SURVEY Towards a global view of dynamical systems, for the <i>C</i><sup>1</sup>-topology

Bonatti, Christian

doi:10.1017/s0143385710000891

Cited by 38 publications

(27 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More precisely, the union of hyperbolic diffeomorphisms and diffeomorphisms with tangencies or heterodimensional cycles are dense in the space of diffeomorphisms, see [39]. Based on the results afterwards, Bonatti and Díaz conjectured that the union of diffeomorphisms that are hyperbolic and those with heterodimensional cycles are dense in the space of diffeomorphisms, see [7,11]. There are many works related to this subject, like [43,19,23,22].…”

Section: Backgrounds and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Hyperbolicity versus weak periodic orbits inside homoclinic classes

Wang

2017

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

We prove that, for $C^1$-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class $H(p)$ have all their Lyapunov exponents bounded away from 0, then $H(p)$ must be (uniformly) hyperbolic. This is in sprit of the works of the stability conjecture, but with a significant difference that the homoclinic class $H(p)$ is not known isolated in advance, hence the "weak" periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an "intrinsic" nature, and the classical proof of the stability conjecture does not pass through. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas

show abstract

Section: Backgrounds and Main Resultsmentioning

confidence: 99%

“…Conjecture 1 ( [7,9]). There is a residual subset R ⊂ Diff 1 (M ), such that for all f ∈ R, if a homoclinic class H(p) is not hyperbolic, then there is a periodic point q ∈ H(p), whose stable dimension is different from that of p.…”

Section: Backgrounds and Main Resultsmentioning

confidence: 99%

Hyperbolicity versus weak periodic orbits inside homoclinic classes

Wang

2017

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…Conjecture 4 ( [9,12,15,24]). For any generic f ∈ Diff 1 (M ), if a homoclinic class H(p) is not hyperbolic, then arbitrarily C 1 -close to f , there is a diffeomorphism g that exhibits a heterodimensional cycle associated to p g , where p g is the continuation of p.…”

Section: Heterodimensional Cycles Inside Non-hyperbolic Homoclinic CLmentioning

confidence: 99%

“…As in case (c), one can expect that the class is contained in a locally invariant submanifold tangent to E. We are thus reduced to the case of a homoclinic class whose periodic points have one-dimensional unstable spaces and sectional dissipative and has no domination corresponding to index dim(M ) − 1. We are thus reduced to a generalized Smale's conjecture for higher dimension, as described in [9,Conjecture 8].…”

Section: Heterodimensional Cycles Inside Non-hyperbolic Homoclinic CLmentioning

confidence: 99%

Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

Cheng¹,

Crovisier²,

Gan³

et al. 2017

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…Partially hyperbolic dynamical systems have received a large amount of attention in recent years. These systems display a wide variety of highly chaotic behaviour [Bon11], but they have enough structure to allow, in some cases, for the dynamics to be understood and classified [CRHRHU15,HP15b].…”

Section: Introductionmentioning

confidence: 99%

Constructing center-stable tori

Hammerlindl¹

2018

Ann. Inst. H. Poincaré Anal. Non Linéaire

View full text Add to dashboard Cite

We show that any weakly partially hyperbolic diffeomorphism on the 2-torus may be realized as the dynamics on a center-stable or centerunstable torus of a 3-dimensional strongly partially hyperbolic system. We also construct examples of center-stable and center-unstable tori in higher dimensions.In the construction in [RHRHU16], the dynamics on the 2-torus tangent to E c ⊕ E u is Anosov. In fact, it is given by a hyperbolic linear map on T 2 , the cat map. It has long been known that a weakly partially hyperbolic system, that is, a diffeomorphism g : T 2 → T 2 with a splitting of the form E c ⊕ E u or E c ⊕ E s , need not be Anosov. Therefore, one can ask exactly which weakly partially hyperbolic systems may be realized as the dynamics on an invariant 2-torus sitting inside a 3-dimensional strongly partially hyperbolic system. We show, in fact, that there are no obstructions on the choice of dynamics.Theorem 1.1. For any weakly partially hyperbolic diffeomorphism g 0 : T 2 → T 2 , there is an embedding i : T 2 → T 3 and a strongly partially hyperbolic diffeomorphism f : T 3 → T 3 such that i (T 2 ) is either a center-stable or center-unstable torus (depending on the splitting of g 0 ) and i −1 • f • i = g 0 .

show abstract

SURVEY Towards a global view of dynamical systems, for the C¹-topology

Cited by 38 publications

References 48 publications

Hyperbolicity versus weak periodic orbits inside homoclinic classes

Hyperbolicity versus weak periodic orbits inside homoclinic classes

Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

Constructing center-stable tori

Contact Info

Product

Resources

About

SURVEY Towards a global view of dynamical systems, for the C1-topology

Cited by 38 publications

References 48 publications

Hyperbolicity versus weak periodic orbits inside homoclinic classes

Hyperbolicity versus weak periodic orbits inside homoclinic classes

Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

Constructing center-stable tori

Contact Info

Product

Resources

About

SURVEY Towards a global view of dynamical systems, for the C¹-topology