Remarks on structurally stable proper foliations

1994 ◽  
Vol 115 (1) ◽  
pp. 111-120 ◽  
Author(s):  
Marco Brunella
Keyword(s):  
Structural Stability ◽  
Closed Manifold ◽  
Codimension One ◽  
Structurally Stable

Let M be a closed manifold of dimension 3 and let Fol(M) be the space of codimension one C∞-foliations on M. A foliation ∈ Fol(M) is said to be Cr- structurally stable if there exists a neighbourhood V of in Fol(M) in the (Epstein) Cr-topology such that every foliation is topologically conjugate to , through a homeomorphism near to the identity. Some background on the problem of structural stability of foliations can be found in [8]. In this paper we shall be concerned with proper foliations, i.e. foliations all of whose leaves are proper.

scholarly journals Scalar field cosmology — geometry of dynamics

2014 ◽  
Vol 11 (02) ◽  
pp. 1460012 ◽  
Author(s):  
Marek Szydłowski ◽  
Orest Hrycyna ◽  
Aleksander Stachowski
Keyword(s):  
Phase Space ◽  
Scalar Field ◽  
Dynamical Systems ◽  
Potential Function ◽  
Structural Stability ◽  
Saddle Points ◽  
Scalar Fields ◽  
Fine Tuning ◽  
Scalar Field Cosmology ◽  
Structurally Stable

We study the Scalar Field Cosmology (SFC) using the geometric language of the phase space. We define and study an ensemble of dynamical systems as a Banach space with a Sobolev metric. The metric in the ensemble is used to measure a distance between different models. We point out the advantages of visualization of dynamics in the phase space. It is investigated the genericity of some class of models in the context of fine tuning of the form of the potential function in the ensemble of SFC. We also study the symmetries of dynamical systems of SFC by searching for their exact solutions. In this context, we stressed the importance of scaling solutions. It is demonstrated that scaling solutions in the phase space are represented by unstable separatrices of the saddle points. Only critical point itself located on two-dimensional stable submanifold can be identified as scaling solution. We have also found a class of potentials of the scalar fields forced by the symmetry of differential equation describing the evolution of the Universe. A class of potentials forced by scaling (homology) symmetries was given. We point out the role of the notion of a structural stability in the context of the problem of indetermination of the potential form of the SFC. We characterize also the class of potentials which reproduces the ΛCDM model, which is known to be structurally stable. We show that the structural stability issue can be effectively used is selection of the scalar field potential function. This enables us to characterize a structurally stable and therefore a generic class of SFC models. We have found a nonempty and dense subset of structurally stable models. We show that these models possess symmetry of homology.

scholarly journals Characterisations of Ω-stability and structural stability via inverse shadowing

2006 ◽  
Vol 74 (2) ◽  
pp. 185-196 ◽  
Author(s):  
Taeyoung Choi ◽  
Keonhee Lee ◽  
Yong Zhang
Keyword(s):  
Structural Stability ◽  
Shadowing Property ◽  
Inverse Shadowing ◽  
Orbital Inverse Shadowing ◽  
Structurally Stable

We give characterisations of Ω-stable diffeomorphisms and structurally stable diffeomorphisms via the notions of weak inverse shadowing and orbital inverse shadowing, respectively. More precisely, it is proved that the C1 interior of the set of diffeomorphisms with the weak inverse shadowing property coincides with the set of Ω-stable diffeomorphisms and the C1 interior of the set of diffeomorphisms with the orbital inverse shadowing property coincides with the set of structurally stable diffeomorphisms.

Shadowing and structural stability for operators

2020 ◽  
pp. 1-20
Author(s):  
NILSON C. BERNARDES ◽  
ALI MESSAOUDI
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Structural Stability ◽  
Invertible Operator ◽  
Hyperbolic Operator ◽  
Weighted Shifts ◽  
Shadowing Property ◽  
Finite Dimensional ◽  
Structurally Stable

A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_{0}(\mathbb{Z})$ and $\ell _{p}(\mathbb{Z})$ ( $1\leq p<\infty$ ) that satisfy the shadowing property.

scholarly journals Coordinated self-interference of wave packets: a new route towards classicality for structurally stable systems

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
M. Ćosić ◽  
S. Petrović ◽  
S. Bellucci
Keyword(s):  
Proton Beam ◽  
Structural Stability ◽  
Phase Function ◽  
Quantum Dynamics ◽  
Wave Packets ◽  
The Self ◽  
Levels Of Reality ◽  
Classical Behavior ◽  
Diffraction Patterns ◽  
Structurally Stable

Abstract This is a study of proton transmission through planar channels of tungsten, where a proton beam is treated as an ensemble of noninteracting wave packets. For this system, the structural stability manifests in an appearance of caustic lines, and as an equivalence of self-interference produced waveforms with canonical diffraction patterns. We will show that coordination between particle self-interference is an additional manifestation of the structural stability existing only in ensembles. The main focus of the analysis was on the ability of the coordination to produce classical structures. We have found that the structures produced by the self-interference are organized in a very different manner. The coordination can enhance or suppress the quantum aspects of the dynamics. This behavior is explained by distributions of inflection, undulation, and singular points of the ensemble phase function, and their bifurcations. We have shown that the coordination has a topological origin which allows classical and quantum levels of reality to exist simultaneously. The classical behavior of the ensemble emerges out of the quantum dynamics without a need for reduction of the quantum to the classical laws of motion.

scholarly journals Perturbation of the hope flow and transverse foliations

1991 ◽  
Vol 122 ◽  
pp. 75-82 ◽  
Author(s):  
Shigenori Matsumoto ◽  
Atsushi Sato
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Closed Manifold ◽  
Codimension One ◽  
Plane Field

Consider a nonsingular vector field X on a closed manifold Mn. As a matter of fact, X always admits a transverse codimension one plane field, which however may fail to be integrable. In fact it is well known that there are many examples of vector fields which do not admit transverse foliations.

Temperature: a nonnegligible factor for the formation of a structurally stable, self-assembled reduced graphite oxide hydrogel

RSC Advances ◽  
2015 ◽  
Vol 5 (1) ◽  
pp. 10-15 ◽  
Author(s):  
Yang Liu ◽  
Cheng-Lu Liang ◽  
Rui-Ying Bao ◽  
Guo-Qiang Qi ◽  
Wei Yang ◽  
...  
Keyword(s):  
Mechanical Strength ◽  
Structural Stability ◽  
Graphite Oxide ◽  
Electrical Resistance ◽  
Reduced Graphite Oxide ◽  
Different Temperatures ◽  
Self Assembled ◽  
The Stability ◽  
Structurally Stable

The stability of rGO hydrogels prepared at different temperatures was investigated. The network of rGO hydrogel formed at 40 °C showed the best structural stability, the lowest electrical resistance and highest mechanical strength.

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