## 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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