text:
\section{Introduction}\label{introduction}

\subsection{Context}

Let $g \geq 1$ and let $\kappa$ be a integer partition of $2g-2$. A stratum $\mathcal{H}_g(\kappa)$ of the moduli space of translation surfaces is the set of isomorphism classes of pairs $(X,\omega)$ where $X$ is a genus $g$ Riemann surface and $\omega$ is a non vanishing holomorphic $1$-form on $X$ whose zeroes have multiplicities given by $\kappa$. This space admits a natural action by the group $\GL$, which is a generalization of the action of $\GL$ on the space of flat tori $\GL / SL(2,\mathbb{Z})$. The classification of the closed invariant sets is a central problem in Teichmüller dynamics. Recently, a deep theorem of Eskin, Mirzakhani and Mohammadi has shed light on the structure of such sets: they are immersed orbifolds cut out by linear equations with real coefficients in period coordinates. These objects are referred to as affine invariant orbifolds. See \cite{eskin2015isolation}.

\medskip

Transverse to the $\GL$-action, there is a local action by $\mathbb{C}^{\vert \kappa \vert-1}$ that fits into a natural holomorphic foliation of the strata of the moduli space. It is usually referred to as the isoperiodic foliation, the kernel foliation or the Rel foliation. The leaf $\mathcal{F}_X$ of a translation surface $(X,\omega)$ is locally obtained as a level set of the absolute periods map. See \cite{hooper2015rel} for a careful definition of this foliation. This foliation has been introduced about 25 years ago by Kontsevich and Eskin, and later by McMullen and Calta before it became a central object in Teichmüller dynamics. See for example how it is involved in the classfication of horocycle orbits of Prym eigenforms in $\mathcal{H}(1,1)$ obtained by Bainbridge, Smillie and Weiss in \cite{bainbridge2016horocycle}.

\medskip

Several papers have been devoted to understanding the dynamics of its leaves. McMullen showed that the foliation is ergodic with respect to the Masur-Veech measure in the principal stratum in genus 2 and 3 using Ratner's theory in \cite{mcmullen2014moduli}. Hooper and Weiss gave the first examples of dense leaves outside the principal stratum in \cite{hooper2015rel}. Shortly after Calsamiglia, Deroin and Francaviglia have obtained a Ratner-like classification of the minimal sets in the principal stratum and obtained the ergodicity with respect to the Masur-Veech measure as a consequence of this classification. See \cite{calsamiglia2015transfer}. Simultaneously, Hamenstädt reproved this ergodicity statement using a different approach, still in the principal strata, in \cite{hamenstadt2018ergodicity}. Both this last two results used McMullen's result as a base case for an induction. Very surprisingly, apart from the examples of Hooper and Weiss, nothing is known for the dynamics of the isoperiodic foliation in strata where at least one zero is not simple.

\medskip

\subsection{Statement of the results}

We define on any affine invariant orbifold $\M$ a foliation $\FM$ that generalizes the isoperiodic foliation on the strata of the moduli space. Those foliations are particularly interesting in the light of Eskin, Mirzakhani and Mohammadi's result as any information on the geometry of affine invariant orbifolds directly translates into information on the dynamics of the $\GL$-action. The dynamics of the foliation $\FM$ also contains information on the dynamics of the isoperiodic foliation itself as any leaf  $\FM_X$ is a connected component of $\mathcal{F}_X \cap \M$.

\medskip

The foliation $\FM$ will be referred to as the $\M$-isoperiodic foliation. For some affine invariant orbifolds, the leaves of $\FM$ have dimension $0$. It is for instance the case for Teichmüller discs or for hyperelliptic loci of non hyperlliptic strata. More sophisticated examples are given in \cite{eskin2018billiards}. The other affine invariant orbifolds are called non absolute. In this text we study the dynamics of the foliation $\FM$ in the case where $\M$ is a non absolute rank 1 affine invariant orbifold. We isolate a property $\mathcal{P}$ that describes cylinder decompositions of a particular type for surfaces in $\M$ and establish the following criterion, key to the rest of the paper:

\begin{theorem}\label{A}
Let $\M$ be a non absolute rank 1 affine invariant orbifold. If $\M$ satisfies property $\mathcal{P}$, then all the leaves of $\FM$ are projectively dense.
\end{theorem}

Here, projectively dense means that the leaf of any surface $(X,\omega)$ is dense in the locus of surfaces in $\M$ that have the same area as $(X,\omega)$. Equivalently, the projection of $\FM_X$ in $\mathbb{P}\M$ is dense. This has to do with the fact that the area of a tranlsation surface can be computed only with its absolute periods and thus the area function is constant along the leaves of the $\M$-isoperiodic foliation. See definition \ref{projectivedensity} for more details. The proof of theorem \ref{A} relies on the fact that for a surface with a cylinder decomposition provided by property $\mathcal{P}$, the associated horocycle orbit is approximable by $\M$-isoperiodic deformations and thus is contained in the closure of its $\M$-isoperiodic leaf. We show that this can be done in a direction where of the horocycle orbit closure coincide with the whole $\SL$ orbit closure, concluding the proof. We then give examples of affine invariant orbifolds that have property $\mathcal{P}$ and others that do not. Special attention is dedicated to Prym eigenforms of genus 3 and we show the following:

\begin{theorem}\label{B}
Let $\kappa = (2,2)^{odd}, (2,1,1)$ or $(1,1,1,1)$ and let $\M$ be a connected component of $\Omega E_D(\kappa)$. Then either all the leaves of $\FM$ are all closed or all the leaves of $\FM$ are projectively dense. The last case occurs if, and only if, $D$ is not a square.
\end{theorem}

The remaining connected components $\mathcal{H}^{odd}(4)$, $\mathcal{H}^{hyp}(4)$ and $\mathcal{H}(2,2)^{hyp}$ are not included in Theorem \ref{B} as the corresponding Prym eigenform loci are union of Teichmüller discs and the leaves have dimension 0, hence closed even when D is not a square. This is clear when there is only a single singularity. See proposition 2.3 \cite{lanneau2014connected} for a proof of that claim in $\mathcal{H}^{hyp}(2,2)$. The genus 2 case is an immediate consequence of Theorem \ref{A} and had already been established by Mcmullen. It is known that Prym eigenforms of discriminant $D$ are primitive if and only if $D$ is not a square. Theorem \ref{B} can thus be understood as a dynamical characterisation of primitivity for Prym eigenforms of genus 3.

\medskip

Finally, we use Theorem \ref{A} to investigate the geometric structure of the closure of isoperiodic leaves and deduce the following:

\begin{theorem}\label{C}
Let $\kappa =(2,1,1)$ or $(1,1,1,1)$ and let $(X,\omega)$ be a Prym eigenform in $\Omega E_D(\kappa)$. If $D$ is not a square, then the leaf $\mathcal{F}_X$ is projectively dense in $\mathcal{H}_3(\kappa)$.
\end{theorem}

In the case of the Prym eigenforms in $\mathcal{H}^{odd}(2,2)$, the leaves are not projectively dense in the stratum, and the computation of their closure is contained in Theorem $\ref{A}$ since in this situation the leaves of $\FM$ with $\M$ being a connected component of $\Omega E_D(\kappa)$ are equal to the leaves of $\mathcal{F}$. It is interesting to note the dichotomy arising with the number of singularities. Theorem \ref{C} gives, to our knowledge, the first examples of dense isoperiodic leaves in $\mathcal{H}(2,1,1)$ and gives a new proof, based on dynamics, of the projective density of the isoperiodic leaves of prym eigenforms in the case $\kappa = (1,1,1,1)$. This was already known as part of the work of Calsamiglia, Deroin and Francaviglia.

\subsection{Organisation of the paper}

In section \ref{framework}, we collect relevant definitions. We review the structures of the strata and of the closed $\GL$-invariant sets, recall the definitions of the Prym eigenform loci, and define the foliation $\FM$. In section \ref{twists}, we give a common framework for cylinder and isoperiodic deformations. In section \ref{p}, we establish the criterion presented in Theorem \ref{A} and draw its first consequences on the structure of the closure of the leaves of $\FM$. In section \ref{examples}, we provide a list of affine invariant orbifolds that have property $\mathcal{P}$, and deduce Theorem \ref{B}. In section \ref{full}, we use the structural result established in section \ref{p} to prove theorem \ref{C}. Appendix A is dedicated to enumerating all the cylinder decompositions of a certain type of Prym eigenforms in $\mathcal{H}(1,1,1,1)$. This list is used to show that Prym eigenforms in this stratum have property $\mathcal{P}$.

