Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow

Journal of Topology and Analysis ◽

10.1142/s1793525321500527 ◽

2021 ◽

pp. 1-57

Author(s):

Alberto Abbondandolo ◽

Pietro Majer

Keyword(s):

Structural Stability ◽

Closed Manifold ◽

Geometric Properties ◽

Unstable Manifolds

We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible invariant stable foliations” that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse–Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse–Smale gradient-like flow on a closed manifold [Formula: see text] are the open cells of a CW-decomposition of [Formula: see text].

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Correction to 'Geometric Properties of Resistive Nonlinear n-Ports:Transversality, Structural Stability, Reciprocity and Anti-Reciprocity'

IEEE Transactions on Circuits and Systems ◽

10.1109/tcs.1981.1084949 ◽

1981 ◽

Vol 28 (2) ◽

pp. 168-168

Author(s):

L. Chua ◽

T. Matsumoto ◽

S. Ichiraku

Keyword(s):

Structural Stability ◽

Geometric Properties

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On the ergodicity of hyperbolic Sinaĭ–Ruelle–Bowen measures II: the low-dimensional case

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.136 ◽

2017 ◽

Vol 38 (8) ◽

pp. 3042-3061

Author(s):

MICHIHIRO HIRAYAMA ◽

NAOYA SUMI

Keyword(s):

Probability Measure ◽

Closed Manifold ◽

Dimensional Case ◽

Anosov Diffeomorphism ◽

Stable And Unstable Manifolds ◽

Higher Dimensional ◽

Unstable Manifolds ◽

Low Dimensional

In this paper, we consider diffeomorphisms on a closed manifold $M$ preserving a hyperbolic Sinaĭ–Ruelle–Bowen probability measure $\unicode[STIX]{x1D707}$ having intersections for almost every pair of stable and unstable manifolds. In this context, we show the ergodicity of $\unicode[STIX]{x1D707}$ when the dimension of $M$ is at most three. If $\unicode[STIX]{x1D707}$ is smooth, then it is ergodic when the dimension of $M$ is at most four. As a byproduct of our arguments, we obtain sufficient (topological) conditions which guarantee that there exists at most one hyperbolic ergodic Sinaĭ–Ruelle–Bowen probability measure. Even in higher dimensional cases, we show that every transitive topological Anosov diffeomorphism admits at most one hyperbolic Sinaĭ–Ruelle–Bowen probability measure.

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Geometric properties of resistive nonlinear n-ports: Transversality, structural stability, reciprocity, and anti-reciprocity

IEEE Transactions on Circuits and Systems ◽

10.1109/tcs.1980.1084866 ◽

1980 ◽

Vol 27 (7) ◽

pp. 577-603 ◽

Cited By ~ 26

Author(s):

L. Chua ◽

T. Matsumoto ◽

S. Ichiraku

Keyword(s):

Structural Stability ◽

Geometric Properties

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Remarks on structurally stable proper foliations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100071954 ◽

1994 ◽

Vol 115 (1) ◽

pp. 111-120 ◽

Cited By ~ 3

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.

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Time-resolved high-resolution electron microscopy of structural stability in mgo clusters

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100165823 ◽

1996 ◽

Vol 54 ◽

pp. 672-673

Author(s):

T. Kizuka ◽

N. Tanaka

Keyword(s):

Electron Microscopy ◽

High Resolution ◽

Magnesium Oxide ◽

Structural Stability ◽

High Resolution Electron Microscopy ◽

Video Tape ◽

Solid State Physics ◽

Time Resolved ◽

Resolution Electron ◽

Structural And Functional Properties

Structure and stability of atomic clusters have been studied by time-resolved high-resolution electron microscopy (TRHREM). Typical examples are observations of structural fluctuation in gold (Au) clusters supported on silicon oxide films, graphtized carbon films and magnesium oxide (MgO) films. All the observations have been performed on the clusters consisted of single metal element. Structural stability of ceramics clusters, such as metal-oxide, metal-nitride and metal-carbide clusters, has not been observed by TRHREM although the clusters show anomalous structural and functional properties concerning to solid state physics and materials science.In the present study, the behavior of ceramic, magnesium oxide (MgO) clusters is for the first time observed by TRHREM at 1/60 s time resolution and at atomic resolution down to 0.2 nm.MgO and gold were subsequently deposited on sodium chloride (001) substrates. The specimens, single crystalline MgO films on which Au particles were dispersed were separated in distilled water and observed by using a 200-kV high-resolution electron microscope (JEOL, JEM2010) equipped with a high sensitive TV camera and a video tape recorder system.

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