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 $C^1$ 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 $\mathbb{T}^2$ are dynamically coherent and leaf conjugate to linear toral endomorphisms. Moreover, we give an example of a partially hyperbolic endomorphism on $\mathbb{T}^2$ 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