\subsection{Acknowledgements}

I am deeply indebted to Erwan Lanneau for introducing to the world of tranlation surfaces and suggesting the questions studied in this text, and many more. I would also like to thank Barak Weiss, Pascal Hubert and Duc Manh Nguyen for having discussed with me those results which have significantly benefited from their insightful comments. This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d'avenir.

\section{Framework}\label{framework}

\subsection{Period coordinates \& the action of $\GL$}

We denote by $\tilde{\mathcal{H}}(\kappa)$ the space of equivalence classes of marked translation surfaces $(X,\omega,f)$ where $f: S \to X$ is a homeomorphism from a fixed genus $g$ surface $S$ such that the preimage of the singularities of $\omega$ by $f$ is a subset $\Sigma$, and the order of the singularities of $\omega$ are prescribed by $\kappa$. The following map is known as the period map :

$$
\Phi :
\begin{matrix}
\tilde{\mathcal{H}}(\kappa) &\to & H^1(S,\Sigma,\mathbb{C}) \\
(X,\omega,f) &\mapsto & (\gamma \mapsto \int_{f\circ \gamma} \omega)
\end{matrix}
$$

There is a complex structure on $\tilde{\mathcal{H}}(\kappa)$ that turns $\Phi$ into a local biholomorphism, and if $MCG(S,\Sigma)$ denotes the relative mapping class group of $S$ that fixes $\Sigma$ globally, then $MCG(S,\Sigma)$ acts almost freely on $\teich$ by precomposition: $\varphi \cdot (X,\omega,f) = (X,\omega, f \circ \varphi^{-1})$. The quotient set is isomorphic to $\moduli$ and the latter is endowed with the complex orbifold structure that turns the canonical projection $\pi : \tilde{\mathcal{H}}(\kappa) \to \mathcal{H}(\kappa)$ into a local biholomorphism (in the orbifold sense, see \cite{caramello2019introduction} for relevant definitions). The space $\tilde{\mathcal{H}}(\kappa)$ is endowed with a group action by $GL_2^{+}(\mathbb{R})$ defined by :
$$
\forall g \in \GL \ \Phi(g\cdot (X,\omega,f)) = g \cdot \Phi (X,\omega,f)
$$

That action descends to an action on $\moduli$ in a way that the canonical projection $\pi$ is $\GL$-equivariant. If $\theta \in S^1$, we denote by $R_{\theta} \in \SL$ the rotation of angle $\theta$. We will use the following notation:

$$
\forall \theta \in S^1 \ \forall t \in \mathbb{R}, \ g_t^{\theta} = R_{\theta} \begin{pmatrix} e^t && 0 \\ 0  && e^{-t} \end{pmatrix} R_{-\theta}, \ \ h_t^{\theta} = R_{\theta} \begin{pmatrix} 1 && t \\ 0  && 1 \end{pmatrix} R_{-\theta}
$$

For any $\theta$, the action of $A_{\theta}=\{g_t^{\theta}, t \geq 0 \}$ and $H_{\theta} = \{h_t^{\theta}, t \geq 0 \}$ are known as the geodesic flow and horocycle flow in direction $\theta$. More details on the structures of these spaces and the action of $\GL$ can be found in \cite{zorich2006flat} or \cite{forni2013introduction}.

\begin{dfn}[affine invariant orbifold]\label{affinemanifold}
An affine invariant orbifold is a closed connected subset $\M$ of $\moduli$ obtained as the image of a complex orbifold $\mathfrak{M}$ by a proper immersion $\iota$ that satisfies the following property: if $x \in \mathfrak{M}$, there is an open set $\mathcal{U}$ around $x$, an orbifold chart $(\mathcal{V},\varphi)$ around $\iota(x)$ and a $\mathbb{R}$-linear vector subspace $V$ of $H^1(S,\Sigma,\mathbb{R})$ such that:

\begin{equation}\label{localmodel}
\iota(\mathcal{U}) \cap \varphi(\mathcal{V}) = \varphi(\mathcal{V} \cap V \otimes \mathbb{C})
\end{equation}
\end{dfn}

In the previous definition, immersion refers to an immersion of orbifold, See \cite{caramello2019introduction} for relevant definitions. Affine invariant orbifolds are invariant under the action of $\GL$ and Eskin, Mirzakhani and Mohammadi proved in a celebrated result that the converse is true. More details can be found in \cite{eskin2015isolation}. We will use the following useful caracterisation.

\begin{prop}\label{characterisation}
Let $\M$ be an affine invariant orbifold, and let $\iota : \mathfrak{M} \to \moduli$ as in definition \ref{affinemanifold}. If $p:\hat{\mathfrak{M}}\to \mathfrak{M}$ is the orbifold universal cover of $\mathfrak{M}$, there is linear subspace $V \in H^1(S,\Sigma,\mathbb{R})$ and a map $\hat{\iota} : \mathfrak{M}\to \teich$ such that $\pi \circ \hat{\iota} = \iota \circ p$ and $\Phi \circ \hat{\iota}$ is a local biholomorphism from $\hat{\mathfrak{M}}$ to $V\otimes \mathbb{C}$.
\end{prop}

\begin{proof}

By construction, the orbifold fundamental group of $\hat{\mathfrak{M}}$ is trivial. The smooth map $\iota \circ p$ can thus be lifted to a smooth map $\hat{\iota} : \hat{\mathfrak{M}} \to \teich{}$ such that $\pi \circ \hat{\iota} = \iota \circ p$. Now, equation \eqref{localmodel} implies that for any $x \in \hat{\mathfrak{M}}$ there are a neighborhood $\mathcal{U}_1$ of $x$, a neighborhood $\mathcal{U}_2$ of $\Phi \circ \hat{\iota}(x)$ and a subspace $V_x \subset H^1(S,\Sigma,\mathbb{R})$ so that:

$$
\Phi \circ \hat{\iota}(\mathcal{U}_1) = \mathcal{U}_2 \cap V_x \otimes \mathbb{C}
$$

By connectedness of $\M$, one deduces that all the $V_x$ are equal, proving that the image of $\Phi \circ \hat{\iota}$ is indeed of the form $V \otimes \mathbb{C}$.

\end{proof}

For details on orbifold fundamental groups, orbifold coverings and lifts of maps, the reader is referred to part $\mathcal{G}$ chapter 1 and 3 of \cite{bridson2013metric}\\

An important numerical invariant associated to these loci is the rank, defined as follows: define $\rho :  H^1(S,\Sigma,\mathbb{C}) \to H^1(S,\mathbb{C})$ to be the canonical restriction map, and chose a local model $V$ of $\M$ as in \eqref{localmodel}. Avila, Eskin and Möller proved in \cite{avila2017symplectic} that $\rho(V \otimes \mathbb{C})$ is a symplectic subspace of $H^1(S,\mathbb{C})$. The rank of $\M$, denoted $rk(\M)$, is then defined as half the complex dimension of this space (this notion is well defined as affine invariant orbifolds are required to be connected). More details can be found in \cite{wright2015cylinder}. The following definition will be important for the remainder of this text :

\begin{dfn}
Let $\M$ be a affine invariant orbifold. The field of definition of $\M$ is the smallest subfield $k(\M)$ of $\mathbb{R}$ such that any $V$ as in \eqref{localmodel} can be written as $V = V_0 \otimes_{k(\M)} \mathbb{R}$, where $V_0$ is a $k(\M)$-linear subspace of $H^1(S,\Sigma,k(\M))$
\end{dfn}

By definition, affine invariant orbifolds are cut out by real linear equations and the field of definition of $\M$ is thus the smallest subfield of $\mathbb{R}$ in which those linear equations take their coefficients. Wright has proved that the field of definition can be computed using periodic translation surfaces: let $(X,\omega)$ be a translation surface. the quadratic form $\omega \otimes \overline{\omega}$ induces a flat metric on $X$, away from the singularities of $\omega$. We say that $(X,\omega)$ is periodic in direction $\theta$ if the geodesic rays in direction $\theta$ are either periodic, or saddle connections (they start and end at singularities of $\omega$). It is well known that if a translation surface is periodic in a given direction, it is decomposed as a union of cylinders (those are subsets isometric to $[0,h] \times \mathbb{R} / c \mathbb{Z}$) whose boundary components are union of saddle connections in direction $\theta$). Wright proved the following in \cite{wright2015cylinder}:

\begin{prop}\label{field}
Let $\M$ be an affine invariant orbifold, and let $(X,\omega)$ be a periodic surface in $\M$ with m cylinders. Then, $k(\M)$ is contained in $\mathbb{Q}[c_2c_1^{-1},\cdots,c_mc_1^{-1}]$ where the $c_i$ are the circumferences of the cylinders.
\end{prop}

Wright actually proved a stronger result and proved the other inclusion, provided one considers only a subclass of cylinders. See \cite{wright2015cylinder} for more details.

