Homoclinic classes for generic C^1 vector fields

By C. M. Carballo, C. Morales, Maria José Pacifico. Ergodic Theory and Dynamical Systems, 2001

67 citations recorded by the source corpus.

We prove that homoclinic classes for a residual set of C^1 vector fields X on closed n-manifolds are maximal transitive, and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We also prove that a homoclinic class of X is isolated if and only if it is \Omega-isolated, and it is the intersection of its stable set with its unstable set. All these properties are well known for structurally stable Axiom A vector fields.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/00548874007c117ad75a24d702dede83a491f914

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. C. Bonatti
2. Lorenzo J. Díaz
3. F. Abdenur
4. E. Pujals
5. S. Crovisier
6. Marcelo Viana
7. Artur Avila
8. Jiagang Yang

In Mathematical Physics

1. C. Bonatti
2. Lorenzo J. Díaz
3. F. Abdenur
4. S. Crovisier
5. E. Pujals
6. Marcelo Viana
7. Jiagang Yang
8. Artur Avila

In Statistical and Nonlinear Physics

1. C. Bonatti
2. Lorenzo J. Díaz
3. F. Abdenur
4. E. Pujals
5. S. Crovisier
6. Marcelo Viana
7. Jiagang Yang
8. Artur Avila

Citations in the corpus

• A C^1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources (2003, background): “2Observe that the first part of the problem was positively answered in [Ar1] and [CMP] (recall footnote 1).”
• Generic robustness of spectral decompositions (2003, methodology): “Several recent papers (including Diaz, Pujals and Ures [10], Bonatti, Diaz and Pujals [5], Carballo and Morales [8], and Carballo, Morales and Pacifico [ 9 ]) explore their “hyperbolic-like” properties, many of which hold only for generic diffeomorphisms.”
• ROBUST HETERODIMENSIONAL CYCLES AND $C^1$-GENERIC DYNAMICS (2007, background): “In particular, since the recurrence classes define a partition of the chain recurrence set, two homoclinic classes of f ∈ R which are non-disjoint coincide (this result was previously stated in [ CMP ]).”