# Dynamic of Pair of some Distributions: Bi-lagrangian structure and its
prolongations on the (co)tangent bundles, and Cherry flow

Bertuel Tangue NDAWA
University Institute of Technology
of the University of Ngaoundere
Box: 455, Ngaoundere (Cameroon)
<EMAIL_ADDRESS>
August 27, 2024

In the memory of Dad NDAWA Joseph, Dad TCHOUATEUN Pierre, and Dad PAHANE
Isaac.

## Abstract

We consider a bi-Lagrangian manifold
$(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$. That is, $\omega$ is a 2-form,
closed and non-degenerate (called symplectic form) on $M$, and
$(\mathcal{F}_{1},\mathcal{F}_{2})$ is a pair of transversal Lagrangian
foliations on the symplectic manifold $(M,\omega)$. In this case,
$(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ is a bi-Lagrangian structure on
$M$.

In this paper, we prolong a bi-Lagrangian structure on $M$ on its tangent
bundle $TM$ and its cotangent bundle $T^{*}M$ in different ways. As a
consequence some dynamics on the bi-Lagrangian structure of $M$ can be
prolonged as dynamics on the bi-Lagrangian structure of $TM$ and $T^{*}M$.
Observe that a pair of transversal vector fields without singularity on the
2-torus $\mathbb{T}^{2}=\mathbb{S}^{1}\times\mathbb{S}^{1}$ endowed with a
symplectic form defines a bi-Lagrangian structure on $\mathbb{T}^{2}$. This
sparked our curiosity. By studying the dynamic of pairs of vector fields on
$\mathbb{T}^{2}$, we found that some circle maps with a flat piece (called
Cherry maps) can be generated by a pair of vector fields. Moreover, the push
forward action of the set of diffeomorphisms $\mathbb{T}^{2}$ on the set of
its vector fields induces a conjugation action on the set of generated Cherry
maps.

Keywords: Symplectic, Symplectomorphism, Bi-Lagrangian, Para-Kähler, Hess
connection, Cherry vector field, Circle map, Flat piece.

MSC2010: 53D05, 53D12.

Acknowledgment:

##  1 Introduction

The bi-Lagrangian manifolds has been intensively explored in the past years,
see [1, 2, 3, 4, 7, 8, 12, 17]. Among the many reasons to study them, they are
the areas of geometric quantization (see [12]) and of Koszul-Vinberg
Cohomology (see [4]). The data of a bi-Lagrangian manifold
$(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ induces in a one-to-one way a
para-Kähler structure $(G,F)$ on $M$ (that is, $G$ is a pseudo-Riemannian
metric on $M$ and $F$ is a para-complex structure on $M$ which permutes with
$G$ in the following sens: $G(F(\cdot),F(\cdot))=-G(\cdot,\cdot)$). The three
tensors $\omega$, $G$ and $F$ are linked by the relation:
$\omega(\cdot,\cdot)=G(F(\cdot),\cdot)$, see [7, 8, 9, 17]. Therefore, a bi-
Lagrangian manifold put together symplectic, semi-Riemannian, and almost
product structures (note that, if $(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$
is a bi-Lagrangian manifold, then $(\mathcal{F}_{1},\mathcal{F}_{2})$ is a bi-
Lagrangian structure on the symplectic manifold $(M,\omega)$). In [12], the
author proves that a bi-Lagrangian manifold
$(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ have a unique torsion free
(torsionless) connection $\nabla$ called Hess (or bi-Lagrangian) connection
which parallelizes $\omega$ ($\nabla$ is a symplectic connection) and
preserves both foliations. The explicit formula of a Hess connection was given
in [1, 2, 3]. In the case of 2-torus $\mathbb{T}^{2}$, a Lagrangian foliation
can be defined by a vector field without singularity. Thus the study of
Lagrangian foliation on the 2-torus is a complement to the study of the
dynamic of some vector fields on $\mathbb{T}^{2}$ with a finite number of
singularities, see [5, 18, 19, 22].

This paper extends on the one hand, the results in [26], and on the other
hand, some results in [18, 22].

Before we can explain more precisely and prove our results, it is necessary to
present some definitions, fix some notations and formulate some known results
we need.

###  1.1 Basics definitions, properties and notations

We assume that all the objects are smooth throughout this paper unless
otherwise stated.

####  1.1.1 Foliation

Let $M$ be an $m$-manifold. We define a $k$-dimensional foliation
$\mathcal{F}$ on $M$ to be a decomposition of $M$ into a union of disjoint,
non-empty, connected, immersed $k$-dimensional submanifolds $\\{S_{x}\\}_{x\in
M}$, called the leaves of the foliation, with the following property (called
completely integrable property): every point $y$ in $M$ has a coordinate chart
$(U,y^{1},\dots,y^{m})$ such that for each leaf $S_{x}$ the components of
$U\cap S_{x}$ are described by the equations
$y^{k+1}\mbox{=constant},\dots,y^{m}=\mbox{constant}$, see [14, p. 370], [15,
p. 501].