\subsection{The $\M$-isoperiodic foliation}

The aim of this subsection is to define a foliation on any affine invariant orbifold. In that perspective, let us first recall the definition of the usual isoperiodic foliation $\mathcal{F}$: the map $\rho \circ \Phi$ is a $MCG(S,\Sigma)$-equivariant submersion. The isoperiodic foliation is defined as the quotient foliation of the foliation of $\teich$ by connected components of the level sets of $\rho \circ \Phi$. We are going to adapt this construction. Let $\M$ be an invariant affine orbifold and $(X,\omega)$ be a surface in $\M$. We denote by $\FM_X$ the connected component of $\mathcal{F}_X \cap \M$ that contains $(X,\omega)$.

\begin{prop}\label{foliation} Let $\iota : \mathfrak{M} \to \moduli$ be an immersion such that $\iota(\mathfrak{M}) = \M$ as in definition \ref{affinemanifold}. There is a foliation $\mathfrak{F}$ on $\mathfrak{M}$ such that for any $(X,\omega) \in \M$ and $x \in \mathfrak{M}$ such that $\iota(x) = (X,\omega)$, there is a neighborhood $\mathcal{U}$ around $x$ in $\mathfrak{M}$ such that:

$$
\iota(\mathcal{U}\cap \mathfrak{F}_x) = \iota(\mathcal{U})\cap \FM_X
$$

\end{prop}

\begin{proof}

Let $\hat{\iota} : \hat{\mathfrak{M}} \to \moduli$ and $V \in H^1(S,\Sigma,\mathbb{R})$ as in proposition \ref{characterisation}. Since $\Phi \circ \hat{\iota}$ is a local biholomorphism and $\rho$ is a linear surjection, the map $\rho \circ \Phi \circ \tilde{\iota}$ is a submersion onto $\rho(V\otimes \mathbb{C})$. The connected components of its level sets induce a foliation on $\hat{\mathfrak{M}}$. The map $\iota$ induces a representation of the orbifold fundamental group of $\M$ in $MCG(S,\Sigma)$ and thus an action on $\rho(V \otimes \mathbb{C})$. By construction the map $\rho \circ \Phi \circ \tilde{\iota}$ is $\pi_1^{orb}(\mathfrak{M})$-equivariant. This proves that the aforementioned foliation descends to a foliation $\mathfrak{F}$ on $\mathfrak{M}$. \\

Now, let $(X,\omega)$ and $x$ be as in the statement and choose a neighborhood $\mathcal{U}$ of $x$, a point $\hat{x} \in \hat{\mathfrak{M}}$ and a $\hat{\mathcal{U}}$ of $\hat{x}$ in $\hat{\mathfrak{M}}$ such that $p(\hat{\mathcal{U}}) = \mathcal{U}$. Up to shrinking $\mathcal{U}$, one can assume that $\Phi \circ \hat{\iota}$ induces a diffeomorphism from $\hat{\mathcal{U}}$ to $\mathcal{V} \cap V \otimes \mathbb{C}$. By definition of $\mathfrak{F}$, we have:

$$
\mathcal{U} \cap \mathfrak{F}_x = p((\Phi \circ \hat{\iota})^{-1}(\xi + Ker \rho \cap V \otimes \mathbb{C} \cap \mathcal{V}))
$$

Consequently, using the relation $\pi \circ \hat{\iota} = \iota \circ p$, we obtain:

\begin{equation*}
\begin{split}
\iota(\mathcal{U} \cap \mathfrak{F}_x) &= \pi (\Phi^{-1}(\xi + Ker \rho \cap V \otimes \mathbb{C} \cap \mathcal{V})) \\
&= \pi(\Phi^{-1}(V \otimes \mathbb{C} \cap \mathcal{V})) \cap \pi(\Phi^{-1}(\xi + Ker \rho)) \ \\
&= \iota(\mathcal{U}) \cap \mathcal{F}_X
\end{split}
\end{equation*}

To conclude, notice that $\iota(\mathcal{U}) = \iota(\mathcal{U}) \cap \M$ and thus $\iota(\mathcal{U})\cap \mathcal{F}_X = \iota(\mathcal{U}) \cap \FM_X$
\end{proof}

\textbf{Remark.} In the previous construction, the foliation $\mathfrak{F}$ is independent of the choices made for $\hat{\mathfrak{\M}}$ and $\hat{\iota}$ as any other choice only differ by the action of an element of $\pi_1^{orb}(\mathfrak{M})$ and the leaves of $\mathfrak{F}$ are obtained after taking quotient by $\pi_1^{orb}(\mathfrak{M})$.\\

Given the nature of proposition \ref{foliation}, we shall say that the $\FM_X$ are the leaves of an immersed foliation, which will be referred to as the \textbf{$\M$-isoperiodic foliation}

\medskip

\textbf{Examples.}

\begin{enumerate}

\item If $\M$ is the whole stratum $\moduli$, the foliation $\FM$ is the usual isoperiodic foliation (or Kernel foliation or Rel foliation).
\item Let $\M$ and $\N$ be two affine invariant orbifolds with $\M \subset \N$. Then the $\M$-isoperiodic leaf of any surface of $\M$ is contained in its $\N$-isoperiodic leaf.
\item Let $(X,\omega)$ be a flat cover of a Veech surface $(X_0,\omega_0)$ with ramification over the singularities of $\omega_0$ and a non periodic point. Let $\M$ be the $\GL$-orbit closure of $(X,\omega)$. Then the $\M$-isoperiodic leaf of any surface in $\M$ is isomorphic to the surface $X_0$ punctured at the singularities of $\omega_0$.
\item Let $\M$ be a Teichmuller disc. Then the $\M$-isoperiodic leaf of any surface $(X,\omega) \in \M$ is the singleton $\{(X,\omega)\}$.
\end{enumerate}

The last example of this list leads us to the following definition:

\begin{dfn}
An affine invariant orbifold $\M$ is said to be non absolute when leaves of $\FM$ have positive dimension. This is equivalent to the fact that any local model $V$ of $\M$ as in \eqref{localmodel} intersects $\mathrm{ker} \rho$ non trivially.
\end{dfn}

Notice that all the local models of $\M$ intersect $\mathrm{ker} \rho$ non trivially if, and only if, one of its model does. An affine invariant orbifold $\M$ is non absolute whenever $dim_{\mathbb{C}}(\M) > 2rk(\M)$. The $\M$-isoperiodic foliation behave nicely with respect to the action of $\GL$, as shows the following:

\begin{prop}\label{product}
Let $\mathcal{M}$ be a non absolute affine invariant orbifold, let $(X,\omega)$ be a translation surface in $\mathcal{M}$ and let $g \in \GL$. The following formulas holds :

\begin{equation}\label{commutation}
g \cdot \mathcal{F}^{\mathcal{M}}_X = \mathcal{F}^{\mathcal{M}}_{g\cdot X}
\end{equation}

and

\begin{equation}\label{product}
\GL \cdot \FM_{X} = \M
\end{equation}
\end{prop}

\begin{proof}
Equation \eqref{commutation} is a consequence of the fact the Period map is $\GL$-equivariant. For equation \eqref{product}, let $(X_1,\omega_1)$ be a translation surface in $\M$. Notice that sets $\GL \cdot \FM_{X}$ and $\GL \cdot \FM_{X_1}$ both contain an open set and since the action of $\GL$ on $\M$ is topologically transitive, we deduce that there is a matrix $g \in \GL$ such that $g \cdot \FM_X \cap \FM_{X_1}$ is non empty. But by \eqref{commutation} $g \cdot \FM_X = \FM_{g \cdot X}$ and thus $g \cdot \FM_X = \FM_{X_1}$. This concludes.
\end{proof}

