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 Eₓ^ u Eₓ^ s, x A, of the space E into the direct sum of the unstable subspace Eₓ^ u and stable subspace Eₓ^ s. 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.
Glyzin et al. (Thu,) studied this question.