The expressions $T\mathcal{F}\subset TM$ and
$\Gamma\left(T\mathcal{F}\right)=\Gamma\left(\mathcal{F}\right)\subset\Gamma\left(TM\right)=\mathfrak{X}(M)$
(the set of vector fields on $M$) denote the tangent bundle of $\mathcal{F}$
and the set of sections of $T\mathcal{F}$ respectively. For each point $y\in
M$ the vector subspace $T_{y}S_{y}\subset T_{y}M$ is called the bundle tangent
of $\mathcal{F}$ over $y$ and is denoted by $\mathcal{F}_{y}$ or
$T_{y}\mathcal{F}$.

The Lie bracket of two vector fields $X$, $Y$ is that defined by
$[X,Y]:=X\circ Y-Y\circ X$.

Note that the completely integrable property of a foliation $\mathcal{F}$
means that $\Gamma\left(\mathcal{F}\right)$ is stable under the Lie bracket;
that is, if $X,Y\in\Gamma\left(\mathcal{F}\right)$ then
$[X,Y]\in\Gamma\left(\mathcal{F}\right)$. This is the Frobenius Theorem, see
[15, p. 496].

Let $\psi:M\longrightarrow N$ be a diffeomorphism. The push forward
$\psi_{*}\mathcal{F}=\\{\psi(S_{x})\\}_{x\in M}$ of $\mathcal{F}$ by $\psi$ is
a foliation on $N$, and

$\Gamma\left(\psi_{*}\mathcal{F}\right):=\\{\psi_{*}X,\;X\in\Gamma\left(\mathcal{F}\right)\\}=\psi_{*}\Gamma\left(\mathcal{F}\right).$
(1.1)

We denote by $Diff(M)$ the set of diffeomorphisms from $M$ to itself.

Let $S\subset M$ be a submanifold. The conormal space at a point $x\in S$ is
defined by

$N^{*}_{x}S=\\{\xi_{x}\in T_{x}M,\;{\xi_{x}}_{|T_{x}S}=0\\},$

and the conormal bundle of $S$ is

$N^{*}S=\\{(\xi_{x},x)\in T^{*}M,\;\xi_{x}\in N^{*}_{x}S\\}.$

The conormal bundle of a foliation $\mathcal{F}=\\{S_{x}\\}_{x\in M}$ is

$N^{*}\mathcal{F}=\\{N^{*}S_{x}\\}_{x\in M}.$

If the manifold $M$ is endowed with a symplectic form $\omega$ (as a
consequence, $m=2n$), a submanifold $S\subset M$ is Lagrangian if for every
$X\in\Gamma\left(TS\right)$, $\omega(X,Y)=0$ if and only if
$Y\in\Gamma\left(TS\right).$ That is, the orthogonal section

$\Gamma\left(TS\right)^{\perp}=\left\\{Y\in\mathfrak{X}(M):\;\omega(X,Y)=0,\;X\in\Gamma\left(TS\right)\right\\}$

of $\Gamma\left(TS\right)$ is equal to $\Gamma\left(TS\right)$.

A foliation $\mathcal{F}$ is Lagrangian if its leaves are lagrangian. That is,
$\Gamma\left(\mathcal{F}\right)^{\perp}=\Gamma\left(\mathcal{F}\right).$

A bi-Lagrangian structure on $M$ consists on a pair
$(\mathcal{F}_{1},\mathcal{F}_{2})$ of transversal Lagrangian foliations
together with a symplectic form $\omega$. As a consequence,
$TM=T\mathcal{F}_{1}\oplus T\mathcal{F}_{2}$.

Let $(\mathcal{F}_{1},\mathcal{F}_{2})$ be a bi-Lagrangian structure on a
symplectic $2n$-manifold $(M,\omega)$. Every point in $M$ has an open
neighborhood $U$ which is the domain of a chart whose local coordinates
$(p^{1},\dots,p^{n},q^{1},\dots,q^{n})$ are such that

$\begin{cases}\Gamma(\mathcal{F}_{1})_{\mid U}=\left<\frac{\partial}{\partial
p^{1}},\dots,\frac{\partial}{\partial p^{n}}\right>,\vspace{0.25cm}\\\
\Gamma(\mathcal{F}_{2})_{\mid U}=\left<\frac{\partial}{\partial
q^{1}},\dots,\frac{\partial}{\partial q^{n}}\right>.\end{cases}$

Such a chart, and such local coordinates, are said to be adapted to the bi-
Lagrangian structure $(\mathcal{F}_{1},\mathcal{F}_{2})$. Moreover, if

$\omega=\sum_{i=1}^{n}dq^{i}\wedge dp^{i},$

then such a chart, and such local coordinates, are said to be adapted to the
bi-Lagrangian structure $(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$.