Recall that the area of a surface $(X,\omega) \in \moduli$ is defined as the integral of $\frac{i}{2} \cdot \omega \wedge \overline{\omega}$ over $X$. It is well known that the area of $(X,\omega)$ only depends upon the integral of $\omega$ over the absolute cycles, and thus the area function is constant along the leaves of the $\M$-isoperiodic foliation. This leads to the following definition.

\begin{dfn}\label{projectivedensity}
Let $\M$ be a non absolute affine invariant orbifold, and let $(X,\omega)$ be a surface in $\M$. The leaf $\FM_X$ is said to be projectively dense if it dense in the set of surfaces in $\M$ that have the same area as $(X,\omega)$.
\end{dfn}

\subsection{Prym eigenform loci}

In this section, we recall a construction giving an infinite family of rank one affine invariant orbifolds discovered by McMullen. Let $(X,\omega)$ be a translation surface endowed with a holomorphic involution $\tau$. We denote by $\Omega(X)$ the set of holomorphic $1$-forms, and by $\Omega^-(X)$ the set of $\tau$-anti invariant holomorphic $1$-forms. We say that $(X,\omega)$ is a Prym form if $\omega \in \Omega^-(X)$, that is $\tau^{\ast} \omega = - \omega$, and $dim \ \Omega^-(X) = 2$. The Prym variety $Prym(X,\tau)$ is defined as the 2-dimensional abelian variety $(\Omega^-(X))^{\ast} / H_1^-(X,\mathbb{Z})$ endowed with the polarization coming from the intersection form on $H_1(X,Z)$. Finally, let $D$ be a positive integer congruent to 0 or 1 mod 4 and let $\mathcal{O}_D \simeq \mathbb{Z}[X]/(X^2+bX+c)$ be the real quadratic order of discrimant $D = b^2-4c$.

\begin{dfn}
A Prym eigenform is a Prym form $(X,\omega)$ corresponding to an involution $\tau$ such that there is an injective ring morphism $\mathfrak{i} : \mathcal{O}_D \to End(Prym(X,\tau))$ satisfying the following properties:

\begin{enumerate}
\item $\mathfrak{i}(\mathcal{O}_D)$ is a proper self adjoint subring of $End(Prym(X),\tau)$.
\item $\omega$ is an eigenvector for the action of $\mathcal{O}_D$ on $\Omega(X)^{-}$
\end{enumerate}
\end{dfn}

We will denote by $\Omega E_D(\kappa)$ the set Prym eigenforms as in the previous definition contained in the stratum $\moduli$. For more details, see \cite{mcmullen2007prym}. McMullen proved the following:

\begin{prop}
The $\GL$-orbit closure of any Prym eigenform is a rank one affine invariant orbifold.
\end{prop}

The Prym eigenforms play a crucial role in McMullen's classification of affine invariant orbifold in genus $2$. It is proved in \cite{mcmullen2007dynamics} that if the orbit of a surface is neither closed nor dense in the stratum in which it belongs, then it is a Prym eigenform, the Prym involution being given by the hyperelliptic involution. In \cite{mcmullen2007prym}, infinite families of Prym eigenforms are constructed in genus up to $5$, and it is a consequence of Riemann-Hurwitz formula that Prym eigenforms can not exist for genus bigger than 5.

\section{Modifying the twist parameters}\label{twists}

In this section, we establish a number of useful results and tools to deform translation surfaces.

\subsection{The twist map}

A classic and useful construction to navigate inside the moduli space is as follows. Suppose $(X,\omega)$ is periodic in direction $\theta$ and decomposed into $m$ cylinders, which we denote by $\mathcal{C}_1, \cdots, \mathcal{C}_m$. We denote by $\gamma_i$ the core curve of the cylinder $\mathcal{C}_i$. One can define a smooth map (in the orbifold sense) $\tau$ from $\mathbb{R}^m$ to $\moduli$ in the following way: choose $(x_1,\cdots,x_m)$ in $\mathbb{R}^m$, cut open the surface along the core curves of its cylinders, rotate the top component of each cylinder $\mathcal{C}_i$ by an amount of $x_ic_i$ and glue back. This construction yields a smooth embedding from the $m$-dimensional torus to $\moduli{}$. See figure 1. This embedding, called the twist map, is of particular interest as it allows to see specific deformations of surfaces as linear flows, of which we have a detailed understanding.

\begin{figure}[h]
    \centering
\begin{tikzpicture}
    \begin{scope}[shift={(0,0)}]
    \draw (1,0) rectangle (2,0.5);
    \draw (0,.5) rectangle (2,1);
    \draw (0,1) rectangle (1,1.5);
    \draw (1,-.5) node{$X$};
    \draw[>=latex,->](2.5,.75) .. controls +(45:.5) and +(135:.5) .. (4.5,.75) ;
    \end{scope}

    \begin{scope}[shift={(5,0)}]
    \draw (1,0.5) -- (.5,0) -- (1.5,0) -- (2,.5);
    \draw (0,.5) rectangle (2,1);
    \draw (0,1) -- (1,1.5) -- (2,1.5) -- (1,1);
    \draw (1,-.5) node{$\tau(c_1,0,-c_3/2)$};
    \draw[fill=black,opacity = .1] (1,1) -- (1,1.5) -- (2,1.5) -- cycle;
    \draw[dashed] (1,1) -- (1,1.5);
    \end{scope}

    \begin{scope}[shift={(8,0)}]
    \draw (-.5,0.75) node{$=$};
    \draw (1,0.5) -- (.5,0) -- (1.5,0) -- (2,.5);
    \draw (0,.5) rectangle (2,1);
    \draw (0,1) rectangle (1,1.5);
    \draw (1,-.5) node{$\tau (0,0,-c_3/2)$};
    \draw[fill=black,opacity = .1] (0,1) -- (0,1.5) -- (1,1.5) -- cycle;
    \draw[dashed] (0,1) -- (1,1.5);
    \end{scope}

    \end{tikzpicture}
	 \caption{twisting cylinders}
    \label{twistmap}
\end{figure}

\medskip

More precisely, let $f : S \to X$ be a marking on $X$, and define $\xi = \Phi(X,\omega,f)$. Recall that the geometric intersection induces a non degenerate form $I : H_1(S-\Sigma,-\mathbb{Z}) \times H_1(S,\Sigma,\mathbb{Z}) \to \mathbb{Z}$. The core curves of the cylinders can be seen as elements of $H_1(S-\Sigma,\mathbb{Z})$ and we denote by $C_i^{\ast}$ the cocyle $I(f^{-1}\circ \gamma_i,\cdot)$. Define $E$ to be the linear subspace of $H^{1}(S,\Sigma,\mathbb{R})$ spanned by the $C_i^{\ast}$ and finally set $\mathcal{C}$ to be the arc-wise component of $\{(Y,\theta,h) \in \teich \ | \ \Phi(Y,\theta,h) \in \xi + E\}$ that contains $(X,\omega,f)$.

\begin{prop}
The period map induces a diffeomorphism from $\mathcal{C}$ to $\xi + E$.
\end{prop}

Now, define a map $\tilde{\tau}$ from $\mathbb{R}^m$ to $\teich$ that sends $(x_1,\cdots,x_m)$ to the surface of $\mathcal{C}$ whose image by the period map is $\xi+ \sum c_ix_iC_i^{\ast}$. The action of $\mathbb{Z}^m$ on $\mathbb{R}^m$ by translation is intertwined by $\tilde{\tau}$ with the action of the subgroup of $\Gamma_g$ spanned the Dehn twists about the core curves of the cylinders. There is also a finite subgroup $\Delta$, perhaps trivial, of automorphisms of $(X,\omega)$ that acts on $X$ by permuting cylinders and on the $m$-dimensional torus $\mathbb{T}^m$ by permuting the coordinates. Consequently, there is a map $\tau : \mathbb{T}^m / \Delta$ to $\moduli$ that fits into the commutative diagram depicted in figure \ref{commutativity}. We shall refer to the map $\tau$ as the twists map associated to $(X,\omega)$. Notice that by construction $\Phi \circ \tilde{\tau}$ is affine, and its constant part is $\xi$. For a similar construction and more details, the reader is referred to the section 6.2 of \cite{hooper2015rel}.

