaller period, such that dist(OX (q), x) > 4r. We now define τ V := X −s B(Xs (q), r) . s=0 Since OX (q) ⊂ V this set is a non-empty neighborhood of OX (q): we just have to use the continuity of the flow X t .

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