Let $\nabla$ be a linear connection. The torsion tensor $T_{\nabla}$ (or
simply $T$ if there is no ambiguity) and curvature tensor $R_{\nabla}$ (or
simply $R$) are given respectively by

$T_{\nabla}(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y],\;X,Y\in\mathfrak{X}(M)$

and

$R_{\nabla}(X,Y)Z=\nabla_{X}{\nabla_{Y}Z}-\nabla_{Y}{\nabla_{X}^{Z}}-\nabla_{[X,Y]},\;X,Y,Z\in\mathfrak{X}(M).$

We say that a bi-Lagrangian structure (manifold) is affine when its Hess
connection $\nabla$ is a curvature-free connection; that is, $\nabla$ is flat.
The set of affine bi-Lagrangian structure is characterized in Theorem 1.2.

We say that a connection $\nabla$

  1. -

parallelizes $\omega$ if $\nabla\omega=0$; this means,

$\omega(\nabla_{X}{Y},Z)+\omega(Y,\nabla_{X}{Z})=X\omega(Y,Z),\;X,Y,Z\in\mathfrak{X}(M);$

  2. -

preserves $\mathcal{F}$ if
$\nabla{\Gamma\left(\mathcal{F}\right)}\subseteq\Gamma\left(\mathcal{F}\right)$;
more precisely,

$\nabla_{X}Y\in\Gamma\left(\mathcal{F}\right),\;(X,Y)\in\mathfrak{X}(N)\times\Gamma\left(\mathcal{F}\right).$

Einstein summation convention: an index repeated as sub and superscript in a
product represents summation over the range of the index. For example,

$\lambda^{j}\xi_{j}=\sum_{j=1}^{n}\lambda^{j}\xi_{j}.$

In the same way,

$X^{j}\frac{\partial}{\partial
y^{j}}=\sum_{j=1}^{n}X^{j}\frac{\partial}{\partial y^{j}}.$

Let $k\in\mathbb{N}$. Instead of $\\{1,2,\dots,k\\}$ we will simply write
$[k]$.

####  1.1.2 Prolongations of some objects to tangent bundle

The prolongations of tensor fields and linear connexion have been already
defined by several authors, see [24, 30, 13]. We are going to present the
vertical and complete lifts of a tensor field, the vertical lift of a linear
connection, and introduce the vertical lift of a foliation. If $T$ is a tensor
field on a manifold $M$, $T^{v}$ and $T^{c}$ mean the vertical and the
complete lifts of $T$ on $TM$ respectively.

Let $\pi:TM\longrightarrow M$ be the natural projection, $f\in C^{\infty}(M)$,
$X\in\mathfrak{X}(M)$ and $\alpha\in\Omega^{1}(M)$ (the set of 1-forms on
$M$).

  1. -

$f^{v}=f\circ\pi$ and $f^{c}=df$;

  2. -

$X^{v}(f^{c})=(Xf)^{v}$ and $X^{c}(f^{c})=(Xf)^{c}$;

  3. -

$\alpha^{v}(Y^{c})=(\alpha(Y))^{v}$ and $\alpha^{c}(Y^{c})=(\alpha(Y))^{c}$,
$Y\in\mathfrak{X}(M)$.

From these definitions, it follows that

$\begin{array}[]{ll}(fg)^{v}=f^{v}g^{v},&(fg)^{c}=f^{c}g^{v}+f^{v}g^{c},\\\
(fX)^{v}=f^{v}X^{v},&(fX)^{c}=f^{c}X^{v}+f^{v}X^{c},\\\
(f\alpha)^{v}=f^{v}\alpha^{v},&(f\alpha)^{c}=f^{c}g^{v}+f^{v}\alpha^{c}\end{array}$
(1.2)

where $f,g\in C^{\infty}(M)$, $X\in\mathfrak{X}(M)$ and
$\alpha\in\Omega^{1}(M)$.

By (1.2), the vertical and complete lifts of a tensor field on $M$ are defined
inductively by using the following formulas:

$\begin{array}[]{ll}(S+T)^{v}=S^{v}+T^{v},&(S+T)^{c}=S^{c}+T^{c},\\\ (S\otimes
T)^{v}=S^{v}\otimes T^{v},&(S\otimes T)^{c}\end{array}$ (1.3)

for all tensor fields $S$ and $T$ on $M$. As a consequence, if $T$ is a
$(r,0)$ tensor, then

$T^{c}(X_{1}^{c},\dots,X_{r}^{c})=(T(X_{1},\dots,X_{r}))^{c}\;\mbox{for all
}X_{1},\dots,X_{r}\in\mathfrak{X}(M).$ (1.4)

In this paper, we just need the complete lifts of a linear connection and a
foliation. The complete lift of a linear connection $\nabla$ is

