Expansive homeomorphisms and hyperbolic diffeomorphisms on 3-manifolds

1996 ◽  
Vol 16 (3) ◽  
pp. 591-622 ◽  
Author(s):  
José L. Vieitez
Keyword(s):  
Basic Result ◽  
Three Dimensional ◽  
Classification Problem ◽  
Topological Manifold ◽  
Anosov Diffeomorphism ◽  
Hyperbolic Diffeomorphisms ◽  
Topological Manifolds ◽  
Stable And Unstable Sets

AbstractThis paper is a contribution to the classification problem of expansive homeomorphisms. Let M be a compact connected oriented three dimensional topological manifold without boundary and f: M → M an expansive homeomorphism.We show that if the topologically hyperbolic period points of f are dense in M then M = , and f is conjugate to an Anosov diffeomorphism. This follows from our basic result: for such a homeomorphism, all stable and unstable sets are (tamely embedded) topological manifolds.

scholarly journals Classification of partially hyperbolic diffeomorphisms under some rigid conditions

2020 ◽  
pp. 1-12
Author(s):  
PABLO D. CARRASCO ◽  
ENRIQUE PUJALS ◽  
FEDERICO RODRIGUEZ-HERTZ
Keyword(s):  
Three Dimensional ◽  
Skew Product ◽  
Anosov Diffeomorphism ◽  
Anosov Flow ◽  
Hyperbolic Diffeomorphisms ◽  
Finite Covers ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract Consider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the second author to this restricted setting.

scholarly journals Retinal vessel segmentation with constrained-based nonnegative matrix factorization and 3D modified attention U-Net

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yang Yu ◽  
Hongqing Zhu
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Computational Cost ◽  
Three Dimensional ◽  
Nonnegative Matrix ◽  
Retinal Vessel ◽  
Classification Problem ◽  
Image Resolution ◽  
Retinal Vessels ◽  
Segmentation Task

AbstractDue to the complex morphology and characteristic of retinal vessels, it remains challenging for most of the existing algorithms to accurately detect them. This paper proposes a supervised retinal vessels extraction scheme using constrained-based nonnegative matrix factorization (NMF) and three dimensional (3D) modified attention U-Net architecture. The proposed method detects the retinal vessels by three major steps. First, we perform Gaussian filter and gamma correction on the green channel of retinal images to suppress background noise and adjust the contrast of images. Then, the study develops a new within-class and between-class constrained NMF algorithm to extract neighborhood feature information of every pixel and reduce feature data dimension. By using these constraints, the method can effectively gather similar features within-class and discriminate features between-class to improve feature description ability for each pixel. Next, this study formulates segmentation task as a classification problem and solves it with a more contributing 3D modified attention U-Net as a two-label classifier for reducing computational cost. This proposed network contains an upsampling to raise image resolution before encoding and revert image to its original size with a downsampling after three max-pooling layers. Besides, the attention gate (AG) set in these layers contributes to more accurate segmentation by maintaining details while suppressing noises. Finally, the experimental results on three publicly available datasets DRIVE, STARE, and HRF demonstrate better performance than most existing methods.

On the integrability of intermediate distributions for Anosov diffeomorphisms

1995 ◽  
Vol 15 (2) ◽  
pp. 317-331 ◽  
Author(s):  
M. Jiang ◽  
Ya B. Pesin ◽  
R. de la Llave
Keyword(s):  
Finite Number ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Stable Foliation

AbstractWe study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

-Hölder linearization of hyperbolic diffeomorphisms with resonance

2014 ◽  
Vol 36 (1) ◽  
pp. 310-334 ◽  
Author(s):  
WENMENG ZHANG ◽  
WEINIAN ZHANG
Keyword(s):  
Hölder Continuity ◽  
Three Dimensional ◽  
Holder Continuity ◽  
Hyperbolic Diffeomorphisms ◽  
Analytic Mapping

Concerning hyperbolic diffeomorphisms, one expects a better smoothness of linearization, but it may be confined by resonance among eigenvalues. Hartman gave a three-dimensional analytic mapping with resonance which cannot be linearized by a Lipschitz conjugacy. Since then, efforts have been made to give the ${\it\alpha}$-Hölder continuity of the conjugacy and hope the exponent ${\it\alpha}<1$ can be as large as possible. Recently, it was proved for some weakly resonant hyperbolic diffeomorphisms that ${\it\alpha}$ can be as large as we expect. In this paper we prove that this result holds for all $C^{\infty }$ weakly resonant hyperbolic diffeomorphisms.

scholarly journals Embedding Topological $n$-Manifolds into Compact Nonstandard Topological $n$-Manifolds with Boundary

2021 ◽  
Author(s):  
Yu-Lin Chou
Keyword(s):  
Topological Manifold ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Topological Manifolds ◽  
One Point Compactification ◽  
Main Message

We show as a main message that there is a simple dimension-preserving way to openly and densely embed every topological manifold into a compact ``nonstandard'' topological manifold with boundary.This class of ``nonstandard'' topological manifolds with boundary contains the usual topological manifolds with boundary.In particular,the Alexandroff one-point compactification of every given topological $n$-manifold is a ``nonstandard'' topological $n$-manifold with boundary.

