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**Homoclinic tangencies and hyperbolicity for surface diffeomorphisms.**
*(English)*
Zbl 0959.37040

The authors prove that in the complement of the closure of the hyperbolic surface diffeomorphisms, the surface diffeomorphisms exhibiting a homoclinic tangency are \(C^1\) dense.

Due to the existence of pathological dynamics, one seeks in the theory of dynamical systems a description of the dynamics of almost all systems (for instance, of systems from a residual set). In the sixties it was thought in the west that this goal might be realized by looking at structurally stable systems. It was however soon shown that structurally stable systems are not dense in the set of all systems. In the nineties J. Palis formulated a conjecture that a \(C^r\) diffeomorphism on a compact manifold can be approximated, in the \(C^r\) topology, by either a hyperbolic one or by one exhibiting a homoclinic bifurcation. For diffeomorphisms on surfaces, a homoclinic bifurcation would be a homoclinic tangency. The conjecture is proved in this paper, in the \(C^1\) topology, for diffeomorphisms on compact surfaces. One ingredient is a generalization to two dimensions of a result in interval or circle dynamics by R. Mañé, in which conditions for hyperbolicity are given. The authors show that a compact invariant set \(\Lambda\) of a \(C^2\) diffeomorphism on a surface, that has a dominated splitting and only hyperbolic periodic points, is the union of a hyperbolic set and a finite number of normally hyperbolic invariant circles with irrational rotation.

Due to the existence of pathological dynamics, one seeks in the theory of dynamical systems a description of the dynamics of almost all systems (for instance, of systems from a residual set). In the sixties it was thought in the west that this goal might be realized by looking at structurally stable systems. It was however soon shown that structurally stable systems are not dense in the set of all systems. In the nineties J. Palis formulated a conjecture that a \(C^r\) diffeomorphism on a compact manifold can be approximated, in the \(C^r\) topology, by either a hyperbolic one or by one exhibiting a homoclinic bifurcation. For diffeomorphisms on surfaces, a homoclinic bifurcation would be a homoclinic tangency. The conjecture is proved in this paper, in the \(C^1\) topology, for diffeomorphisms on compact surfaces. One ingredient is a generalization to two dimensions of a result in interval or circle dynamics by R. Mañé, in which conditions for hyperbolicity are given. The authors show that a compact invariant set \(\Lambda\) of a \(C^2\) diffeomorphism on a surface, that has a dominated splitting and only hyperbolic periodic points, is the union of a hyperbolic set and a finite number of normally hyperbolic invariant circles with irrational rotation.

Reviewer: Ale Jan Homburg (Amsterdam)