$\nabla^{c}_{X^{c}}Y^{c}=(\nabla_{X}Y)^{c},$ (1.5)

and if $T$ is a tensor

$\nabla^{c}T=(\nabla T)^{c},$ (1.6)

If $\mathcal{F}$ is a foliation, then $\mathcal{F}^{c}$ is the foliation whose
$\Gamma\left(\mathcal{F}^{c}\right)$ the set of sections is

$\Gamma\left(\mathcal{F}^{c}\right):=\\{X^{c},\;X\in\Gamma\left(\mathcal{F}\right)\\}=\left(\Gamma\left(\mathcal{F}\right)\right)^{c}.$
(1.7)

####  1.1.3 Cherry vector field and map

Let $X$ be a vector field on the 2-torus $\mathbb{T}^{2}$ and let $x$ be a
point of $\mathbb{T}^{2}$.

We denote by:

  1. -

$Sin(X)$ the set of singularities of $X$;

  2. -

$t\mapsto\Phi^{t}_{X}$ the flow of $X$ on a neighborhood of $x$. That is,
there is a neighborhood $U$ of $x$ such that

$\left.\dfrac{d}{dt}\right|_{t=0}(\Phi^{t}_{X}(y))=X_{y}\;\mbox{ for every
}y\in U;$

  3. -

$\gamma^{-}(x)=\\{\Phi^{t}_{X},\;t\leq 0\\}$ and
$\gamma^{+}(x)=\\{\Phi^{t}_{X},\;t\geq 0\\}$ the negative and positive semi-
trajectories of $x$ respectively;

  4. -

$\gamma(x)=\gamma^{-}(x)\cup\gamma^{+}(x)$ the one-dimensional trajectory of
$x$.

Note that if $\gamma(x)$ and the circle $\mathbb{S}^{1}$ are homeomorphic,
then $\gamma(x)$ is a closed trajectory or is a periodic trajectory. the
trajectory $\gamma(x)$ is said to be non-closed if it is neither a fixed point
nor periodic trajectory.

A vector field without closed trajectory is said to be Cherry on
$\mathbb{T}^{2}$ if it has two singularities, a sink and a saddle, both
hyperbolic. We denote by $\mathfrak{X}_{c}(\mathbb{T}^{2})$ the set of Cherry
vector field on $\mathbb{T}^{2}$.

The first example of such a vector field was given by Cherry, see [5].

In this paper, We link pairs of vector fields with a class $\mathscr{L}$ of
circle maps with a flat piece noted.

We consider $\mathbb{S}^{1}$ as the interval $[0,1]$ where we identify 0 with
1. We fix $\ell_{1},\ell_{2}\geq 0$, an interval $U=(a,b)\subset[0,1]$ and, a
low dimensional order preserving map $f$ belongs to $\mathscr{L}$ with
critical exponents $(\ell_{1},\ell_{2})$ and flat piece $U$ if the following
holds:

  1. 1.

The image of $U$ is one point.

  2. 2.

The restriction of $f$ to $[0,1]\setminus\overline{U}$ is a diffeomorphism
onto its image.

  3. 3.

On a left-sided neighborhood of $a$, $f$ equals

$h_{l}((x-a)^{\ell_{1}})$

where $h_{l}$ is a diffeomorphism on a two-sided neighbourhood of $a$.
Analogously, on some right-sided neighborhood of $b$, $f$ can be represented
as

$h_{r}((x-b)^{\ell_{2}}).$

The interval $U$ is called the flat piece (or interval) of $f$. A map in
$\mathscr{L}$ is called a Cherry map.

We say that a map $f\in\mathscr{L}$ with $(\ell_{1},\ell_{2})$ as its critical
exponents is symmetric if $\ell_{1}=\ell_{2}$.

Let $F$ be a lift of $f\in\mathscr{L}$ on the real line. The rotation number
$\rho(f)$ of $f$ is defined (independently of $F$) by

$\rho(f):=\lim_{n\rightarrow\infty}\dfrac{F^{n}(0)}{n}(mod\>1).$

Note that a map $f$ in $\mathscr{L}$ has a periodic point if and only if
$\rho(f)$ is a rational number. The class of maps in $\mathscr{L}$ with
irrational rotation number is more interesting, see [18, 10, 11, 22, 25, 26].

###  1.2 Technical tools

In this part, we present results that we will need in the following.

####  1.2.1 Symplectic manifolds

Symplectic manifolds have been studied since 1780. Among many results on the
theory, the cotangent bundle of a manifold is endowed with a so-called
tautological 2-form. This part is devoted to the precise formulation of this
result. For more familiarization with the concepts covered here, the reader is
referred to [6, 16].

