scholarly journals The closing lemma and the planar general density theorem for Sobolev maps

2021 ◽  
Vol 149 (4) ◽  
pp. 1687-1696
Author(s):  
Assis Azevedo ◽  
Davide Azevedo ◽  
Mário Bessa ◽  
Maria Joana Torres
Keyword(s):  
Density Theorem ◽  
Closing Lemma ◽  
Sobolev Maps

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.

On Isbell's density theorem for bitopological pointfree spaces I

2020 ◽  
Vol 273 ◽  
pp. 106962
Author(s):  
M. Andrew Moshier ◽  
Imanol Mozo Carollo ◽  
Joanne Walters-Wayland
Keyword(s):  
Density Theorem

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.

A new proof for the residual set dimension of the apollonian packing

1984 ◽  
Vol 96 (3) ◽  
pp. 413-423 ◽  
Author(s):  
Claude Tricot
Keyword(s):  
Hausdorff Dimension ◽  
Density Theorem ◽  
Residual Set ◽  
Curvilinear Triangle

AbstractLet (Dn) be the apollonian packing of a curvilinear triangle T, ρn the radius of Dn, E = T—U Dn the residual set, dim (E) its Hausdorff dimension. In this paper we give a new proof of the equality dim proved by Boyd [2]. Our technique is to construct a sequence of regular triangles covering E, and suitable measures μkcarried by E which allow us to apply a density theorem.

Sobolev maps with integer degree and applications to Skyrme’s problem

1992 ◽  
Vol 436 (1896) ◽  
pp. 197-201 ◽  
Keyword(s):  
Finite Energy ◽  
Natural Class ◽  
Sobolev Maps

Let ɸ : R 3 → S 3 ⊂ R 4 , ∣ A ( ɸ )∣ 2 ═ Ʃ 3 α,β═1 │∂ ɸ /∂ x α ∧ ∂ ɸ /∂ x β ∣ 2 and let k ϵ Z . Skyrme's problem consists in minimizing the energy ε( ɸ ) : ═ ∫ R 3 ∣∇ ɸ ∣ 2 + ∣ A ( ɸ )∣ 2 d x among maps with degree k ═ d ( ɸ ) : ═ 1/2π 2 ∫ R 3 det ( ɸ , ∇ ɸ ) d x . We show that for all ɸ with finite energy d ( ɸ ) is an integer and then obtain existence of a minimizer of ε in the natural class of maps with finite energy.

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.

scholarly journals A Converse of Artin′s Density Theorem: The Case of Cubic Fields

1993 ◽  
Vol 45 (1) ◽  
pp. 28-44 ◽  
Author(s):  
I. Delcorso ◽  
R. Dvornicich
Keyword(s):  
Density Theorem ◽  
Cubic Fields
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