\begin{figure}[h]
    \centering
    \begin{tikzpicture}
    \draw[>=latex,->] (1,0) -- (2,0) ;
    \draw[>=latex,->] (1,-1.5) -- (2,-1.5);
    \draw[>=latex,->] (0,-.5) -- (0,-1) ;
    \draw[>=latex,->] (3,-.5) -- (3,-1) ;
    \draw (0,0) node{$\mathbb{R}^m$};
    \draw (0,-1.5) node{$\mathbb{T}^m / \Delta $};
    \draw (3,0) node{$\teich$};
    \draw (3,-1.5) node{$\moduli$};
    \draw (1.5,0) node[above]{$\tilde{\tau}$};
    \draw (1.5,-1.5) node[above]{$\tau$};
    \draw (-.1,-.75) node[left]{$q$};
    \draw (3.1,-.75) node[right]{$\pi$};
    \end{tikzpicture}
    \caption{the twist map}
    \label{commutativity}
\end{figure}

\medskip

\subsection{Navigating inside affine invariant orbifolds with twists}

Let $\mathcal{M}$ is a non absolute affine invariant orbifold that contains the surface $(X,\omega)$. We denote by $k$ the rank of $\M$ and by $r$ the dimension of the foliation $\FM$. This last quantity is known as the Rel of $\M$. From now on, we assume that the cylinder decomposition of $(X,\omega)$ is \textbf{$\mathcal{F}^\mathcal{M}$-stable}: this means that any every horizontal saddle connection of X vanishes as an element of $T^{\ast}_X\mathcal{F}_X^{\mathcal{M}}$. Equivalently, this means that there are no saddle connections joining different singularities along a boundary of a cylinder, or if there is one it has to remain so along any isoperiodic deformation inside $\mathcal{M}$ that preserves the cylinder decomposition. The stable decomposition is relevant in our context as they allow to recognize some isoperiodic deformation as cylinder deformations. The following figure shows an example of isoperiodic deformation that is not expressed as a cylinder deformation:

\begin{center}
    \begin{tikzpicture}[scale = 0.7]
    \begin{scope}[shift={(0,0)}]
    \draw (0,0) rectangle (2,1);
    \draw (.7,1) rectangle (2.7,2);
    \draw (0,0) node{$\times$};
    \draw (.7,0) node{$\bullet$};
    \draw (2,0) node{$\times$};
    \draw (0,1) node{$\times$};
    \draw (.7,1) node{$\bullet$};
    \draw (2,1) node{$\times$};
    \draw (2,2) node{$\times$};
    \draw (.7,2) node{$\bullet$};
    \draw (2.7,1) node{$\bullet$};
    \draw (2.7,2) node{$\bullet$};
     \draw[>=latex,->](3.5,1) .. controls +(45:.5) and +(135:.5) .. (4.5,1) ;
    \end{scope}

    \begin{scope}[shift={(5.3,0)}]
    \draw (0,0) rectangle (2,1);
    \draw (1.6,1) rectangle (3.6,2);
    \draw (0,0) node{$\times$};
    \draw (1.6,0) node{$\bullet$};
    \draw (2,0) node{$\times$};
    \draw (0,1) node{$\times$};
    \draw (1.6,1) node{$\bullet$};
    \draw (2,1) node{$\times$};
    \draw (2,2) node{$\times$};
    \draw (1.6,2) node{$\bullet$};
    \draw (3.6,1) node{$\bullet$};
    \draw (3.6,2) node{$\bullet$};

    \end{scope}

    \end{tikzpicture}
\end{center}

Let $\hat{\iota} : \mathfrak{M} \to \moduli$ and $V \in H^1(S,\Sigma,\mathbb{C})$ as in proposition \ref{characterisation} such that $(X,\omega,f)$ belongs to the image of $\hat{\iota}$. Define $\V$ and $\K(X)$ to be the preimages of $V$ and $\xi + Ker \rho \cap V$ by the map $\Phi \circ \tilde{\tau}$. Those spaces are linear as preimages of affine subspaces by an affine map that contain $0$. Notice that by construction $\tau \circ q(\V)$ is contained in $\M$ and $\tau \circ q(\K(X))$ is contained in $\FM_X$. We emphasize that $\V$ and $\K(X) $ are defined up to the choice of the pair $(\hat{\iota},V)$.

\begin{lem}\label{VK}
The space $\V(X)$ is rational and its dimension is at most $k+r$, the dimension of $\K(X)$ is exactly $r$.
\end{lem}

\begin{proof}

\textbf{1.} Choose a neighborhood $\mathcal{U}$ of $(X,\omega,f)$ in $\teich$ such that any point point in $\mathcal{U}$ represents a surface in $\M$ if, and only if, its image by the period map is contained in $V$. Now, let $\V(X)^{\mathbb{Q}}$ be the smallest rational linear subspace that contains $\V(X)$. It is known that $q(\V(X))$ is dense in $q(\V(X)^{\mathbb{Q}})$ and since $\M$ is closed, $\tau \circ q(\V(X)^{\mathbb{Q}})$ is contained in $\M$. As a consequence, $\Phi \circ \tilde{\tau}(\V(X)^{\mathbb{Q}}) \cap \Phi(\mathcal{U})$ is contained in $V$. By definition of $\V$, this implies that $\V^{\mathbb{Q}}$ is contained in $\V$, hence $\V$ is rational. Now, notice that $\rho (d_0\tau(\mathbb{R}^m))$ is isotropic as it is spanned by the $C_i^{\ast}$ and the $\gamma_i$ do not intersect. By \cite{avila2017symplectic}, we know that $\rho(V)$ is symplectic and has dimension $2k$ by definition. This implies that $\rho (d_0\tau(\V(X))$ has dimension at most $k$ and since the dimension of $ker \rho \cap V$ is $r$ by definition, we deduce that $V$ has dimension at most $k+r$. \\

\textbf{2.} We shall prove that the linear part of $\Phi \circ \tilde{\tau}$ is onto $Ker \rho \cap V $. Let $u$ be in $Ker \rho \cap V$. Since the cylinder decomposition of $X$ is stable, it implies that $u$ vanishes on the cycle forming the boundaries of the cylinders. It also vanishes on the core curves of the cylinders as they are absolute cycles. Consequently, the only cycles on which it may not vanish are the cross curves of the cylinders (those are path inside a cylinder that connects the bottom component to the top component), which we denote by $\sigma_i$. Consequently, we get that $u = \sum u(\sigma_i)C_i^{\ast}$, and thus $u = \Phi \circ \tilde{\tau}(u(\sigma_1)c_i^{-1},\cdots,u(\sigma_m)c_m^{-1})-\xi$. As the dimension of $Ker \rho \cap V$ is $r$ by definition and $\Phi \circ \tilde{\tau}$ is injective, we get that the dimension of $\K(X)$ is $r$.
\end{proof}

The case of equality for the dimension of $\V $ in the previous proposition is obtained precisely when the two subspaces $Pres(X,\M)$ and $Twist(X,\M)$ defined in \cite{wright2015cylinder} are equal. Reciprocally, if we do not require the decomposition of $X$ to be $\FM$-stable, the dimension of $\K(X)$ is only bounded above by $r$. This expresses the fact that some isoperiodic deformations that preserve that cylinder decomposition of $(X,\omega)$ will not be recognized as twist deformations. \\

\textbf{Examples.}

\begin{itemize}
\item If $\M = \moduli$ then $\V(X) = \mathbb{R}^m$ and is indeed rational.
\item If $\M$ is a connected component of $\Omega E_D(\kappa)$ then $v \in \K(X)$ if and only if $\tau(v)$ is in $\mathcal{F}_X$ and the Prym involution of $(X,\omega)$ persits on $\tau(v)$. This comes from the fact that the condition defining Prym eigenforms among Prym forms does not involve relative periods.
\end{itemize}

\begin{dfn}
The support of $u \in \K(X)$ is the collection of indices $i$ so that $u_i \neq 0$. If there are no other vector in $\K(X)$ whose support is stricly contained the one of $u$, we say that u is a minimal $\M$-isoperiodic deformation. Finally, two minimal $\M$-isoperiodic deformations are said to be transverse if their supports are disjoint.
\end{dfn}

The support of $u$ records the information of which cylinders have been twisted on $(X,\omega)$ to get to the surface $\tau(u)$, and $u$ is a minimal $\M$-isoperiodic deformation if it is not possible to twist some cylinders in the support of $u$ while staying inside $\FM_X$ without twisting the other cylinders of the support of $u$. Finally, we define the degree of $u \in \mathbb{R}^m$ as the $ dim \ \mathrm{Span}_{\mathbb{Q}} \langle u_i \rangle - 1$

\begin{dfn}[Property $\mathcal{P}$]
The surface $(X,\omega)$ is said to have property $\mathcal{P}$ if it has a horizontal $\FM$-stable cylinder decomposition such that $\K(X)$ contains a minimal $\M$-isoperiodic deformation of degree at least $1$. By extension, we say that an affine invariant orbifold $\M$ has property $\mathcal{P}$ if it contains a surface that has property $\mathcal{P}$.
\end{dfn}

\textbf{Remark.} Having property $\mathcal{P}$ does not depend on the choice of the pair $(\hat{\iota},V)$ even though the spaces $\V(X)$ and $\K(X)$ do, as we mentionned. This is due to the fact $MCG(S,\Sigma)$ acts on $H^1(S,\Sigma,\mathbb{C})$ by integer matrices.

