scholarly journals Measure-Expansive Homoclinic Classes for C1 Generic Flows

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1232
Author(s):  
Manseob Lee
Keyword(s):  
Vector Field ◽  
Smooth Manifold ◽  
Closed Orbit ◽  
Homoclinic Class ◽  
Homoclinic Classes

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.

HYPERBOLICITY OF HOMOCLINIC CLASSES OF VECTOR FIELDS

2014 ◽  
Vol 98 (3) ◽  
pp. 375-389 ◽  
Author(s):  
KEONHEE LEE ◽  
MANSEOB LEE ◽  
SEUNGHEE LEE
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Closed Orbit ◽  
Homoclinic Class ◽  
Shadowing Property ◽  
Homoclinic Classes

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.

CHAIN COMPONENTS WITH THE STABLE SHADOWING PROPERTY FOR C1 VECTOR FIELDS

2021 ◽  
pp. 1-17
Author(s):  
MANSEOB LEE ◽  
LE HUY TIEN
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Vector Fields ◽  
Closed Orbit ◽  
Homoclinic Class ◽  
Shadowing Property ◽  
Chain Component

Let M be a closed n-dimensional smooth Riemannian manifold, and let X be a $C^1$ -vector field of $M.$ Let $\gamma $ be a hyperbolic closed orbit of $X.$ In this paper, we show that X has the $C^1$ -stably shadowing property on the chain component $C_X(\gamma )$ if and only if $C_X(\gamma )$ is the hyperbolic homoclinic class.

scholarly journals Lyapunov stable homoclinic classes for smooth vector fields

2019 ◽  
Vol 17 (1) ◽  
pp. 990-997
Author(s):  
Manseob Lee
Keyword(s):  
Vector Fields ◽  
Smooth Vector ◽  
Homoclinic Class ◽  
Shadowing Property ◽  
Homoclinic Classes

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.

scholarly journals Newhouse phenomenon and homoclinic classes

2010 ◽  
Vol 31 (5) ◽  
pp. 1537-1562 ◽  
Author(s):  
JIAGANG YANG
Keyword(s):  
Saddle Point ◽  
Infinite Sequence ◽  
Homoclinic Class ◽  
Generic Subset ◽  
Codimension One ◽  
Homoclinic Tangencies ◽  
Homoclinic Classes

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.

scholarly journals Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

2009 ◽  
Vol 29 (5) ◽  
pp. 1479-1513 ◽  
Author(s):  
LORENZO J. DÍAZ ◽  
ANTON GORODETSKI
Keyword(s):  
Lyapunov Exponents ◽  
Residual Subset ◽  
Homoclinic Class ◽  
Zero Measure ◽  
Ergodic Measures ◽  
Homoclinic Classes

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.

scholarly journals Countably Expansiveness for Continuous Dynamical Systems

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1228
Author(s):  
Manseob Lee ◽  
Jumi Oh
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Stability Theory ◽  
Smooth Manifold ◽  
The Stability ◽  
Expansive Flows ◽  
Dense Set

Expansiveness is very closely related to the stability theory of the dynamical systems. It is natural to consider various types of expansiveness such as countably-expansive, measure expansive, N-expansive, and so on. In this article, we introduce the new concept of countably expansiveness for continuous dynamical systems on a compact connected smooth manifold M by using the dense set D of M, which is different from the weak expansive flows. We establish some examples having the countably expansive property, and we prove that if a vector field X of M is C 1 stably countably expansive then it is quasi-Anosov.

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

2019 ◽  
Vol 63 (1) ◽  
pp. 217-228
Author(s):  
Mário Bessa ◽  
Maria Joana Torres
Keyword(s):  
Homoclinic Class ◽  
Area Preserving Maps ◽  
Area Preserving ◽  
Homoclinic Classes

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.

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

2012 ◽  
Vol 33 (3) ◽  
pp. 739-776 ◽  
Author(s):  
CH. BONATTI ◽  
S. CROVISIER ◽  
L. J. DÍAZ ◽  
N. GOURMELON
Keyword(s):  
Periodic Orbit ◽  
Chain Recurrence ◽  
Homoclinic Class ◽  
Dynamical Properties ◽  
Self Replication ◽  
Homoclinic Classes

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.

On the existence of non-trivial homoclinic classes

2007 ◽  
Vol 27 (5) ◽  
pp. 1473-1508 ◽  
Author(s):  
CHRISTIAN BONATTI ◽  
SHAOBO GAN ◽  
LAN WEN
Keyword(s):  
Periodic Orbit ◽  
Dominated Splitting ◽  
Homoclinic Class ◽  
Partially Hyperbolic ◽  
Homoclinic Classes

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.

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