On two different approaches to hyperbolicity definition

We consider a diffeomorphism f acting in a Banach space E that has a closed invariant set A (such that $f(A)=A$). We perform a comparative analysis of the two currently known definitions of hyperbolicity of the set A. The first is Anosov's classical definitions, stated in terms of the $Df$-invariant decomposition Eₓᵘ⊕ Eₓˢ, x∈ A, of the space E into the direct sum of the unstable subspace Eₓᵘ and stable subspace Eₓˢ. The second definition, based on works by Zelik with coauthors, is stated in terms of the uniform regularity of a certain difference operator. We show that these definitions are equivalent. On the way we establish results on the uniform boundedness and uniform continuity in x∈ A of the projections corresponding to the above decomposition of E. We also present sufficient conditions ensuring that the restriction f|A has the property of essential dependence of trajectories xₙ=fⁿ(x), n∈ N, on the initial point x∈ A, which is characteristic for chaotic dynamics. Bibliography: 35 titles.