\section{Property $\mathcal{P}$ and density of the leaves}\label{p}

\subsection{Proof of Theorem \ref{A}}

\begin{prop}\label{containshorocycle}
Let $\mathcal{M}$ be a non absolute rank one affine invariant orbifold and suppose $(X,\omega)$ is a surface in $\M$ that has property $\mathcal{P}$. Then the closure of $\FM$ contains the horocycle orbit of $(X,\omega)$.
\end{prop}

\begin{proof}
Let $u \in \K(X)$ be a minimal deformation of positive degree. There is a rational subspace $V_u \subset \mathbb{R}^m$ of dimension $1+deg(u)$ so that $q(V_u)$ is the closure of $\{q(t\cdot u), \ t \in \mathbb{R}\}$. By proposition \ref{VK}, $V_u$ is contained in $\V$. However, it is not contained in $\K(X)$. Indeed, the support of any vector in $V_u$ is contained in the one of $u$, this is due to the fact that vanishing of a coordinate is a rational linear relation that has to be satisfied by any vector in $V_u$. But since the degree of $u$ is at least $2$, there is a vector for which this inclusion is strict. By definition, such a vector can not be in $\K(X)$. Now, the subpspace $\K(X)+ V_u$ contains $\K(X)$ and this inclusion is strict. Consequently, its dimension is at least $k+1$. But, since $\M$ is rank $1$, the dimension of $\V(X)$ is at most $k+1$. Hence, $\V(X) = K + V_u$ and the dimension of $\V(X)$ is $k+1$. As a consequence of this equality, we deduce that $\tau \circ q (V_{\mathcal{M}})$ is contained in the closure of  $\mathcal{F}_X^{\mathcal{M}}$. Indeed, let $w = k + v \in V_{\mathcal{M}}$, with $v \in V_u$. Then there is a growing sequence $t_n > 0$ so that $q (t_n \cdot u)$ converges to $v$. But then, by continuity, $\tau \circ q (k + t_n \cdot v)$ is a sequence of points in $\mathcal{F}_X^{\mathcal{M}}$ converging to $\tau \circ q (w)$. To conclude, notice that $h_t \cdot (X,\omega) = \tau(t\cdot(h_1c_1^{-1},\cdots,h_mc_m^{-1}))$ is contained in $\tau \circ q(\V(X))$.
\end{proof}

\begin{prop}\label{whole}
Let $\mathcal{M}$ be a non absolute rank 1 affine invariant orbifold and $(X,\omega) \in \mathcal{M}$. If the horocycle orbit of $(X,\omega)$ is contained in its isoperiodic leaf, then so is its $\SL$-orbit.
\end{prop}

\begin{proof}
We will need the following lemma:

\begin{lem}
The $\M$-isoperiodic leaf of $(X,\omega)$ contains a surface fixed by a non trivial hyperbolic matrix.
\end{lem}

\begin{proof}[proof of the lemma]
It is proved in, for instance, \cite{wright2014field} that any affine invariant orbifold contains a surface fixed by a non trivial hyperbolic matrix, using a closing lemma for the geodesic flow. Let $(X_1,\omega_1)$ be such a surface associated to a hyperbolic matrix $g$. Now corollary \ref{product} applied to $(X_1,\omega_1)$ shows that there is a matrix $u$ that sends $(X_1,\omega_1)$ to the $\M$-isoperiodic leaf of $(X,\omega)$. The surface $u \cdot (X_1,\omega_1)$ is fixed by the hyperbolic matrix $ugu^{-1}$.
\end{proof}

Now, pick a surface $(X_0,\omega_0) \in \FN_X$ as in the previous lemma, with $g \cdot (X_0,\omega_0) = (X_0,\omega_0)$ for some hyperbolic matrix $g$ and denote by $\theta$ the angle between the horizontal line and the expanding line of this matrix. Using proposition \eqref{commutation} and the fact that the closure of $\FM_X$ is saturated by $\FM$ we show that the horocycle orbit of $(X_0,\omega_0)$ is also contained in the closure of the leaf of $(X,\omega)$. Let $\theta_n$ be the angle between the horizontal line and the image under $g^n$ of this horizontal line. Note that this sequence converges to $\theta$. If $H_{\theta_n} \cdot (X,\omega)$ is contained in the closure of $\FM_X$, using proposition \ref{commutation}, one gets that $g \cdot H_{\theta_n} \cdot (X,\omega)$ is also contained in the closure of $\FM_X$. But since $g\cdot H_{\theta_n} = H_{\theta_{n+1}} \cdot  g$, the latter is exactly $H_{\theta_{n+1}} \cdot g \cdot (X,\omega)$. As $g \cdot (X,\omega) = (X,\omega)$, this gives that for all $n >0 \ H(\theta_n)\cdot (X,\omega)$ is contained in the closure of $\FM_X$. By continuity, this implies that the horocycle orbit in direction $\theta$ of $(X,\omega)$ also is contained in the closure of its $\M$-isoperiodic leaf. To conclude, use proposition 5.2 in \cite{hooper2015rel} to deduce that the closure of $\FM_X$ contains the $\SL$-orbit of $(X_1,\omega_1)$. Once again, we use proposition \ref{commutation} to deduce that the $\SL$-orbit of $(X,\omega)$ is also contained in the closure of its $\M$-isoperiodic leaf.
\end{proof}

We can now prove:
\begin{thm}[Theorem \ref{A}]\label{crit1}
Let $\mathcal{M}$ be a non absolute rank one affine invariant orbifold having property $\mathcal{P}$. Then the $\M$-isoperiodic foliation is projectively minimal.
\end{thm}

\begin{proof}
Let $(X,\omega)$ be a surface in $\M$ that has property $\mathcal{P}$. According to propostion \ref{containshorocycle}, the $\SL$-orbit of $(X,\omega)$ is contained in the closure of its $\M$-isoperiodic leaf. Using equation \eqref{commutation} and the fact the closure of $\FM_X$ saturated by $\FM$, we deduce that the closure $\FM_X$ is invariant under the action of $\SL$. Using equation \eqref{product}, we conclude that the $\M$-isoperiodic leaf of $(X,\omega)$ is projectively dense. Finally, another utilisation of equation \eqref{product} together with \ref{commutation} proves that there is an element of $\GL$ that sends the $\M$-isoperiodic leaf of $(X,\omega)$ to the $\M$-isoperiodic leaf of any other surface in $\M$. Since this action is continuous, all the leaves are projectively dense.
\end{proof}

In section 5, we will give examples of rank 1 affine invariant orbifolds that have property $\mathcal{P}$. However, not all of them have it, as shows the following example.

\begin{prop}\label{counterexample} Let $(X_0,\omega_0)$ be a Veech surface, that is a surface whose $\GL$-orbit is closed, and let $p: (X,\omega) \to (X_0,\omega_0)$ be a flat cover: a holomorphic map such that $p^{\ast}\omega_0 = \omega$. Finally, let $\M$ be the $\GL$-orbit closure of $(X,\omega)$. Then, $\M$ is a rank 1 affine invariant orbifold such that $\FM_X$ is closed. In particular, if $\N$ is an arithmetic rank 1 affine invariant orbifold, then the leaves of $\FN$ are closed.
\end{prop}

\begin{proof} Let us prove the first assertion first. Let $\M_0$ be the $\GL$-orbit of $(X_0,\omega_0)$. It is easy to show that any surface in $\M$ is also a flat cover over a surface in $\M_0$. Thus there is a $\GL$-equivariant map $\Psi$ from $\M$ to $\M_0$ that maps a surface to the unique surface it covers, and this map is continuous. To conclude, it remains to notice that the leaf $\FM_X$ is a connected component of the preimage of $(X_0,\omega_0)$ by $\Psi$, which is closed.

\medskip

For the second assertion, notice that in this case, $\N$ contains a surface whose period are rational and thus is a flat cover over a square torus, which is a Veech surface. Up to flowing along $\FN$, we get a new surface that still is a flat cover of a torus and whose $\GL$-orbit closure is $\M$. Then the second assertion follows from the first one.
\end{proof}

\textbf{Remark.} The affine invariant orbifold $\M$ in proposition \ref{counterexample} is non absolute if and only if the set of ramification points of $p$ cointains a non periodic point. See \cite{Gutkin2003affine} for more details.

\subsection{Structure of the closure of the leaves}

Theorem \ref{A} allows us to compute the closure of the leaves of the full isoperiodic foliation $\mathcal{F}$ in $\moduli$.

\begin{thm}\label{structure}

If $(X,\omega)$ is a surface in a rank 1 affine invariant orbifold $\M$ with property $\mathcal{P}$, then the isoperiodic leaf $\mathcal{F}_X$ is projectively dense in an affine invariant orbifold $\N$. Furthermore, if $K_{\moduli}(X)$ contains $m$ transverse minimal isoperiodic deformations of degree $(d_i)_{i \leq m}$, then :

$$
rk(\mathcal{N}) \geq \sum d_i
$$

\end{thm}

\begin{proof}
We show as in the proof of Theorem \ref{A} that the closure of $\mathcal{F}_X$ is invariant under the action of $\SL$. Therefore the leaf $\mathcal{F}_X$ is projectively dense in a affine invariant orbifold $\mathcal{N}$. Now, notice that $K_{\moduli}(X) = K_{\mathcal{N}}(X)$, and denote by $u_i$ be the family of transverse minimal deformations of the statement. Denote also by $V_i$ the associated subspace of $V_{\mathcal{N}}$ that gives the closure of $\{q(t\cdot u_i), \ t > 0\}$ , and by $p_{\mathcal{N}}$ the canonical projection from $\VN$ to $\VN / \KN$. We claim that the $p_{\mathcal{N}}(V_i)$ form a direct sum. Indeed, let $v_i \in V_i$ so that $\sum p_{\mathcal{N}}(v_i) = 0$. Then, there is a $k \in \KN$ so that $\sum v_i = k$. Let $k_i \in \KN \cap V_i$ so that $k = \sum k_i$. We then have $\sum (v_i - k_i) = 0$. Notice that since the $u_i$ are transverse, if two vectors belong to different $V_i$, their support are disjoint. This implies that $v_i = k_i$, and thus $p_{\mathcal{N}}(v_i) = 0$. We are thus left with proving that the dimension of $p_{\mathcal{N}}(V_i)$ is $m_i$. Proposition 1.4.1 of \cite{katok1995introduction} implies that for any $i$, there is an independant family $v_{i,j}$ of $m_i + 1$ deformations such that $v_{i,0} = v_i$ and the support of $v_{i,j+1}$ is strictly contained in the one of $v_{i,j}$. By definition of being minimal, this implies that the $v_{i,j}$ are not contained in $\KN$, for every $j \geq 1$. This gives that the dimension of $\KN / \VN$ is bounded below by $\sum d_i$, and since by lemma \ref{VK} the dimension of $\KN / \VN$ is bounded above by $rk (\mathcal{N})$.
\end{proof}

\section{Examples of rank 1 affine invariant orbifold with property $\mathcal{P}$}\label{examples}

\subsection{The genus 2 case}

\begin{prop}
Let $\M$ be a non arithmetic rank one affine invariant orbifold contained in $\mathcal{H}(1,1)$. Then $\M$ has property $\mathcal{P}$.
\end{prop}

\begin{proof}
Let $(X,\omega)$ be a horizontally periodic surface in $\M$, and up to flowing along the isoperiodic foliation, we can assume that this decomposition is $\FM$-stable. That means that $(X,\omega)$ is cylinder equivalent (meaning up to adjusting height and twist parameter of the cylinders) to the following surface:

\begin{center}
    \begin{tikzpicture}

    \draw (1,0) rectangle (2,.5);
    \draw (0,.5) rectangle (2,1);
    \draw (0,1) rectangle (1,1.5);

    \draw (1,0) node{$\bullet$};
    \draw (2,0) node{$\bullet$};
    \draw (0,1) node{$\bullet$};
    \draw (1,1) node{$\bullet$};
    \draw (2,1) node{$\bullet$};

    \draw (0,.5) node{$\times$};
    \draw (1,.5) node{$\times$};
    \draw (2,.5) node{$\times$};
    \draw (0,1.5) node{$\times$};
    \draw (1,1.5) node{$\times$};

    \draw (1.5,.25) node{1};
    \draw (1,.75) node{2};
    \draw (.5,1.25) node{3};

    \draw (1,-1) node{fig. The only stable cylinder decomposition in genus 2};

    \end{tikzpicture}
\end{center}

This affirmation can be checked very easily as there are only two possibilities for the diagram of separatrices in this case, and they are isomorphic. See Appendix \ref{list} for more details. Then the vector $(\gamma_1,-\gamma_2,\gamma_3)$ belongs to $\K(X)$, and it is necessarily of positive degree or else proposition \ref{field} would imply that $\M$ is arithmetic.
\end{proof}

We recall that the non absolute rank one affine invariant orbifolds in genus 2 have been classified by McMullen in \cite{mcmullen2007dynamics}, and they are connected components of prym eigenfom loci. In this case $\mathcal{F}$ and $\FM$ are the same as the $\Omega E_D(1,1)$ are saturated by the isoperiodic foliation. These remarks, together with Theorem \ref{A} and proposition \ref{counterexample} then implies:

\begin{thm}
Let $(X,\omega)$ be a Prym eigenform in $\Omega E_D(1,1)$, then the leaf $\mathcal{F}_X$ is either closed or projectively dense in $\Omega E_D(1,1)$. The latter case occurs if, and only if, $D$ is not a square.
\end{thm}

This result was already known prior to that work. It is for instance part of the classification established in \cite{calsamiglia2015transfer}. However it was interesting to obtain an elementary proof of this result that does not make use of degeneration of translation surfaces as it is the case in the aforementioned paper.
\subsection{Prym eigenform loci in genus 3}

\begin{prop}\label{pforeigen}
Let $\kappa = (2,2)^{odd}, (2,1,1)$ or $(1,1,1,1)$ and let $\M$ be a connected component of $\Omega E_D(\kappa)$. Then $\M$ has property $\mathcal{P}$ if and only if D is not a square.
\end{prop}

\begin{proof}
First, notice that D is a square if and only if $\M$ is arithmetic as $k(\M) = \mathbb{Q}(\sqrt{D})$. The case where $D$ is a square has already been proved in proposition \ref{counterexample}. We assume from now on that $D$ is not a square. We treat separately the different strata:

\begin{itemize}
\item If $\M$ is contained in $\Omega E_D^{odd}(2,2)$, theorem $B$ of \cite{lanneau2014connected} states that $\M$ contains a translation surface that is cylinder equivalent to the following surface:

    \begin{center}
\begin{tikzpicture}

\draw (1,0) rectangle (2,.5) ;
\draw (0,.5) rectangle (3,1) ;
\draw (0,1) rectangle (1,1.5);
\draw (2,1) rectangle (3,1.5);

\draw (0,.5) node{$\times$};
\draw (1,.5) node{$\times$};
\draw (2,.5) node{$\times$};
\draw (3,.5) node{$\times$};
\draw (0,1.5) node{$\times$};
\draw (1,1.5) node{$\times$};
\draw (2,1.5) node{$\times$};
\draw (3,1.5) node{$\times$};

\draw (1,0) node{$\bullet$};
\draw (2,0) node{$\bullet$};
\draw (0,1) node{$\bullet$};
\draw (1,1) node{$\bullet$};
\draw (2,1) node{$\bullet$};
\draw (3,1) node{$\bullet$};

\draw (1.5,0.25) node{1};
\draw (1.5,0.75) node{2};
\draw (.5,1.25) node{3};
\draw (2.5,1.25) node{4};

\draw (.5,.25) node[scale = 0.7] {A};
\draw (.5,1.75) node[scale = 0.7] {A};
\draw (2.5,.25) node[scale = 0.7] {B};
\draw (2.5,1.75) node[scale = 0.7] {B};

\end{tikzpicture}
\end{center}

Clearly, the vector $v = (c^{-1}_1,-c^{-1}_2, c^{-1}_3, c^{-1}_4) \in \K(X)$ as the Prym involution persists on $\tau(v)$, and it is a minimal $\M$-isoperiodic deformation. If its degree is 0, that means that all the cylinders have a commensurable circumferences and thus that $\M$ is arithmetic by proposition \ref{field}, which is not. Then the degree of $v$ is at least one, and $\M$ has property $\mathcal{P}$.

