On maximal transitive sets of generic diffeomorphisms.

*(English)*Zbl 1032.37011Locally generic \(C^1\)-diffeomorphisms \(f\) of \(3\)-manifolds with maximal transitive Cantor sets without periodic points are constructed. The locally generic diffeomorphisms constructed also exhibit strongly pathological features generalizing the Newhouse phenomenon (coexistence of infinitely many sinks or sources). These features are the following: coexistence of infinitely many nontrivial (hyperbolic and nonhyperbolic) attractors and repellors, and coexistence of infinitely many nontrivial (nonhyperbolic) homoclinic classes.

These phenomena are proved to be associated to the existence of a homoclinic class \(H(P,f)\) (\(P\) is a periodic point of \(f\)) with the following properties:

– in a \(C^1\)-robust way, the homoclinic class \(H(P,f)\) does not admit any dominated splitting,

– there is a periodic point \(P'\) homoclinically related to \(P\) such that the Jacobians of \(P'\) and \(P\) are greater than and less than one, respectively.

The paper’s main result is the following:

Theorem. Given any compact \(3\)-manifold \(M\) there is a locally residual subset \(\mathcal F(M)\) of \(\text{Diff}^1(M)\) of diffeomorphisms \(f\) having maximal transitive Cantor sets \(\Lambda _f\) without periodic points. In particular, the maximal transitive Cantor sets \(\Lambda _f\) are not homoclinic classes.

These phenomena are proved to be associated to the existence of a homoclinic class \(H(P,f)\) (\(P\) is a periodic point of \(f\)) with the following properties:

– in a \(C^1\)-robust way, the homoclinic class \(H(P,f)\) does not admit any dominated splitting,

– there is a periodic point \(P'\) homoclinically related to \(P\) such that the Jacobians of \(P'\) and \(P\) are greater than and less than one, respectively.

The paper’s main result is the following:

Theorem. Given any compact \(3\)-manifold \(M\) there is a locally residual subset \(\mathcal F(M)\) of \(\text{Diff}^1(M)\) of diffeomorphisms \(f\) having maximal transitive Cantor sets \(\Lambda _f\) without periodic points. In particular, the maximal transitive Cantor sets \(\Lambda _f\) are not homoclinic classes.

Reviewer: Eugene Ershov (St.Peterburg)

##### MSC:

37C20 | Generic properties, structural stability of dynamical systems |

37D30 | Partially hyperbolic systems and dominated splittings |

37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |