Abstract:
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 $E_x^{\mathrm u}\oplus E_x^{\mathrm s}$, $x\in A$, of the space $E$ into the direct sum of the unstable subspace $E_x^{\mathrm u}$ and stable subspace $E_x^{\mathrm s}$. 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 $x\in A$ 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 $x_n=f^n(x)$, $n\in \mathbb{N}$, on the initial point $x\in A$, which is characteristic for chaotic dynamics.
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