Introduction to the Closing Lemma

1978 ◽  
pp. 225-230 ◽  
Author(s):  
Clark Robinson
Keyword(s):  
Closing Lemma

A closing lemma for polynomial automorphisms of C²

Astérisque ◽  
2020 ◽  
Vol 415 ◽  
pp. 35-43
Author(s):  
Romain DUJARDIN
Keyword(s):  
Closing Lemma ◽  
Polynomial Automorphisms

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

scholarly journals Hadamard–Perron theorems and effective hyperbolicity

2014 ◽  
Vol 36 (1) ◽  
pp. 23-63 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
YAKOV PESIN
Keyword(s):  
Closing Lemma ◽  
Stable And Unstable Manifolds ◽  
Local Dynamics ◽  
Local Diffeomorphisms ◽  
Finite Amount ◽  
Amount Of Information ◽  
Unstable Manifolds

We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

scholarly journals Quasi-shadowing for partially hyperbolic diffeomorphisms

2014 ◽  
Vol 35 (2) ◽  
pp. 412-430 ◽  
Author(s):  
HUYI HU ◽  
YUNHUA ZHOU ◽  
YUJUN ZHU
Keyword(s):  
Spectral Decomposition ◽  
Hyperbolic Systems ◽  
Decomposition Theorem ◽  
Closing Lemma ◽  
Shadowing Property ◽  
Hyperbolic Diffeomorphisms ◽  
Uniformly Hyperbolic Systems ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

AbstractA partially hyperbolic diffeomorphism $f$ has the quasi-shadowing property if for any pseudo orbit $\{x_{k}\}_{k\in \mathbb{Z}}$, there is a sequence of points $\{y_{k}\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_{k})$ by a motion ${\it\tau}$ along the center direction. We show that any partially hyperbolic diffeomorphism has the quasi-shadowing property, and if $f$ has a $C^{1}$ center foliation then we can require ${\it\tau}$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^{0}$-perturbation. When $f$ has a uniformly compact $C^{1}$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems which hold for uniformly hyperbolic systems, such as the Anosov closing lemma, the cloud lemma and the spectral decomposition theorem.

The Closing Lemma

1967 ◽  
Vol 89 (4) ◽  
pp. 956 ◽  
Author(s):  
Charles C. Pugh
Keyword(s):  
Closing Lemma

scholarly journals The planar closing lemma for chain recurrence

1994 ◽  
Vol 341 (1) ◽  
pp. 173-192 ◽  
Author(s):  
Maria Lúcia Alvarenga Peixoto ◽  
Charles Chapman Pugh
Keyword(s):  
Closing Lemma ◽  
Chain Recurrence

The C1 closing lemma for non-singular endomorphisms

1991 ◽  
Vol 11 (2) ◽  
pp. 393-412 ◽  
Author(s):  
Lan Wen
Keyword(s):  
Closing Lemma

AbstractA slightly improved version of the (idealized) C1 closing lemma is proved. This turns out to generalize the C1 closing lemma from diffeomorphisms to nonsingular endomorphisms.

Versions of the closing lemma for certain dynamical systems on tori

2003 ◽  
Vol 4 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Samuil Aranson ◽  
Mikhail Malkin ◽  
Vladislav Medvedev ◽  
Evgeny Zhuzhoma
Keyword(s):  
Dynamical Systems ◽  
Closing Lemma

An improved sectional-Anosov closing lemma

2010 ◽  
Vol 268 (1-2) ◽  
pp. 317-327 ◽  
Author(s):  
C. A. Morales
Keyword(s):  
Closing Lemma
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