Dynamics on Bi-Lagrangian Structures and Cherry maps

Abstract

We consider a bi-Lagrangian structure $(\omega ,{\mathcal{F}}_1,{\mathcal{F}}_2)$ on a manifold $M$; that is, $(M,\omega ,{\mathcal{F}}_1,{\mathcal{F}}_2)$ is a bi-Lagrangian manifold. We prolong bi-Lagrangian structures on $M$ and lift a given dynamics to its tangent and cotangent bundles in several ways. In some cases, we show that the lifted structures are affine. In the case of the two-dimensional torus ${\mathbb{T}}^2$, we prove that an extension of the same dynamics to pairs of so-called Cherry vector fields induces a conjugation action on a subset of Cherry maps, namely circle maps with a flat interval. Furthermore, we define linear connections for certain Cherry maps using bi-Lagrangian data. Finally, we show that maps generated by a bi-Lagrangian structure belong a priori to different Hölder homeomorphism classes.

14 pages