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.

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.

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 Tame dynamics and robust transitivity chain-recurrence classes versus homoclinic classes

2014 ◽  
Vol 366 (9) ◽  
pp. 4849-4871 ◽  
Author(s):  
C. Bonatti ◽  
S. Crovisier ◽  
N. Gourmelon ◽  
R. Potrie
Keyword(s):  
Chain Recurrence ◽  
Homoclinic Classes

scholarly journals Hyperbolicity versus weak periodic orbits inside homoclinic classes

2017 ◽  
Vol 38 (6) ◽  
pp. 2345-2400 ◽  
Author(s):  
XIAODONG WANG
Keyword(s):  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Homoclinic Class ◽  
Intrinsic Nature ◽  
Significant Difference ◽  
Stability Conjecture ◽  
The Stability ◽  
Classical Proof ◽  
Homoclinic Classes

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 zero, then$H(p)$must be (uniformly) hyperbolic. This is in the spirit of the works on 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 work. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas.

scholarly journals Dynamical properties of maps derived from maps with strong negative Schwarzian derivative

1984 ◽  
Vol 7 (4) ◽  
pp. 803-808
Author(s):  
Abraham Boyarsky
Keyword(s):  
Periodic Orbit ◽  
Periodic Point ◽  
Schwarzian Derivative ◽  
Stable Periodic Orbit ◽  
Stable Periodic ◽  
Dynamical Properties

A strong Schwarzian derivative is defined, and it is shown that the convolution of a function with a map from an interval into itself having negative strong Schwarzian derivative is a function with negative Schwarzian derivative. Such convolutions have0as a stable periodic point and at most one other stable periodic orbit in the interior of the domain.

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