Let $M$ be a $m$-manifold and let ${}^{*}\pi:TM\longrightarrow M$ be the
natural projection. The tautological 1-form or Liouville 1-form $\theta$ is
defined by

$\theta_{(x,\alpha_{x})}(v)=\alpha_{x}\left(T_{(x,\alpha_{x})}^{*}\pi(v)\right),\;(x,\alpha_{x})\in
T^{*}M,\,v\in T_{(x,\alpha_{x})}T^{*}M,$

and its exterior differential $d\theta$ is called the canonical symplectic
form or Liouville 2-form on the cotangent bundle $T^{*}M$.

Note that for any coordinate chart $(U,x^{1},\dots,x^{m})$ on $M$, with
associated cotangent coordinate chart
$(T^{*}U,x^{1},\dots,x^{m},\xi_{1},\dots,\xi_{m})$ we have

$\theta=\sum_{1}^{m}\xi_{i}dx_{i},$

and

$d\theta=\sum_{1}^{m}d\xi_{i}\wedge dx_{i}.$

######  Proposition 1.1.

Let $M$ be a manifold. The cotangent bundle $T^{*}M$ of $M$ endowed with the
canonical symplectic form $d\theta$ is a symplectic manifold.

####  1.2.2 Bi-Lagrangian (Para-Kähler) manifolds

In this part, we briefly give some needed results concerning Hess affine bi-
Lagrangian structures and the push forward of a bi-Lagrangian structure. The
result characterizes affine bi-Lagrangian structures.

######  Theorem 1.2.

[12, Theor. 2:, p. 159] Let $(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ be a
bi-Lagrangian structure on a $2n$-manifold $M$ with $\nabla$ as its Hess
connection. Then the following assertions are equivalent.

  1. a.

The connection $\nabla$ is flat.

  2. b.

Each point of $M$ has a coordinate chart adapted to
$(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$.

The following result shows how to push forward a bi-Lagrangian structure.

######  Lemma 1.3.

[27, Lem. 2.2., p. 6] Let $(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ be a
bi-Lagrangian manifold with $\nabla$ as its Hess connection, and let $N$ be a
manifold which is diffeomorphic to $M$. Then for any diffeomorphism
$\psi:M\longrightarrow N$, the structure
$((\psi^{-1})^{*}\omega,\psi_{*}\mathcal{F}_{1},\psi_{*}\mathcal{F}_{2})$ is
bi-Lagrangian on $N$.

####  1.2.3 Cherry (vector) map

Cherry vectors have been studied by several authors. Among the many results
found on the theory, we have the following one found in [22, §6.2, 6.3, p.
148-149].

######  Proposition 1.4.

Let $X$ be a Cherry vector field on
$\mathbb{T}^{2}\approx\mathbb{S}^{1}\times[0,1]$ with $t\mapsto\Phi_{X}^{t}$
as its flow at the neighborhood of a point $s\in\mathbb{T}^{2}$, let $\Delta$
be the set of points $x\in\mathbb{S}^{1}\times\\{0\\}$ such that
$\Phi_{X}^{t}(x)\in\mathbb{S}^{1}\times\\{1\\}$ for some $t>0$, and let $t(x)$
be the minimal $t>0$ such that
$\Phi_{X}^{t}(x)\in\mathbb{S}^{1}\times\\{1\\}$. Then the map

$f:\Delta\longrightarrow\mathbb{S}^{1},\;x\mapsto\Phi_{X}^{t(x)}(x)$

belongs to $\mathscr{L}$.

##  2 Statements and proofs of results

###  2.1 Statements of results

Our first result presents lifted bi-Lagrangian structures on the trivial
bundles of some manifolds.

######  Theorem 2.1.

Let $(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ be a bi-Lagrangian with
$\nabla$ as its Hess connection. We have:

  1. 1.

the quadruplet $(T^{*}M,d\theta,N^{*}\mathcal{F}_{1},N^{*}\mathcal{F}_{2})$ is
an affine bi-Lagrangian manifold;

  2. 2.

the expression
$(T^{*}M,\tilde{\omega}:=^{*}\pi^{*}\omega+d\theta,N^{*}\mathcal{F}_{1},N^{*}\mathcal{F}_{2})$
is a bi-Lagrangian manifold.

  3. 3.

the object $(TM,\omega^{c},\mathcal{F}_{1}^{c},\mathcal{F}_{2}^{c})$ is an bi-
Lagrangian manifold with $\nabla^{c}$ as its Hess connection. Moreover, if
$(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ is affine, then so is
$(M,\omega^{c},\mathcal{F}_{1}^{c},\mathcal{F}_{2}^{c})$.

######  Remark 2.1.

Let $(M,\omega)$ be a symplectic manifold. Since $TM$ is diffeomorphic to
$T^{*}M$ (via the map $X\mapsto\omega(X,\cdot)$), then by Lemma 1.3, a bi-
Lagrangian structure on $TM$ induces a bi-Lagrangian structure on $T^{*}M$,
and vice-versa.

