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Robust heterodimensional cycles and \(C^1\)-generic dynamics. (English) Zbl 1156.37004

A diffeomorphism \(f\) has a heterodimensional cycle if there exist (transitive) hyperbolic sets \(\Lambda\) and \(\Sigma\) with different indices (dimension of the unstable bundle) such that the unstable manifold of \(\Lambda\) meets the stable one of \(\Sigma\) and vice versa. This cycle is said to have a co-index 1 if \(\text{index}( \Lambda) =\text{index}( \Sigma) \pm1.\) The cycle is called robust if, for every \(g\) close to \(f,\) the continuations of \(\Lambda\) and \(\Sigma\) for \(g\) have heterodimensional cycle. One of the goals of the paper is to demonstrate that blenders (hyperbolic sets whose embeddings in the ambient manifolds possess certain geometric properties) and robust heterodimensional cycles appear naturally in the unfolding of heterodimensional cycles associated with two saddles.
The main result states that, for a \(C^{1}\)-diffeomorphism \(f\) with a co-index 1 cycle associated with a pair of saddles, there exist diffeomorphisms arbitrarily \(C^{1}\)-close to \(f\) having robust (heterodimensional) co-index 1 cycles. Therefore, every diffeomorphism \(f\) defined on a 3-manifold with a heterodimensional cycle associated with a pair of saddles belongs to the \(C^{1}\)-closure of the set of diffeomorphisms possessing \(C^{1}\)-robust heterodimensional cycles. Several interesting consequences of the main result are discussed. The first characterizes hyperbolicity of tame diffeomorphisms (diffeomorphisms whose chain recurrence classes are robustly isolated) in terms of existence of robust cycles. It asserts that there exists an open and dense subset \(\mathcal{O}\) of the set \(\mathcal{T}\) of tame diffeomorphisms such that every \(f\in\mathcal{O}\) is either hyperbolic (Axiom A and the no-cycles condition) or it has a \(C^{1} \)-robust heterodimensional cycle.
The second result is concerned with generation of co-index 1 cycles by arbitrarily small perturbations. Namely, there exists a residual subset \(\mathcal{R}\) of \(\text{Diff}^{1}(M)\) such that any diffeomorphism \(f\in\mathcal{R}\) possessing a chain recurrence class with periodic saddles of different indices has a robust heterodimensional cycle. Finally, the third result relates existence of \(C^{1}\)-robust co-index 1 cycles in terms of the shadowing property. Recall that a diffeomorphism \(f\) satisfies the shadowing property if, for any \(\delta>0,\) there exists an \(\varepsilon>0\) such that any finite \(\varepsilon \)-pseudo-orbit of \(f\) is \(\delta\)-shadowed by a true orbit, that is, for a \(\delta\)-pseudo-orbit \(( x_{i}) _{i=0}^{n}\) there exists an \(x\) such that \(d( f^{( i) }( x) ,x_{i}) <\varepsilon\) for all \(i=0,1,\ldots,n.\) A consequence of the main theorem is that for a diffeomorphism \(f\) with a co-index 1 cycle, there exists an open set \(\mathcal U\subset\text{Diff}^{1}(M),\) whose closure contains \(f,\) such that \(\mathcal{U}\) contains diffeomorphisms that do not have the shadowing property.

MSC:

37C20 Generic properties, structural stability of dynamical systems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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