29 research outputs found
On expanding foliations
Certain families of manifolds which support Anosov flows do not support
expanding, quasi-isometric foliations.Comment: 10 page
Partial hyperbolicity on 3-dimensional nilmanifolds
Every partially hyperbolic diffeomorphism on a 3-dimensional nilmanifold is
leaf conjugate to a nilmanifold automorphism.Comment: 30 pages, 6 figure
Ergodic components of partially hyperbolic systems
This paper gives a complete classification of the possible ergodic
decompositions for certain open families of volume-preserving partially
hyperbolic diffeomorphisms. These families include systems with compact center
leaves and perturbations of Anosov flows under conditions on the dimensions of
the invariant subbundles. The paper further shows that the non-open
accessibility classes form a lamination and gives results about the
accessibility classes of non-volume-preserving systems.
Note: this document has been modified slightly from earlier preprints. The
numbering of sections was changed to match the published version and an erratum
has been added to the end.Comment: 56 pages, 3 figure
Polynomial Global Product Structure
An Anosov diffeomorphism is topologically conjugate to an infranilmanifold
automorphism if and only if it has polynomial Global Product Structure.Comment: 5 page
Partial hyperbolicity and classification: a survey
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.Comment: 49 pages, 6 figure
Partially hyperbolic surface endomorphisms
We prove that a class of weakly partially hyperbolic endomorphisms on
are dynamically coherent and leaf conjugate to linear toral
endomorphisms. Moreover, we give an example of a partially hyperbolic
endomorphism on which does not admit a centre foliation.Comment: 13 pages, 1 figur
Horizontal vector fields and Seifert fiberings
This paper gives a classification of the topology of vector fields which are
nowhere tangent to the fibers of a Seifert fibering.Comment: 35 pages, 1 figure. The v2 version of this preprint significantly
differs from v1. It is over twice as long and gives much more detail. Some
statements involving manifolds with multiple Seifert fiberings were also
correcte
Center bunching without dynamical coherence
We answer a question of Burns and Wilkinson, showing that there are open
families of volume-preserving partially hyperbolic diffeomorphisms which are
accessible and center bunched and neither dynamically coherent nor Anosov. We
also show in the volume-preserving setting that any diffeomorphism which is
partially hyperbolic and Anosov may be isotoped to a diffeomorphism which is
partially hyperbolic and not Anosov.Comment: 8 page
A computational method to extract macroscopic variables and their dynamics in multiscale systems
This paper introduces coordinate-independent methods for analysing multiscale
dynamical systems using numerical techniques based on the transfer operator and
its adjoint. In particular, we present a method for testing whether an
arbitrary dynamical system exhibits multiscale behaviour and for estimating the
time-scale separation. For systems with such behaviour, we establish techniques
for analysing the fast dynamics in isolation, extracting slow variables for the
system, and accurately simulating these slow variables at a large time step. We
illustrate our method with numerical examples and show how the reduced slow
dynamics faithfully represents statistical features of the full dynamics which
are not coordinate dependent.Comment: 30 page
Classification of partially hyperbolic surface endomorphisms
We show that in the absence of periodic centre annuli, a partially hyperbolic
surface endomorphism is dynamically coherent and leaf conjugate to its
linearisation. We proceed to characterise the dynamics in the presence of
periodic centre annuli. This completes a classification of partially hyperbolic
surface endomorphisms.Comment: 24 pages, 2 figures, amended reference