A volume preserving flow with essential coexistence of zero and non-zero Lyapunov exponents

2012 ◽  
Vol 33 (6) ◽  
pp. 1748-1785 ◽  
Author(s):  
JIANYU CHEN ◽  
HUYI HU ◽  
YAKOV PESIN
Keyword(s):  
Lyapunov Exponents ◽  
Smooth Manifold ◽  
Dense Subset ◽  
Three Dimensional ◽  
Frequency Vector ◽  
Almost Everywhere ◽  
Invariant Submanifolds ◽  
Hyperbolic Behavior ◽  
Volume Preserving ◽  
Full Volume

AbstractWe demonstrate essential coexistence of hyperbolic and non-hyperbolic behavior in the continuous-time case by constructing a smooth volume preserving flow on a five-dimensional compact smooth manifold that has non-zero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of three-dimensional invariant submanifolds, and on each of these submanifolds the flow is linear with Diophantine frequency vector.

scholarly journals -openness of non-uniform hyperbolic diffeomorphisms with bounded -norm

2019 ◽  
Vol 40 (11) ◽  
pp. 3078-3104
Author(s):  
CHAO LIANG ◽  
KARINA MARIN ◽  
JIAGANG YANG
Keyword(s):  
Lyapunov Exponents ◽  
Dense Subset ◽  
Topological Properties ◽  
Hyperbolic Diffeomorphisms ◽  
Uniform Hyperbolicity ◽  
Partially Hyperbolic ◽  
The Stability ◽  
Bounded Norm ◽  
Volume Preserving

We study the $C^{1}$-topological properties of the subset of non-uniform hyperbolic diffeomorphisms in a certain class of $C^{2}$ partially hyperbolic symplectic systems which have bounded $C^{2}$ distance to the identity. In this set, we prove the stability of non-uniform hyperbolicity as a function of the diffeomorphism and the measure, and the existence of an open and dense subset of continuity points for the center Lyapunov exponents. These results are generalized to the volume-preserving context.

The Lyapunov exponents of generic zero divergence three-dimensional vector fields

2007 ◽  
Vol 27 (5) ◽  
pp. 1445-1472 ◽  
Author(s):  
MÁRIO BESSA
Keyword(s):  
Vector Field ◽  
Lyapunov Exponents ◽  
Vector Fields ◽  
Dense Subset ◽  
Three Dimensional ◽  
Invariant Set ◽  
Compact Manifolds ◽  
Dimensional Vector ◽  
Dominated Splitting

AbstractWe prove that for a C1-generic (dense Gδ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point p∈M that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C1-dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. p∈M that either the Lyapunov exponents at p are zero or p belongs to a compact invariant set with dominated splitting for the linear Poincaré flow.

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.

scholarly journals Deep learning based MRI contrast synthesis using full volume prediction

2021 ◽  
Author(s):  
Jose V Manjon ◽  
Jose E Romero ◽  
Pierrick Coupé
Keyword(s):  
Three Dimensional ◽  
Patient Comfort ◽  
Segmentation Method ◽  
Resonance Imaging ◽  
Complementary Information ◽  
Mri Contrast ◽  
Ambiguity Reduction ◽  
Magnetic Resonance Imaging Mri ◽  
Time Limitations ◽  
Full Volume

Abstract In Magnetic Resonance Imaging (MRI), depending on the image acquisition settings, a large number of image types or contrasts can be generated showing complementary information of the same imaged subject. This multi-spectral information is highly beneficial since can improve MRI analysis tasks such as segmentation and registration, thanks to pattern ambiguity reduction. However, the acquisition of several contrasts is not always possible due to time limitations and patient comfort constraints. Contrast synthesis has emerged recently as an approximate solution to generate other image types different from those acquired originally. Most of the previously proposed methods for contrast synthesis are slice-based which result in intensity inconsistencies between neighbor slices when applied in 3D. We propose the use of a 3D convolutional neural network (CNN) capable of generating T2 and FLAIR images from a single anatomical T1 source volume. The proposed network is a 3D variant of the UNet that processes the whole volume at once breaking with the inconsistency in the resulting output volumes related to 2D slice or patch-based methods. Since working with a full volume at once has a huge memory demand we have introduced a spatial-to-depth and a reconstruction layer that allows working with the full volume but maintain the required network complexity to solve the problem. Our approach enhances the coherence in the synthesized volume while improving the accuracy thanks to the integrated three-dimensional context-awareness. Finally, the proposed method has been validated with a segmentation method, thus demonstrating its usefulness in a direct and relevant application.

Passive scalars, three-dimensional volume-preserving maps, and chaos

1988 ◽  
Vol 50 (3-4) ◽  
pp. 529-565 ◽  
Author(s):  
Mario Feingold ◽  
Leo P. Kadanoff ◽  
Oreste Piro
Keyword(s):  
Three Dimensional ◽  
Passive Scalars ◽  
Volume Preserving
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