scholarly journals Asymptotic orbital shadowing property for diffeomorphisms

2019 ◽  
Vol 17 (1) ◽  
pp. 191-201 ◽  
Author(s):  
Manseob Lee
Keyword(s):  
Riemannian Manifold ◽  
Anosov Diffeomorphism ◽  
Shadowing Property ◽  
Orbital Shadowing ◽  
Volume Preserving

Abstract Let M be a closed smooth Riemannian manifold and let f : M → M be a diffeomorphism. We show that if f has the C1 robustly asymptotic orbital shadowing property then it is an Anosov diffeomorphism. Moreover, for a C1 generic diffeomorphism f, if f has the asymptotic orbital shadowing property then it is a transitive Anosov diffeomorphism. In particular, we apply our results to volume-preserving diffeomorphisms.

scholarly journals Orbital Shadowing for -Generic Volume-Preserving Diffeomorphisms

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Manseob Lee
Keyword(s):  
Shadowing Property ◽  
Orbital Shadowing ◽  
Volume Preserving

We show that -generically, if a volume-preserving diffeomorphism has the orbital shadowing property, then the diffeomorphism is Anosov.

Weak stability of the geodesic flow and Preissmann's theorem

2000 ◽  
Vol 20 (4) ◽  
pp. 1231-1251
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Geodesic Flow ◽  
Abelian Subgroup ◽  
Conjugate Points ◽  
Weak Stability ◽  
Shadowing Property

Let $(M,g)$ be a compact, differentiable Riemannian manifold without conjugate points and bounded asymptote. We show that, if the geodesic flow of $(M,g)$ is either topologically stable, or satisfies the $\epsilon$-shadowing property for some appropriate $\epsilon > 0$, then every abelian subgroup of the fundamental group of $M$ is infinite cyclic. The proof is based on the existence of homoclinic geodesics in perturbations of $(M,g)$, whenever there is a subgroup of the fundamental group of $M$ isomorphic to $\mathbb{Z}\times \mathbb{Z}$.

scholarly journals On stable transitivity of finitely generated groups of volume-preserving diffeomorphisms

2017 ◽  
Vol 39 (2) ◽  
pp. 554-576
Author(s):  
ZHIYUAN ZHANG
Keyword(s):  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Dense Subset ◽  
Duke Math ◽  
Large Integer ◽  
Finitely Generated Groups ◽  
Partially Hyperbolic ◽  
New Criterion ◽  
Partially Hyperbolic Diffeomorphism ◽  
Volume Preserving

In this paper, we provide a new criterion for the stable transitivity of volume-preserving finite generated groups on any compact Riemannian manifold. As one of our applications, we generalize a result of Dolgopyat and Krikorian [On simultaneous linearization of diffeomorphisms of the sphere. Duke Math. J. 136 (2007), 475–505] and obtain stable transitivity for random rotations on the sphere in any dimension. As another application, we show that for $\infty \geq r\geq 2$, for any $C^{r}$ volume-preserving partially hyperbolic diffeomorphism $g$ on any compact Riemannian manifold $M$ having sufficiently Hölder stable or unstable distribution, for any sufficiently large integer $K$ and for any $(f_{i})_{i=1}^{K}$ in a $C^{1}$ open $C^{r}$ dense subset of $\text{Diff}^{r}(M,m)^{K}$, the group generated by $g,f_{1},\ldots ,f_{K}$ acts transitively.

scholarly journals Orbital shadowing property on chain transitive sets for generic diffeomorphisms

2020 ◽  
Vol 12 (1) ◽  
pp. 146-154
Author(s):  
Manseob Lee
Keyword(s):  
Maximal Chain ◽  
Dimensional Manifold ◽  
Dominated Splitting ◽  
Shadowing Property ◽  
Orbital Shadowing ◽  
Locally Maximal ◽  
Chain Transitive Sets

AbstractLet f : M → M be a diffeomorphism on a closed smooth n(≥ 2) dimensional manifold M. We show that C1 generically, if a diffeomorphism f has the orbital shadowing property on locally maximal chain transitive sets which admits a dominated splitting then it is hyperbolic.

Volume-Preserving Diffeomorphisms with Various Limit Shadowing

2015 ◽  
Vol 25 (02) ◽  
pp. 1550018 ◽  
Author(s):  
Manseob Lee
Keyword(s):  
Shadowing Property ◽  
Limit Shadowing ◽  
Volume Preserving

We present the following: (1) if a volume-preserving diffeomorphism has C1-robustly various limit shadowing property, then it is Anosov; (2) C1-generically, if a volume-preserving diffeomorphism has various limit shadowing property, then it is Anosov.

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.

Symplectic diffeomorphisms with limit shadowing

2016 ◽  
Vol 10 (02) ◽  
pp. 1750068
Author(s):  
Manseob Lee
Keyword(s):  
Riemannian Manifold ◽  
Closed Formula ◽  
Shadowing Property ◽  
Symplectic Diffeomorphisms ◽  
Limit Shadowing ◽  
Symplectic Diffeomorphism ◽  
Weak Shadowing

Let [Formula: see text] be a symplectic diffeomorphism on a closed [Formula: see text][Formula: see text]-dimensional Riemannian manifold [Formula: see text]. In this paper, we show that [Formula: see text] is Anosov if any of the following statements holds: [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the limit shadowing property or [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the limit weak shadowing property or [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the s-limit shadowing property.

Transitive Anosov flows and Axiom-A diffeomorphisms

2009 ◽  
Vol 29 (3) ◽  
pp. 817-848 ◽  
Author(s):  
CHRISTIAN BONATTI ◽  
NANCY GUELMAN
Keyword(s):  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Anosov Diffeomorphism ◽  
Anosov Flow ◽  
Axiom A ◽  
Anosov Flows

AbstractLet M be a smooth compact Riemannian manifold without boundary, and ϕ:M×ℝ→M a transitive Anosov flow. We prove that if the time-one map of ϕ is C1-approximated by Axiom-A diffeomorphisms with more than one attractor, then ϕ is topologically equivalent to the suspension of an Anosov diffeomorphism.

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