Homoclinic bifurcations, dominated splitting, and robust transitivity.

*(English)*Zbl 1130.37354
Hasselblatt, B.(ed.) et al., Handbook of dynamical systems. Volume 1B. Amsterdam: Elsevier (ISBN 0-444-52055-4/hbk). 327-378 (2006).

From the introduction: For a long time (mainly since PoincarĂ©) it has been a goal in the theory of dynamical systems to describe the dynamics from a general viewpoint, that is, describing the dynamics of “big sets” of the space of all dynamical systems.

Uniform hyperbolicity was soon realized to be a property less universal than initially thought: It was shown that there are open sets in the space of dynamics that are nonhyperbolic. Weaker notions of hyperbolicity, such as partial hyperbolicity and dominated splitting, and their consequences, became a major field of research.

The results of Newhouse pushed some aspects of the theory of dynamical systems in different directions:

1. The study of the dynamical phenomena obtained from homoclinic bifurcations.

2. The characterization of universal mechanism that could lead to robustly nonhyperbolic behavior.

3. The study and characterization of isolated transitive sets that remain transitive for all nearby systems.

4. The dynamical consequences of weaker forms of hyperbolicity.

As we will show, these problems are all related to each other. In many cases, such relations provide a conceptual framework, as the hyperbolic theory did in the case of transverse homoclinic orbits.

Besides, in the early 80’s Palis conjectured that homoclinic tangencies and heterodimensional cycles were \(C^r\)-dense in the complement of the hyperbolic diffeomorphisms in \(\mathrm{Diff}^r(M)\) for a compact manifold \(M\). This conjecture was proved for the case of surfaces and the \(C^1\)-topology. The case of higher topologies is far from being solved.

We would like to emphasize that this chapter attempts to give some insight into the problems but is not meant to be a complete survey.

For the entire collection see [Zbl 1081.00006].

Uniform hyperbolicity was soon realized to be a property less universal than initially thought: It was shown that there are open sets in the space of dynamics that are nonhyperbolic. Weaker notions of hyperbolicity, such as partial hyperbolicity and dominated splitting, and their consequences, became a major field of research.

The results of Newhouse pushed some aspects of the theory of dynamical systems in different directions:

1. The study of the dynamical phenomena obtained from homoclinic bifurcations.

2. The characterization of universal mechanism that could lead to robustly nonhyperbolic behavior.

3. The study and characterization of isolated transitive sets that remain transitive for all nearby systems.

4. The dynamical consequences of weaker forms of hyperbolicity.

As we will show, these problems are all related to each other. In many cases, such relations provide a conceptual framework, as the hyperbolic theory did in the case of transverse homoclinic orbits.

Besides, in the early 80’s Palis conjectured that homoclinic tangencies and heterodimensional cycles were \(C^r\)-dense in the complement of the hyperbolic diffeomorphisms in \(\mathrm{Diff}^r(M)\) for a compact manifold \(M\). This conjecture was proved for the case of surfaces and the \(C^1\)-topology. The case of higher topologies is far from being solved.

We would like to emphasize that this chapter attempts to give some insight into the problems but is not meant to be a complete survey.

For the entire collection see [Zbl 1081.00006].