Semi-hyperbolicity and hyperbolicity (Q946970)

This paper is devoted to show several results on semi-hyperbolicity which support the idea that semi-hyperbolicity implies hyperbolicity. One of the main results of the paper establishes that for \(C^1\) diffeomorphisms semi-hyperbolicity of an invariant set implies its hyperbolicity. Let us recall that hyperbolicity is defined upon the splitting of the dynamics on a stable and an unstable invariant manifolds in such a way that this splitting is continuous and with bounded derivatives. Thus, as it is well-known, a dynamical system possesses a highly stable behavior in a neighborhood of a hyperbolic invariant set. However, the verification of when an invariant set is hyperbolic is very hard by strict numerical computation. The definition of semi-hyperbolicity is weaker than hyperbolicity so that it is easier to be numerically verified, whereas it keeps some properties of hyperbolicity. Roughly speaking, the exact invariance and continuity of the splitting are not assumed in this definition. Some exact estimations of hyperbolicity constants are provided via the semi-hyperbolicity approach. These estimations may allow the strict numerical verification of hyperbolicity.