Minimal non-hyperbolicity and index-completeness
Let \(M\) be a compact \(d\)-dimensional \(C^\infty\)-manifold without boundary and \(\text{Diff}^1(M)\) the space of \(C^1\)-diffeomorphisms from \(M\) into itself endowed with the \(C^1\)-topology. Let \(p\in M\) be a hyperbolic periodic point and denote by \(H(p,f)\) (the homoclinic class of \(p\)) defined as the closure of all transverse intersections of the stable and unstable manifolds of the orbit of \(p\). Also the index of \(p\), denoted by \(\text{ind}(p)\) is the dimension of its stable manifold. For a \(C^1\) generic \(f\) (which means, for an \(f\) belonging to a \(C^1\)-residual set of \(\text{Diff}^1(M)\)) it was proved by \textit{F. Abdenur, Ch. Bonatti, S. Crovisier, L J. Díaz} and \textit{L. Wen} [Ergodic Theory Dyn. Syst. 27, No. 1, 1--22 (2007; Zbl 1144.37007)] that every homoclinic class \(H(p,f)\) containing hyperbolic periodic points of indices \(\alpha\) and \(\beta\) contains also a dense set of hyperbolic periodic points of index \(\gamma\) for all \(\alpha< \gamma< \beta\). Such result means that for a \(C^1\)-generic \(f\), every homoclinic class is index-complete in the sense that the indexes of periodic points contained in \(H(p,f)\) form an interval in \(\mathbb N\). Since whether \(H(f,p)\) is an hyperbolic set, it would contain points with a unique index, therefore the former result is referred to a non-hyperbolic setting. A preliminary result was given by \textit{C. Bonatti, L. J. Díaz, E. R. Pujals} and \textit{J. Rocha} [Astérisque 286, 187--222 (2003; Zbl 1056.37024)], in the sense that if \(f\) is a \(C^1\) generic diffeomorphism, then every robust transitive set (that is, an isolated invariant set remaining transitive by small perturbations) is index-complete. Another related problem is to consider the set of indexes of hyperbolic periodic orbits that approach to, but not necessarily contained in a given chain transitive set \(\Lambda\). One chain transitive set is a compact and invariant set contained in \(M\) for which for every \(\varepsilon > 0\) and any \(x,y \in \lambda\), there is an \(\varepsilon\)-chain connecting \(x\) and \(y\). It is also known that for a \(C^1\)-generic \(f\) and any chain transitive set \(\Lambda \subset M\) of \(f\), there exist hyperbolic periodic orbits of \(f\) that are arbitrarily close to \(\Lambda\) in the Hausdorff metric. Since this result it is evident the set of indices of hyperbolic periodic orbits approaching \(\Lambda\) in the Hausdorff metric is not empty. But, do they form an interval?. When it is the case we have the notion of limit -index-complete. The authors state the problem raised by Abdenur et al. [loc. cit.]: For \(C^{1}\)-generic diffeomorphisms, is every chain transitive set limit-index-complete?. Currently the question is open even for chain recurrent classes (equivalent classes by a equivalence relation on the set of chain recurrent points of \(f\)), which are some type of chain transitive sets. The equivalent classes are mutually disjoint compact invariant sets that can not be decompose into disjoint compact invariant sets. In general, they are infinite in number and has been proved that, \(C^1\) generically, a chain recurrent class containing a hyperbolic periodic point \(p\) with the homoclinic class \(H(p,f)\) previously introduced. Using such result, for \(C^1\) generic diffeomorphisms, chain recurrent classes are decomposed into two categories, the homoclinic classes and the aperiodic (not containing periodic points). Since the problem on indexes is essentially for the non-hyperbolic case, in the paper under review, the authors consider the case of minimally non-hyperbolic sets (a compact invariant set \(\Lambda\) that is non-hyperbolic, but any compact invariant proper subset of it is hyperbolic). Such non-hyperbolic sets are divided into two types, transitive and non-transitive. The first class is characterized as a non-transverse heteroclinic connection of two hyperbolic sets and it is easy to handle. The transitive case is far of being characterized. At this point, recall that \(x \in M\) is a homoclinic tangency of a hyperbolic periodic orbit \(P\) if \(x \in W^s(P) \bigcap W^u(P)\) and if the intersection of \(W^s(P)\) and \(W^u(P)\) at \(x\) is not transverse. Let \[ HT(M) = \{f \in \text{Diff}^{1}(M): f \text{ has a homoclinic tangency} \} \] In the paper, the authors prove the following results: {\parindent=5mm \begin{itemize}\item[1)] There is a \(C^1\) residual subset \({\mathfrak G}\) in \(\text{Diff}^1(M) \backslash \overline{HT(M)}\) such that every transitive minimally non-hyperbolic set of every \(f \in{\mathfrak G}\) is limit-index-complete. \item[2)] There is a \(C^1\) residual subset \({\mathfrak G}\) in \(\text{Diff}^1(M) \backslash \overline{HT(M)}\) such that a transitive minimally non-hyperbolic set of every \(f \in{\mathfrak G}\) has a finely partially hyperbolic splitting. \end{itemize}} Proofs of such results are made using the type of perturbation theory used when dealing with generical results. Mainly trying to make estimations at gaps of the index sets produced in all situations.
scientific article; zbMATH DE number 5648564