Simultaneous dense and non-dense orbits for certain partially hyperbolic diffeomorphisms

2018 ◽  
Vol 40 (4) ◽  
pp. 1083-1107
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Space ◽  
Anosov Diffeomorphism ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Dense Orbits ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism ◽  
Stable Subspace

Let$g:M\rightarrow M$be a$C^{1+\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on$M$. We show that, if$f:M\rightarrow M$is a$C^{1+\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of$f$and$g$span the whole tangent space at some point on$M$, the set of points that equidistribute under$g$but have non-dense orbits under$f$has full Hausdorff dimension. The same result is also obtained when$M$is the torus and$f$is a toral endomorphism whose center-stable subspace does not contain the stable subspace of$g$at some point.

A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

1993 ◽  
Vol 48 (4) ◽  
pp. 535-550 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Classification Problem ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Point Symmetry Group ◽  
Relativistic Case ◽  
Lie Point Symmetry

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group Gare an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra Gof the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of Gvary in case of 3 ≤ s ≤ r, depending on the solvability of G. If the r-dimensional Lie algebra Gof the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of Gone has to determine the optimal subsystems of semisimple subalgebras of Gin order to construct the full optimal systems of s-dimensional subalgebras of Gwith 3 ≤ s ≤ r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra Gof the eight-dimensional Lie point symmetry group Gadmitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.

scholarly journals Automated classification of blood cell neutrophils.

1977 ◽  
Vol 25 (7) ◽  
pp. 633-640 ◽  
Author(s):  
J K Mui ◽  
K S Fu ◽  
J W Bacus
Keyword(s):  
Blood Cell ◽  
White Blood Cell ◽  
Classification Accuracy ◽  
Three Dimensional ◽  
Classification Problem ◽  
Differential Count ◽  
Two Dimensional ◽  
Automated Classification ◽  
Exact Shape

The classification of white blood cell neutrophils into band neutrophils (bands) and segmented neutrophils (segs) is a subproblem of the white blood cell differential count. This classification problem is not well defined for at least two reasons: (a) there are no unique quantitative definitions for bands and segs and (b) existing definitions use the shape of the nucleus as the only discriminating criterion. When cells are classified on a slide, decisions are made from the two-dimensional views of these three-dimensional cells. A problem arises because the exact shape of the nucleus becomes indeterminate when the nucleus overlaps so that the filament is hidden. To assess the importance of this problem, this paper quantitates the classification errors due to overlapped nuclei (ON). The results indicate that, using only neutrophils without ON, the classification accuracy is 89%. For neutrophils with ON, the classification accuracy is 65%. This suggests a classification strategy of first classifying neutrophils into three categories: (a) bands without ON, (b) segs without ON and (c) neutrophils with ON. Category III can then be further classified into segs and bands by other stretegies.

scholarly journals Partial hyperbolicity and classification: a survey

2016 ◽  
Vol 38 (2) ◽  
pp. 401-443 ◽  
Author(s):  
ANDY HAMMERLINDL ◽  
RAFAEL POTRIE
Keyword(s):  
Fundamental Group ◽  
Three Dimensional ◽  
Group Classification ◽  
Partial Hyperbolicity ◽  
Hyperbolic Diffeomorphisms ◽  
Higher Dimensional ◽  
Partially Hyperbolic ◽  
Anosov Flows ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Solvable Fundamental Group

This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples derived from Anosov flows.

scholarly journals Tubular neighbourhoods for submersions of topological manifolds

1973 ◽  
Vol 8 (1) ◽  
pp. 93-102
Author(s):  
David B. Gauld
Keyword(s):  
Topological Manifold ◽  
Main Application ◽  
Compact Neighbourhood ◽  
Topological Manifolds

Let φ: M → N be a submersion from a metrizable manifold to any (topological) manifold, let B ⊂ M be compact, y є N and C ⊂ φ−1(y) be a compact neighbourhood (in φ−1(y)) of B ∩ φ−1(y). It is proven that there is a neighbourhood U of y in N and an embedding ε: U × C → M such that φε is projection on the first factor, ε(y, x) = x for each x ε C, and B ∩ φ−1 (U) ⊂ ε(U×C). The main application given is to topological foliations, it being shown that if C is a compact regular leaf of a foliation F on M then every neighbourhood of C contains a saturated neighbourhood which is the union of compact regular leaves of F.

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