Kamlesh Parwani

On 3-manifolds that support partially hyperbolic diffeomorphisms

Let M be a closed 3-manifold that supports a partially hyperbolic diffeomorphism f. If π1(M) is nilpotent, the induced action of f* on is partially hyperbolic. If π1(M) is almost nilpotent or if π1(M) has subexponential growth, M is finitely covered by a circle bundle over the torus. If π1(M) is almost solvable, M is finitely covered by a torus bundle over the circle. Furthermore, there exist infinitely many hyperbolic 3-manifolds that do not support dynamically coherent partially hyperbolic diffeomorphisms; this list includes the Weeks manifold.If f is a strong partially hyperbolic diffeomorphism on a closed 3-manifold M and if π1(M) is nilpotent, then the lifts of the stable and unstable foliations are quasi-isometric in the universal cover of M. It then follows that f is dynamically coherent.We also provide a sufficient condition for dynamical coherence in any dimension. If f is centre-bunched and if the centre-stable and centre-unstable distributions are Lipschitz, then the partially hyperbolic diffeomorphism f must be dynamically coherent.

Symmetric random walks on Homeo+(R)

Dedicated to John Milnor on his 80th anniversary We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of “global stability at a finite distance”: roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for the study of group actions on the line. For instance, we show that under a suitable change of the coordinates, the drift of every point becomes zero provided that the action is minimal. As a byproduct, we recover the fact that every finitely generated group of homeomorphisms of the real line is topologically conjugate to a group of (globally) Lipschitz homeomorphisms. Moreover, we show that such a conjugacy may be chosen in such a way that the displacement of each element is uniformly bounded. 1. Introduction. In this article, we study symmetric random walks on finitely generated groups of (orientation-preserving) homeomorphisms of the real line. The results presented here fit into the general framework of systems of iterated random functions [8]. However, besides the lack of compactness of

C^1 actions of the mapping class group on the circle

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C 1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C 1 faithful actions on the circle. We also prove that for n ≥ 6, any C 1 action of Aut(Fn) or Out(Fn) on the circle factors through an action of Z/2Z.

Rotation sets for networks of circle maps.

We consider continuous maps of the torus, homotopic to the identity, that arise from systems of coupled circle maps and discuss the relationship between network architecture and rotation sets. Our main result is that when the map on the torus is invertible, network architecture can force the set of rotation vectors to lie in a low-dimensional subspace. In particular, the rotation set for an all-to-all coupled system of identical cells must be a subset of a line.

Monotone Periodic Orbits for Torus Homeomorphisms

Let f be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector ( p , r ), then f has a topologically monotone periodic orbit with the same rotation vector.

Harmonic functions on ℝ-covered foliations

Let ( M ,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that ( M ,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

Zero entropy subgroups of mapping class groups

Given a group action on a surface with a finite invariant set we investigate how the algebraic properties of the induced group of permutations of that set affects the dynamical properties of the group. Our main result shows that in many circumstances if the induced permutation group is not solvable then among the homeomorphisms in the group there must be one with a pseudo-Anosov component. We formulate this in terms of the mapping class group relative to the finite set and show the stronger result that in many circumstances (e.g. if the surface has boundary) if this mapping class group has no elements with pseudo-Anosov components then it is itself solvable.