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arXiv · 2508.12350

Dynamics on Bi-Lagrangian Structures and Cherry maps

Bertuel Tangue Ndawa
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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 H\"{o}lder homeomorphism classes.

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Bertuel Tangue Ndawa. 2025-08-17. Dynamics on Bi-Lagrangian Structures and Cherry maps. https://arxiv.org/abs/2508.12350

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