The Research of the Periodic Deviation Phenomenon on the Chain Hyperbolic Homoclinic Class

Abstract In this paper, we show that for generic C1, if a flow Xt has the shadowing property on a bi-Lyapunov stable homoclinic class, then it does not contain any singularity and it is hyperbolic.

Newhouse phenomenon and homoclinic classes

10.1017/s0143385710000465 ◽

2010 ◽

Vol 31 (5) ◽

pp. 1537-1562 ◽

Cited By ~ 1

Author(s):

JIAGANG YANG

Keyword(s):

Saddle Point ◽

Infinite Sequence ◽

Homoclinic Class ◽

Generic Subset ◽

Codimension One ◽

Homoclinic Tangencies ◽

AbstractWe show that for a C1 generic subset of diffeomorphisms far from homoclinic tangencies, any infinite sequence of sinks or sources must accumulate on a homoclinic class of some saddle point with codimension one.

HYPERBOLICITY OF HOMOCLINIC CLASSES OF VECTOR FIELDS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788714000640 ◽

2014 ◽

Vol 98 (3) ◽

pp. 375-389 ◽

Cited By ~ 3

Author(s):

KEONHEE LEE ◽

MANSEOB LEE ◽

SEUNGHEE LEE

Keyword(s):

Vector Field ◽

Vector Fields ◽

Closed Orbit ◽

Homoclinic Class ◽

Shadowing Property ◽

Let${\it\gamma}$be a hyperbolic closed orbit of a$C^{1}$vector field$X$on a compact$C^{\infty }$manifold$M$and let$H_{X}({\it\gamma})$be the homoclinic class of$X$containing${\it\gamma}$. In this paper, we prove that if a$C^{1}$-persistently expansive homoclinic class$H_{X}({\it\gamma})$has the shadowing property, then$H_{X}({\it\gamma})$is hyperbolic.

Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

10.1017/s0143385708000849 ◽

2009 ◽

Vol 29 (5) ◽

pp. 1479-1513 ◽

Cited By ~ 15

Author(s):

LORENZO J. DÍAZ ◽

ANTON GORODETSKI

Keyword(s):

Lyapunov Exponents ◽

Residual Subset ◽

Homoclinic Class ◽

Zero Measure ◽

Ergodic Measures ◽

AbstractWe prove that there is a residual subset 𝒮 in Diff1(M) such that, for everyf∈𝒮, any homoclinic class offcontaining saddles of different indices (dimension of the unstable bundle) contains also an uncountable support of an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure off.

There are no proper topological hyperbolic homoclinic classes for area-preserving maps

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091519000282 ◽

2019 ◽

Vol 63 (1) ◽

pp. 217-228

Author(s):

Mário Bessa ◽

Maria Joana Torres

Keyword(s):

Homoclinic Class ◽

Area Preserving Maps ◽

Area Preserving ◽

AbstractWe begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class Λ associated with an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then Λ = M and f is an Anosov homeomorphism.

Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication

10.1017/s0143385712000028 ◽

2012 ◽

Vol 33 (3) ◽

pp. 739-776 ◽

Cited By ~ 8

Author(s):

CH. BONATTI ◽

S. CROVISIER ◽

L. J. DÍAZ ◽

N. GOURMELON

Keyword(s):

Periodic Orbit ◽

Chain Recurrence ◽

Homoclinic Class ◽

Dynamical Properties ◽

Self Replication ◽

AbstractConditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes have the property of generating copies of themselves producing wild dynamics (systems with infinitely many homoclinic classes with some persistence). Such wild dynamics also exhibits uncountably many aperiodic chain recurrence classes. A scenario (related with non-dominated dynamics) is presented where viral homoclinic classes occur. A key ingredient are adapted perturbations of a diffeomorphism along a periodic orbit. Such perturbations preserve certain homoclinic relations and prescribed dynamical properties of a homoclinic class.

Measure-Expansive Homoclinic Classes for C1 Generic Flows

Mathematics ◽

10.3390/math8081232 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1232

Author(s):

Manseob Lee

Keyword(s):

Vector Field ◽

Smooth Manifold ◽

Closed Orbit ◽

Homoclinic Class ◽

In this paper, we prove that for a generically C1 vector field X of a compact smooth manifold M, if a homoclinic class H(γ,X) which contains a hyperbolic closed orbit γ is measure expansive for X then H(γ,X) is hyperbolic.

On the existence of non-trivial homoclinic classes

10.1017/s0143385707000090 ◽

2007 ◽

Vol 27 (5) ◽

pp. 1473-1508 ◽

Cited By ~ 11

Author(s):

CHRISTIAN BONATTI ◽

SHAOBO GAN ◽

LAN WEN

Keyword(s):

Periodic Orbit ◽

Dominated Splitting ◽

Homoclinic Class ◽

Partially Hyperbolic ◽

AbstractWe show that, for C1-generic diffeomorphisms, every chain recurrent class C that has a partially hyperbolic splitting $E^s\oplus E^c\oplus E^u$ with dimEc=1 either is an isolated hyperbolic periodic orbit, or is accumulated by non-trivial homoclinic classes. We also prove that, for C1-generic diffeomorphisms, any chain recurrent class that has a dominated splitting $E\oplus F$ with dim(E)=1 either is a homoclinic class, or the bundle E is uniformly contracting. As a corollary we prove in dimension three a conjecture of Palis, which announces that any C1-generic diffeomorphism is either Morse–Smale, or has a non-trivial homoclinic class.