######  Corollary 2.2.

Let $M$ be a manifold equipped with a bi-Lagrangian structure. The action
$(\psi,(\mathcal{F}_{1},\mathcal{F}_{2}))\mapsto(\psi_{*}\mathcal{F}_{1},\psi_{*}\mathcal{F}_{2})$
of the symplectomorphism group of $(M,\omega)$ on the set of its bi-Lagrangian
structures defined in [27, The. 2.3., p. 6] can be lifted on $TM$ and $T^{*}M$
in different ways.

######  Theorem 2.3.

  1. 1.

Some pairs of Cherry vector fields generate maps belonging in $\mathscr{L}$.

  2. 2.

The left action

$*:Diff(\mathbb{T}^{2})\times\mathfrak{X}_{c}(\mathbb{T}^{2})\longrightarrow\mathfrak{X}_{c}(\mathbb{T}^{2}),\;(\varphi,X)\mapsto\varphi_{*}X$
(act1)

induces the following conjugation action

$\circ:Diff(\mathbb{S}^{1})\times\mathscr{L}\longrightarrow\mathscr{L},\;(\varphi,f)\mapsto\varphi\circ
f\circ\varphi^{-1}.$ (act2)

Moreover, for every $X\in\mathfrak{X}_{c}(\mathbb{T}^{2})$ and for every
$\varphi\in Diff(\mathbb{T}^{2})$, if $X$ generates the map $f$, then
$\varphi_{*}X$ generates the map

$\varphi\circ
f\circ\varphi^{-1}:\varphi(\Delta)\longrightarrow\mathbb{S}^{1},\;x\mapsto\varphi\circ\Phi_{X}^{t(x)}\circ\varphi^{-1}(x)$

where $\Delta$ is the set of points $x\in\mathbb{S}^{1}\times\\{0\\}$ such
that $\Phi_{X}^{t}(x)\in\mathbb{S}^{1}\times\\{1\\}$ for some $t>0$, and
$t(x)$ is the minimal $t>0$ such that
$\Phi_{X}^{t}(\varphi^{-1}(x))\in\mathbb{S}^{1}\times\\{1\\}$.

The point 1 of this result extends results in [18, Theo. B., p. 533] and [22,
§6.2, 6.3, p. 148-149]. Now, we are going to characterize the orbit of a
Cherry vector field with respect to $*$ (see, (act1)) from the orbit via
$\circ$ (see, (act2)) of a map in $\mathscr{L}$. For every
$X\in\mathfrak{X}_{c}(\mathbb{T}^{2})$ we denote by $f^{X}$ the map in
$\mathscr{L}$ generated by $X$. We have the following observation.

######  Remark 2.2.

The orbit with respect to $\circ$ (see, (act2)) of a map $f\in\mathscr{L}$
noted $\mathcal{O}_{\circ}(f)$ is defined by

$\mathcal{O}_{\circ}(f):=\\{\varphi\circ f\circ\varphi^{-1},\;\varphi\\}.$

Since two maps belonging in $\mathscr{L}$ with the same irrational rotation
number belong in the same orbit (see, [18, Theo. D., p. 535]), then

$\mathcal{O}_{\circ}(f):=\\{g\in\mathscr{L}:\;\rho(f)=\rho(g)\\}.$

As a consequence, for every $X\in\mathfrak{X}_{c}(\mathbb{T}^{2})$ such that
$\rho(f^{X})$ is a irrational number, we have

$\mathcal{O}_{*}(X):=\\{Y\in\mathfrak{X}_{c}(\mathbb{T}^{2}):\;\rho\left(f^{X}\right)=\rho\left(f^{Y}\right)\\}.$

Note that these observations still hold for $C^{2}$ Cherry vector field (or
Cherry map).

###  2.2 Proofs of results

####  2.2.1 Lifted bi-Lagrangian structures

###### Proof of Theorem 2.1.

######  Lemma 2.4.

Let $\mathcal{F}=\\{S_{x}\\}_{x\in M}$ be a $k$ foliation on a $m$ manifold
$M$. Then $N^{*}\mathcal{F}$ is a Lagrangian foliation on $(T^{*}M,d\theta)$.
Moreover, if $M$ is endowed with a symplectic form $\omega$ and $\mathcal{F}$
is Lagrangian on $(M,\omega)$, so is $N^{*}\mathcal{F}$ on $(M,\omega)$.

###### Proof.

First note that if $S\subset M$ is a sub-manifold, then $N^{*}S$ is a
Lagrangian sub-manifold of $(T^{*}M,d\theta)$ (this is a general result, see
[6, Cor. 3.7., p. 18]). As a consequence,
$N^{*}\mathcal{F}=\\{N^{*}S_{x}\\}_{x\in M}$ is Lagrangian on
$(T^{*}M,d\theta)$. Thus it remains to show that $N^{*}\mathcal{F}$ is
completely integrable. This means

$d\theta([X,Y],Z)=0\;\mbox{ for all }X,Y,Z\in\Gamma(N^{*}\mathcal{F})$

since $N^{*}\mathcal{F}$ is Lagrangian.

