Auteurism

On the essential hyperbolicity of sectional-Anosov flows

“…Let {A 1 , • • • , A r } and {σ 1 , • • • , σ l } be the collection of saddle-type hyperbolic attractors and singularities of X whose basins form a dense subset of M . As is wellknown (p.9 in [27]) for every i = 1, • • • , r there is β i > 0 such that X restricted to B βi (A i ) is sensitive to the initial conditions. Let δ i be the corresponding sensitivity constant for i = 1, • • • , r.…”

“…We say that a vector field X of a manifold M is sensitive to the initial conditions if there is δ > 0 such that for every x ∈ M and every neighborhood U of x there are y ∈ U and t ≥ 0 such that d(X t (x), X t (y)) > δ. The number δ will be referred to as a sensitivity constant of X This is a basic property of chaotic systems widely studied in the literature [4], [13], [18], [27], [28], [29], [30]. The following corollary asserts that this property holds for all sectional-Anosov flows on compact 3-manifolds.…”

“…A smooth vector field on a differentiable manifold is called essentially hyperbolic if it exhibits a finite collection of hyperbolic attractors whose basins form an open and dense subset of the manifold [2], [12]. Basic examples are the Axiom A ones (by the spectral decomposition theorem [16], [27]), including the Anosov flows, but not the geometric Lorenz attractor [1], [14]. On the other hand, there is a class of systems, the sectional-Anosov flows [21], whose representative examples are the Anosov flows, the geometric Lorenz attractors, the saddle-type hyperbolic attracting sets, the multidimensional Lorenz attractors [10] and the examples in [22], [23].…”