Abundance of C 1 -robust homoclinic tangencies
By C. Bonatti, Lorenzo J. Díaz. 2009
67 citations recorded by the source corpus.
A diffeomorphism f has a C 1 -robust homoclinic tangency if there is a C 1 -neighbourhood U of f such that every diffeomorphism in g ∈ U has a hyperbolic setg, depending contin- uously on g, such that the stable and unstable manifolds ofg have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a lo- cal mechanism (blender-horseshoes) generating diffeomorphisms with C 1 -robust homoclinic tangencies. Using blender-horseshoes, we prove that homoclinic classes of C 1 -generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display C 1 -robust homoclinic tangencies.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/303d72c08fad4ddcb2efde885cc36f5bdc05e069
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In Geometry and Topology
- C. Bonatti
- Lorenzo J. Díaz
- Jairo Bochi
- Tom A. Fisher
- Marcelo Viana
- S. Crovisier
- Artur Avila
- F. Abdenur
In Mathematical Physics
- C. Bonatti
- Lorenzo J. Díaz
- Tom A. Fisher
- Jairo Bochi
- Marcelo Viana
- Artur Avila
- S. Crovisier
- Jiagang Yang
In Statistical and Nonlinear Physics
- C. Bonatti
- Lorenzo J. Díaz
- Tom A. Fisher
- Jairo Bochi
- Marcelo Viana
- Artur Avila
- S. Crovisier
- Jiagang Yang
Citations in the corpus
- Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures (2015, background): “ that the strong stable (resp. unstable) manifold of such nonhyperbolic periodic point intersects transversely the unstable manifold of p g (resp. the stable manifold of q g). With the terminology in [BD3], this nonhyperbolic periodic point is strong intermediate. As p gand q gare robustly in the sa”
- Symbolic extensions for partially hyperbolic diffeomorphisms (2009, background): “h that Λf(U) = T n∈Z f n(U¯) is a robustly transitive set whose finest dominated splitting has some (indecomposable) center bundle with dimension at least 2. We begin by reviewing the constructions in [4, 6, 9, 2]. Define N(U) as the set of diffeomorphisms f ∈ Diff(M) such that Λf(U) is robustly transit”