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.

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.

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.

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.

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 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.

Random entropy expansiveness for diffeomorphisms with dominated splittings

2019 ◽  
Vol 20 (02) ◽  
pp. 2050014
Author(s):  
Zeya Mi
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Random Dynamical Systems ◽  
Local Entropy ◽  
Dominated Splitting ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Bowen Balls ◽  
Partially Hyperbolic Diffeomorphisms

We study the local entropy of typical infinite Bowen balls in random dynamical systems, and show the random entropy expansiveness for [Formula: see text] partially hyperbolic diffeomorphisms with multi one-dimensional centers. Moreover, we consider [Formula: see text] diffeomorphism [Formula: see text] with dominated splitting [Formula: see text] such that [Formula: see text] for every [Formula: see text], and all the Lyapunov exponents are non-negative along [Formula: see text] and non-positive along [Formula: see text], we prove the asymptotically random entropy expansiveness for [Formula: see text].

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.

scholarly journals Compact dynamical foliations

2014 ◽  
Vol 35 (8) ◽  
pp. 2474-2498 ◽  
Author(s):  
PABLO D. CARRASCO
Keyword(s):  
Periodic Orbit ◽  
Compact Manifold ◽  
Negative Answer ◽  
Uniform Bound ◽  
Smooth Flow ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphism ◽  
Dennis Sullivan

According to the work of Dennis Sullivan [A counterexample to the periodic orbit conjecture.Publ. Math. Inst. Hautes Études Sci. 46(1976), 5–14], there exists a smooth flow on the five-sphere all of whose orbits are periodic, although there is no uniform bound on their periods. The question addressed in this article is whether these type of examples can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact but there is no uniform bound for their volumes? We develop tools to attack the previous question and show that it has a negative answer provided that all periodic leaves have finite holonomy.

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