Homoclinic tangencies and hyperbolicity for surface diffeomorphisms

By E. Pujals, Mart'in Sambarino. 2000

252 citations recorded by the source corpus.

We prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms, the ones exhibiting a homoclinic tangency are C 1 dense. This represents a step towards the global understanding of dynamics of surface diffeomorphisms.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/127cc8e1d805d935bf9337da6aa85561c4d474fd

How researchers see this paper

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

In Mathematical Physics

1. S. Crovisier
2. C. Bonatti
3. Lorenzo J. Díaz
4. Shaobo Gan
5. Jairo Bochi
6. Dawei Yang
7. E. Pujals
8. Marcelo Viana

In Statistical and Nonlinear Physics

1. S. Crovisier
2. C. Bonatti
3. Lorenzo J. Díaz
4. Jairo Bochi
5. Shaobo Gan
6. Nicolas Gourmelon
7. Dawei Yang
8. Tom A. Fisher

Citations in the corpus

• ROBUST HETERODIMENSIONAL CYCLES AND $C^1$-GENERIC DYNAMICS (2007, methodology): “This conjecture was proved for surface diffeomorphisms by Pujals and Sambarino in [ PS ] (note that for surface diffeomorphism heterodimensional cycles can be omitted).”
• Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures (2015, background): “ted by dieomorphisms that are either hyperbolic or display homoclinic bifurcations (homoclinic tangencies or heterodimensional cycles). This conjecture is true for C1-dieomorphisms of surfaces, see [PS]. See also [CP] for important progress in higher dimensions. Note that the homoclinic bifurcations”
• Adapted metrics for dominated splittings (2007, methodology): “This notion is a key tool for understanding non-hyperbolic systems: • In dimension 2, Pujals and Sambarino [PS00] proved that a diffeomorphism with a dominated splitting may be C1-approached by hyperbolic ones, and diffeomorphisms without dominated splitting may be approached by diffeomorphisms exhi”
• Symbolic extensions and smooth dynamical systems (2005, background): “We note that if we replace the word “residual” by the word “dense”, then this conjecture has been proved for r = 1 by Pujals and Samborino [35].”
• Morse-Smale systems and horseshoes for three dimensional singular flows (2013, methodology): “This was proved by Pujals-Sambarino [52] in the C1 topology.”
• Abundance of C 1 -robust homoclinic tangencies (2009, background): “By [26], for surface diffeomorphisms, the open set N (1)(M(2)) def = Diff(1)(M(2)) \ Hyp(1)(M2) is a Newhouse domain.”