Note that

$d\theta([X,Y],Z)=[X,Y]\theta(Z)-Z\theta([X,Y])-\theta([[X,Y],Z]).$

Let $(U,p^{1},\dots,p^{m})$ be a coordinate system chart adapted to the
foliation $\mathcal{F}$ with
$(T^{*}U,p^{1},\dots,p^{m},\xi_{1},\dots,\xi_{m})$ as its associated bundle
coordinate chart. Observe that

$\Gamma(N^{*}\mathcal{F})|_{T^{*}U}=\left<\frac{\partial}{\partial
p^{1}},\dots,\frac{\partial}{\partial
p^{k}},\frac{\partial}{\partial\xi_{k+1}},\dots,\frac{\partial}{\partial\xi_{m}}\right>$
(2.1)

Let us write

$\begin{cases}(y^{i})_{i=1,\dots,m}=((p^{i})_{i=1,\dots,m},(\xi_{i})_{i=k+1,\dots,m}),\\\
X=X^{i}\frac{\partial}{\partial y^{i}},Y=Y^{j}\frac{\partial}{\partial
y^{j}}\;\mbox{ and }\;Z=Z^{k}\frac{\partial}{\partial y^{k}}.\end{cases}$

Then

$\begin{cases}[X,Y]=\mu^{j}\frac{\partial}{\partial y^{j}},\\\
[[X,Y],Z]=\lambda^{j}\frac{\partial}{\partial y^{j}},\end{cases}$

where

$\begin{cases}\mu^{j}=X^{i}\frac{\partial Y^{j}}{\partial
y^{i}}-Y^{i}\frac{\partial X^{j}}{\partial y^{i}},\\\
\lambda^{j}=\mu^{i}\frac{\partial Z^{j}}{\partial
y^{i}}-Z^{i}\frac{\partial\mu^{j}}{\partial y^{i}}.\end{cases}$

Thus,

$\displaystyle[X,Y]\theta(Z)$ $\displaystyle=\mu^{i}\frac{\partial}{\partial
y^{i}}(Z^{k}\xi_{k}),$ ($e_{1}$) $\displaystyle\theta([[X,Y],Z])$
$\displaystyle=\lambda^{j}\xi_{j},$ ($e_{2}$) $\displaystyle Z\theta([X,Y])$
$\displaystyle=Z^{k}\frac{\partial}{\partial y^{k}}(\mu^{i}\xi_{i}).$
($e_{3}$)

Therefore

$d\theta([X,Y],Z)=\mbox{(\ref{Bieq5})}-\mbox{(\ref{Bieq6})}-\mbox{(\ref{Bieq7})}=0.$

So $N^{*}\mathcal{F}$ is a Lagrangian foliation on $(T^{*}M,d\theta)$.

Now, suppose that $\mathcal{F}$ is a Lagrangian foliation on $(M,\omega)$.
Observe that for every $X\in\Gamma(N^{*}\mathcal{F})$,
${}^{*}\pi_{*}X\in\Gamma(\mathcal{F})$. Thus for all
$X,Y\in\Gamma(N^{*}\mathcal{F})$,

$\tilde{\omega}(X,Y)=\omega(^{*}\pi_{*}X,^{*}\pi_{*}Y)+d\theta(X,Y)=0$

where we use the fact that $\mathcal{F}$ is Lagrangian and the definition of
$d\theta$. So $N^{*}\mathcal{F}$ is a Lagrangian foliation on
$(T^{*}M,\tilde{\omega})$. This ends the proof of Lemma 2.4. ∎

Now we are ready to prove Theorem 2.1. Let
$(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ be a bi-Lagrangian manifold.

  1. 1.

By Lemma 2.4, $(N^{*}\mathcal{F}_{1},N^{*}\mathcal{F}_{2})$ is a pair of
Lagrangian foliation on $(T^{*}M,d\theta)$. From (2.1), it follows that if
$(U,p^{1},\dots,p^{n},q^{1},\dots,q^{n})$ is a coordinate chart adapted to the
bi-Lagrangian structure $(\mathcal{F}_{1},\mathcal{F}_{2})$, with
$(T^{*}U,p^{1},\dots,p^{n},q^{1},\dots,q^{n},\xi_{1},\dots,\xi_{2n})$ as its
associated bundle coordinate chart, then

