Rational Polygons as Rotation Sets of Generic Homeomorphisms of the Two-Torus

We prove the existence of an open and dense set D\subset? Homeo0(T2) (set of toral homeomorphisms homotopic to the identity) such that the rotation set of any element in D is a rational polygon. We also extend this result to the set of axiom A dif- feomorphisms in Homeo0(T2). Further we observe the existence of minimal sets whose rotation set is a non-trivial segment, for an open set in Homeo0(T2)

New Rotation Sets in a Family of Torus Homeomorphisms

We construct a family $\{\Phi_t\}_{t\in[0,1]}$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $\rho(\Phi_t)$ can be described explicitly. We analyze the bifurcations and typical behavior of rotation sets in the family, providing insight into the general questions of toral rotation set bifurcations and prevalence. We show that there is a full measure subset of $[0,1]$, consisting of infinitely many mutually disjoint non-trivial closed intervals, on each of which the rotation set mode locks to a constant polygon with rational vertices; that the generic rotation set in the Hausdorff topology has infinitely many extreme points, accumulating on a single totally irrational extreme point at which there is a unique supporting line; and that, although $\rho(t)$ varies continuously with $t$, the set of extreme points of $\rho(t)$ does not. The family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set, or by any directional ergodic measure.Comment: Author's accepted version. The final publication is available at Springer via http://dx.doi.org/10.1007/s00222-015-0628-

A toral diffeomorphism with a non-polygonal rotation set

We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon

Laminations and groups of homeomorphisms of the circle

If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that pi_1(M) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions on circles. In particular, we show this for tight essential laminations with solid torus guts. We also show that pseudo-Anosov flows induce actions on circles. In all cases, these actions can be made into faithful ones, so pi_1(M) is isomorphic to a subgroup of Homeo(S^1). In addition, we show that the fundamental group of the Weeks manifold has no faithful action on S^1. As a corollary, the Weeks manifold does not admit a tight essential lamination, a pseudo-Anosov flow, or a taut foliation. Finally, we give a proof of Thurston's universal circle theorem for taut foliations based on a new, purely topological, proof of the Leaf Pocket Theorem.Comment: 50 pages, 12 figures. Ver 2: minor improvement

Rotation sets and almost periodic sequences

We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segment as well as non-convex and even plane-separating continua. This shows that the restriction which hold for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences