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We consider skew products over subshifts of finite type in which the fibers are copies of the real line, and we study their mixing properties with respect to any infinite invariant measure given by the product of a Gibbs measure on the base and Lebesgue measure on the fibers. Assuming that the system is accessible, we prove a quantitative version of Krickeberg mixing for a class of observables which is dense in the space of continuous functions vanishing at infinity., Comment: 25 pages

Assuming it preserves an orientation of its stable bundle, any three-dimensional partially hyperbolic diffeomorphism can be used to construct a four-dimensional partially hyperbolic diffeomorphism which is dynamically incoherent. Under the same assumption, the time-one map of any three-dimensional Anosov flow can be used to construct a four-dimensional diffeomorphism which is both absolutely partially hyperbolic and dynamically incoherent. Further results hold in higher dimensions under additional assumptions., Comment: 23 pages, 4 figures

We define a family of $C^1$ functions which we call "nowhere coexpanding functions" that is closed under composition and includes all $C^3$ functions with non-positive Schwarzian derivative. We establish results on the number and nature of the fixed points of these functions, including a generalisation of a classic result of Singer., Comment: 9 pages, 3 figures

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 references

We explicitly construct a dynamically incoherent partially hyperbolic endomorphisms of $\mathbb{T}^2$ in the homotopy class of any linear expanding map with integer eigenvalues. These examples exhibit branching of centre curves along countably many circles, and thus exhibit a form of coherence that has not been observed for invertible systems., Comment: 18 pages, 6 figures

This paper shows any non-Anosov partially hyperbolic diffeomorphism on the 3-torus which is homotopic to Anosov must be accessible.

We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic., Comment: 1 figure

A heterodimensional cycle consists of a pair of heteroclinic connections between two saddle periodic orbits with unstable manifolds of different dimensions. Recent theoretical work on chaotic dynamics beyond the uniformly hyperbolic setting has shown that heterodimensional cycles may occur robustly in diffeomorphisms of dimension at least three. We study a concrete example of a heterodimensional cycle in the continuous-time setting, specifically in a four-dimensional vector field model of intracellular calcium dynamics. By employing advanced numerical techniques, Zhang, Krauskopf and Kirk [Discr. Contin. Dynam. Syst. A 32(8) 2825--2851 (2012)] found that a heterodimensional cycle exists in this model. We investigate the geometric structure of the associated stable and unstable manifolds in the neighbourhood of this heterodimensional cycle, consisting of a single connecting orbit of codimension one and an entire cylinder of structurally stable connecting orbits between two saddle periodic orbits. We employ a boundary-value problem set-up to compute their stable and unstable manifolds, which we visualize in different projections of phase space and as intersection sets with a suitable three-dimensional Poincar\'e section. We show that, locally near the intersection set of the heterodimensional cycle, the manifolds interact as described by the theory for three-dimensional diffeomorphisms. On the other hand, their global structure is more intricate, which is due to the fact that it is not possible to find a Poincar\'e section that is transverse to the flow everywhere. Our results show that the abstract concept of a heterodimensional cycle arises and can be studied in continuous-time models from applications., Comment: 26 pages, 12 figures

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 figure

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 corrected

Let $M$ be a closed 3-manifold which admits an Anosov flow. In this paper we develop a technique for constructing partially hyperbolic representatives in many mapping classes of $M$. We apply this technique both in the setting of geodesic flows on closed hyperbolic surfaces and for Anosov flows which admit transverse tori. We emphasize the similarity of both constructions through the concept of $h$-transversality, a tool which allows us to compose different mapping classes while retaining partial hyperbolicity. In the case of the geodesic flow of a closed hyperbolic surface $S$ we build stably ergodic, partially hyperbolic diffeomorphisms whose mapping classes form a subgroup of the mapping class group $\mathcal{M}(T^1S)$ which is isomorphic to $\mathcal{M}(S)$. At the same time we show that the totality of mapping classes which can be realized by partially hyperbolic diffeomorphisms does not form a subgroup of $\mathcal{M}(T^1S)$. Finally, some of the examples on $T^1S$ are absolutely partially hyperbolic, stably ergodic and robustly non-dynamically coherent, disproving a conjecture by F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures., Comment: 31 pages, 4 figures. To appear in G&T

This paper gives a classification of partially hyperbolic systems in dimension 3 which have at least one torus tangent to the center-stable bundle., Comment: 20 pages, 1 figure

We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow., Comment: 32 pages, 4 figures. Some revisions after referee report. To appear in JMD

We show that a partially hyperbolic system can have at most a finite number of compact center-stable submanifolds. We also give sufficient conditions for these submanifolds to exist and consider the question of whether they can intersect each other., Comment: 16 pages, 2 figures

We show that any weakly partially hyperbolic diffeomorphism on the 2-torus may be realized as the dynamics on a center-stable or center-unstable torus of a 3-dimensional strongly partially hyperbolic system. We also construct examples of center-stable and center-unstable tori in higher dimensions., Comment: 28 pages, 2 figures

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 figures

We develop algorithms built around properties of the transfer operator and Koopman operator which 1) test for possible multiscale dynamics in a given dynamical system, 2) estimate the magnitude of the time-scale separation, and finally 3) distill the reduced slow dynamics on a suitably designed subspace. By avoiding trajectory integration, the developed techniques are highly computationally efficient. We corroborate our findings with numerical simulations of a test problem., Comment: 20 pages

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 pages

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 figures

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 pages

A classification of partially hyperbolic diffeomorphisms on 3-dimensional manifolds with (virtually) solvable fundamental group is obtained. If such a diffeomorphism does not admit a periodic attracting or repelling two-dimensional torus, it is dynamically coherent and leaf conjugate to a known algebraic example. This classification includes manifolds which support Anosov flows, and it confirms conjectures by Rodriguez Hertz--Rodriguez Hertz--Ures and Pujals in the specific case of solvable fundamental group., Comment: 36 pages, 3 figures. To appear in the Journal of Topology

Certain families of manifolds which support Anosov flows do not support expanding, quasi-isometric foliations., Comment: 10 pages

We show the existence of a family of manifolds on which all (pointwise or absolutely) partially hyperbolic systems are dynamically coherent. This family is the set of 3-manifolds with nilpotent, non-abelian fundamental group. We further classify the partially hyperbolic systems on these manifolds up to leaf conjugacy. We also classify those systems on the 3-torus which do not have an attracting or repelling periodic 2-torus. These classification results allow us to prove some dynamical consequences, including existence and uniqueness results for measures of maximal entropy and quasi-attractors., Comment: 28 pages, 1 figure

If a non-ergodic, partially hyperbolic diffeomorphism on the 3-torus is homotopic to an Anosov diffeomorphism $A$, it is topologically conjugate to $A$., Comment: 20 pages

An Anosov diffeomorphism is topologically conjugate to an infranilmanifold automorphism if and only if it has polynomial Global Product Structure., Comment: 5 pages

If the stable, center, and unstable foliations of a partially hyperbolic system are quasi-isometric, the system has Global Product Structure. This result also applies to Anosov systems and to other invariant splittings. If a partially hyperbolic system on a manifold with abelian fundamental group has quasi-isometric stable and unstable foliations, the center foliation is without holonomy. If, further, the system has Global Product Structure, then all center leaves are homeomorphic., Comment: 18 pages, 1 figure

Every partially hyperbolic diffeomorphism on a 3-dimensional nilmanifold is leaf conjugate to a nilmanifold automorphism., Comment: 30 pages, 6 figures