$\begin{cases}\Gamma(N^{*}\mathcal{F}_{1})|_{T^{*}U}=\left<\frac{\partial}{\partial
p^{1}},\dots,\frac{\partial}{\partial
p^{n}},\frac{\partial}{\partial\xi_{n+1}},\dots,\frac{\partial}{\partial\xi_{2n}}\right>,\vspace{0.25cm}\\\
\Gamma(N^{*}\mathcal{F}_{2})|_{T^{*}U}=\left<\frac{\partial}{\partial
q^{1}},\dots,\frac{\partial}{\partial
q^{n}},\frac{\partial}{\partial\xi_{1}},\dots,\frac{\partial}{\partial\xi_{n}}\right>\end{cases}$
(2.2)

and

$d\theta|_{T^{*}U}=\sum_{i=1}^{n}(d\xi_{i}\wedge dp^{i}+d\xi_{n+i}\wedge
dq^{i}).$ (2.3)

By (2.2), we get that $N^{*}\mathcal{F}_{1}$ and $N^{*}\mathcal{F}_{2})$ are
transverse. so $(N^{*}\mathcal{F}_{1},N^{*}\mathcal{F}_{2})$ is a bi-
Lagrangian structure on $(T^{*}M,d\theta)$. By combining equalities in (2.2),
and equality (2.3), it follows from Theorem 1.2 that
$(N^{*}\mathcal{F}_{1},N^{*}\mathcal{F}_{2})$ is affine.

  2. 2.

The point 2 of Theorem 2.1 follows directly by combining Lemma 2.4 and
equalities in (2.2).

  3. 3.

By (1.4), we have

$\omega^{c}(X^{c},Y^{c})=(\omega(X,Y))^{c}\;\mbox{ for all
}X,Y\in\mathfrak{X}(M).$

Thus, since the structure $(\mathcal{F}_{1},\mathcal{F}_{2})$ is bi-Lagrangian
on $(M,\omega)$, then $(\mathcal{F}_{1}^{c},\mathcal{F}_{2}^{c})$ is a pair of
transversal Lagrangian foliation on $(TM,\omega^{c})$.

Let $\nabla$ be the Hess connection of
$(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$. We are going to show that
$\nabla^{c}$ is that of
$(\omega^{c},\mathcal{F}_{1}^{c},\mathcal{F}_{2}^{c})$.

By combining the fact that $\nabla$ is the Hess connection of
$(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$, (1.4), (1.6) and (1.5) we have

$T^{c}(X^{c},Y^{c})=T^{c}(X^{c},Y^{c})=0$ (2.4)
$\nabla^{c}\omega^{c}=(\nabla\omega)^{c}=0$ (2.5)
$\nabla_{X^{c}}Y^{c}=(\nabla_{X}Y)^{c}\in\Gamma(\mathcal{F}^{c}_{i})\;X,Y\in\Gamma(\mathcal{F}i)\;i=1,2.$
(2.6)

By combining (2.4), (2.5) and (2.6), it follows that $\nabla^{c}$ is the Hess
connection of $(\omega^{c},\mathcal{F}_{1}^{c},\mathcal{F}_{2}^{c})$. This
ends the proof of Theorem 2.1.

∎

###### Theorem 2.3.

  1. 1.

Let $X_{1}$ $X_{2}$ be two Cherry vector field such that the following holds:

    1. -

the vector field $X_{1}$ generates $f_{1}\in\mathscr{L}$ with
$U_{1}=(a_{1},b_{1})$ as its flat piece, and $(\ell_{1},\ell_{1})$ as its
critical exponents,

    2. -

the vector field $X_{2}$ generates $f_{1}\in\mathscr{L}$ with
$U_{2}=(a_{2},b_{2})$ as its flat piece, and $(\ell_{2},\ell_{2})$ as its
critical exponents,

    3. -

$f_{1}(U_{1})=f_{2}(U_{2})$,

    4. -

the intervals $(a_{1},b_{1})$ and $(a_{2},b_{2})$ are such that
$a_{1}<a_{2}\leq b_{1}<b_{2}$.

The map

$f(x)=\begin{cases}\begin{array}[]{lll}f_{1}(x)&\mbox{if }&x\in[0,a_{1}]\\\
f_{2}(x)&\mbox{if }&x\in[a_{2},1]\end{array}\end{cases}$

belongs to $\mathscr{L}$ with flat piece $U=(a_{1},b_{2})$ and critical
exponents $(\ell_{1},\ell_{2})$.

  2. 2.

Let $X\in\mathbb{T}^{2}$ with $\Phi_{X}:\mapsto\Phi_{X}^{t}$ as its flow on an
open $U$. Let $\varphi$ be a diffeomorphism. observe that

$\left.\dfrac{d}{dt}\right|_{t=0}(\varphi\circ\Phi^{t}_{X}\circ\varphi^{-1}(y))=(\varphi_{*}X)_{y}\;\mbox{
for every }y\in U.$

This means $\varphi\circ\Phi^{t}_{X}\circ\varphi^{-1}$ is the flow of
$\varphi_{*}X$. This ends the proof of Theorem 2.3.

∎

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