We prove the existence of an open and dense set 𝔇⊂Homeo0(𝕋2) (where Homeo0(𝕋2) denotes the 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 diffeomorphisms 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).

We show that if f is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of f is positive. Further, the entropy is shown to be associated to a C 0 -robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is uniformly large but the entropy approaches zero as much as desired.The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.

For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincaré-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set U vanishes, then either there is a fixed point in the boundary, or the boundary of U is contained in the basin of a finite family of topological "rotational" attractors. This description strongly improves a previous result by Cartwright and Littlewood, by passing from the prime ends compactification to the ambient space. Moreover, the dynamics in a neighborhood of the boundary is semiconjugate to a very simple model dynamics on a planar graph. Other applications involve the decomposability of invariant continua, and realization of rotation numbers by periodic points on circloids.

Abstract. In the context of the Franks-Misiurewicz Conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterise these maps by the existence of an invariant 'foliation' by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalising a well-known result of M. Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector.As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.

We prove that for every ǫ > 0 there exists a minimal diffeomorphism f : T 2 → T 2 of class C 3−ǫ and semiconjugate to an ergodic traslation, with the following properties: zero entropy, sensitivity with respect to initial conditions, and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Mañé's example of derived from Anosov diffeomorphism on T 3 . On Mañe's Derived from Anosov diffeomorphismIn [M1] R.Mañé construct an example on T 3 which is robustly-transitive but not Anosov. This is known as Mañe's Derive from Anosov diffeomorphisms due to the construction: it begins with an Anosov linear map on T 3 with partially hyperbolic structure E s ⊕ E c ⊕ E u and modifies it in a neighborhood of the fixed point in order to change the unstable index of it (and preserving the partially hyperbolic structure). See Figure 1.Let us be more precise. Let T 3 = R 3 / Z 3 be the three dimensional torus and denote by π : R 3 → T 3 the canonical projection, and set p = π(0). PSfrag replacements