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On the dynamics of dominated splitting. (English) Zbl 1178.37032

Let \(f:M\to M\) be a diffeomorphism of a compact Riemannian manifold \(M\). A fundamental problem is to understand how the tangent map \(Df:TM\to TM\) controls or determines the underlying dynamics of \(f\). This problem has been solved for hyperbolic dynamics in the so-called Spectral Decomposition Theorem, where it is given a satisfactory description of the dynamics of a system within the assumption that the tangent map \(Df\) has a hyperbolic structure. In the paper, the problem is solved for surface diffeomorphisms having dominated splitting. An \(f\)-invariant set \(\Lambda\) is said to have dominated splitting if its tangent bundle is decomposed in two continuous invariant subbundles \(T_{\Lambda}=E\oplus F\), such that \[ \|Df^n_{E(x)}\|\|Df^{-n}_{F(f^n(x))}\|\leq C\lambda^n,\text{ for all } x\in\Lambda, n\geq 0 \] with \(C>0\) and \(0<\lambda<1\). Among other interesting results, the paper presents the following theorem which gives a global and complete description of the dynamics of \(f\) when dominated splitting exists on the limit set (the minimum closed invariant set that contains the \(\omega\) and \(\alpha\) limit set of any orbit) \(L(f)\).
Spectral Decomposition Theorem. Let \(f\in\text{Diff}^2(M^2)\) and assume that \(L(f)\) has a dominated splitting. Then \(L(f)\) can be decomposed into \(L(f)=\mathcal I\cup\tilde L(f)\cup\mathcal R\) such that
1. \(\mathcal I\) is a set of periodic points with bounded periods and is contained in a disjoint union of finitely many normally hyperbolic arcs or simple closed curves.
2. \(\mathcal R\) is a finite union of normally hyperbolic periodic simple closed curves supporting an irrational rotation.
3. \(\tilde L(f)\) can be decomposed into a disjoint union of finitely many compact invariant and transitive sets. The periodic points are dense in \(\tilde L(f)\) and contain at most finitely many nonhyperbolic periodic points. The (basic) sets above are the union of finitely many (nontrivial) homoclinic classes. Furthermore \(f/\tilde L(f)\) is expansive.
For systems with dominated splitting on the limit set, all the bifurcations that they can exhibit and different types of dynamics that could follow for small \(C^r\)-perturbations are described.

MSC:

37D30 Partially hyperbolic systems and dominated splittings
37C29 Homoclinic and heteroclinic orbits for dynamical systems
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