\item If $\M$ is contained in $\Omega E_D(2,1,1)$, then once again theorem $B$ of \cite{lanneau2014connected} states that $\M$ contains a translation surface that is cylinder equivalent to the following surface:

\begin{center}
\begin{tikzpicture}

\draw (0,0) rectangle (1,.5);
\draw (0,.5) rectangle (2,1);
\draw (1,1) rectangle (2,1.5);
\draw (1,1.5) rectangle (3,2);
\draw (2,2) rectangle (3,2.5);

\draw (0,0) node{$\bullet$};
\draw (1,0) node{$\bullet$};
\draw (0,1) node{$\bullet$};
\draw (1,1) node{$\bullet$};
\draw (2,1) node{$\bullet$};

\draw (1,1.5) node{$\times$};
\draw (2,1.5) node{$\times$};
\draw (3,1.5) node{$\times$};
\draw (2,2.5) node{$\times$};
\draw (3,2.5) node{$\times$};

\draw (0,.5) node{$\otimes$};
\draw (1,.5) node{$\otimes$};
\draw (0,.5) node{$\otimes$};
\draw (1,2) node{$\otimes$};
\draw (2,2) node{$\otimes$};
\draw (3,2) node{$\otimes$};

\draw (.5,-.25) node[scale = .7]{A};
\draw (.5,1.25) node[scale = .7]{A};
\draw (2.5,1.25) node[scale = .7]{B};
\draw (2.5,2.75) node[scale = .7]{B};

\draw (.5,.25) node{1};
\draw (1,.75) node{2};
\draw (1.5,1.25) node{3};
\draw (2,1.75) node{4};
\draw (2.5,2.25) node{5};

\end{tikzpicture}
\end{center}

In this, the vector $v = (c^{-1}_1,-c^{-1}_2,2 c^{-1}_3,-c^{-1}_4, c^{-1}_5)$ lies in $\K(X)$, and it is minimal. Note that the Prym involution exchanges $1$ with $5$ and $2$ with $4$. For the same reason as in the previous case, the degree of this deformation is at least one.

\item If $\M$ is contained in $\Omega E_D(1^4)$, let $(X,\omega)$ be a horizontally periodic surface in $\M$. It is well known that those surfaces always exist in affine invariant orbifolds, without any assumption on the rank. See for instance corollary 6 of \cite{smillie2004minimal}. Up to flowing along $\FM$, we can assume that the corresponding cylinder decomposition is $\FM$-stable. We claim that we can even assume it is stable: it is not the case that there are two different singularities on the same component of a cylinder. Indeed, if there are two singularities on the same component, then these singularities are not swapped by the Prym involution, or else their cone angle would be $2\pi$. They can thus be disaligned by an $\M$-isoperiodic deformation. This implies that there are 6 cylinders, as can be seen when computing the Euler characteristic. \\

Now, an application of Riemann-Hurwtiz formula reveals that the corresponding Prym form have 4 fixed points. They are either on core curves of cylinders or on their boundary components. Notice that whenever a cylinder is fixed, there are 2 fixed points of the involution on the core curve of this cylinder. For parity reason, there are either 0 or 2 fixed cylinders. Consider first the case where no cylinders are fixed by the Prym form. There are thus 3 pairs of cylinders exchanged by the Prym involution and 4 fixed points. This means that there is a pair of exchanged cylinders that contain two fixed points on the union of their boundary components. But when a saddle connection contains such a point, this saddle connection is shared by the two cylinders and thus there is a cylinder in X that has a fixed point on its two boundary components. Chose a geodesic segment $\alpha$  inside this cylinder that joins those fixed points. The concatenation of $\alpha$ with its image by the prym involution is a geodesic loop fixed by the involution. By the complete periodicity of rank 1 affine invariant orbifolds (Theorem 1.5 of \cite{wright2015cylinder}), the direction of $\alpha$ is a new periodic direction for which there is at least one cylinder fixed by the prym involution. This direction can be assumed to be stable by the same argument that we used before. We can thus consider only the second case where two cylinders are fixed by the involution. Appendix \ref{list}, provides a list of such decompositions and thus $(X,\omega)$ is cylinder equivalent to one of the following surfaces:

\begin{center}
    \begin{tikzpicture}

    \begin{scope}

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        \draw (1,-.5) rectangle (2,0);

        \draw (0,1.25) node[scale = .7]{A};
        \draw (2.5,.75) node[scale = .7]{A};
        \draw (.5,-.25) node[scale = .7]{B};
        \draw (1.5,2.25) node[scale = .7]{B};
        \draw (1.5,-.75) node[scale = .7]{C};
        \draw (2.5,1.75) node[scale = .7]{C};

        \draw[color = blue] (1,-.5) node{$\bullet$};
        \draw[color = blue] (2,-.5) node{$\bullet$};
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        \draw[color = blue] (3,1.5) node{$\bullet$};

        \draw[color = blue] (0,0) node{$\times$};
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        \draw[color = blue] (0,0) node{$\times$};
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        \draw[color = blue] (2,2) node{$\times$};

        \draw[color = red] (0,.5) node{$\bullet$};
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        \draw[color = red] (.5,1) node{$\times$};
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        \draw[color = red] (2,1) node{$\times$};
        \draw[color = red] (3,1) node{$\times$};

        \draw (1.5,-.25) node{1};
        \draw (1,.25) node{2};
        \draw (.3,.75) node{3};
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        \draw (2,1.25) node{5};
        \draw (1.5,1.75) node{6};

        \draw (1.5,-1.5) node{1};

        \end{scope}

        \begin{scope}[shift = {(1,-4)}]
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        \draw (0,0) rectangle (2,.5);
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        \draw (.5,-.25) node[scale = .7]{B};
        \draw (4,2.25) node[scale = .7]{B};
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        \draw[color = red] (3,2) node{$\times$};
        \draw[color = red] (3.5,2) node{$\times$};
        \draw[color = red] (4.5,2) node{$\times$};

        \draw[color = red] (2,1.5) node{$\bullet$};
        \draw[color = red] (3,1.5) node{$\bullet$};
        \draw[color = red] (4,1.5) node{$\bullet$};

        \draw (2,-1) node{4};

        \draw (1,.25) node{1};
        \draw (3.3,.25) node{2};
        \draw (2,.75) node{3};
        \draw (3,1.25) node{4};
        \draw (2.5,1.75) node{5};
        \draw (3.8,1.75) node{6};

        \end{scope}

        \begin{scope}[shift = {(9,0)}]

        \draw[fill = black, opacity = .1] (0,.5) rectangle (2,1);
        \draw[fill = black, opacity = .1] (1,1.5) rectangle (3,2);

        \draw (0,0) rectangle (1,.5);
        \draw (0,.5) rectangle (2,1);
        \draw (0,1) -- (-.5,1.5) -- (.5,1.5) -- (1,1) -- cycle;
        \draw (1,1) rectangle (2,1.5);
        \draw (1,1.5) rectangle (3,2);
        \draw (1,2) rectangle (2,2.5);

        \draw (.5,-.25) node[scale = .7]{B};
        \draw (2.5,2.25) node[scale = .7]{B};
        \draw (1.5,.25) node[scale = .7]{C};
        \draw (1.5,2.75) node[scale = .7]{C};
        \draw (0,1.75) node[scale = .7]{A};
        \draw (2.5,1.25) node[scale = .7]{A};

        \draw[color = blue] (0,0) node{$\bullet$};
        \draw[color = blue] (1,0) node{$\bullet$};
        \draw[color = blue] (1,2) node{$\bullet$};
        \draw[color = blue] (2,2) node{$\bullet$};
        \draw[color = blue] (3,2) node{$\bullet$};

        \draw[color = blue] (-.5,1.5) node{$\times$};
        \draw[color = blue] (.5,1.5) node{$\times$};
        \draw[color = blue] (1,1.5) node{$\times$};
        \draw[color = blue] (1,1.5) node{$\times$};
        \draw[color = blue] (2,1.5) node{$\times$};
        \draw[color = blue] (3,1.5) node{$\times$};

        \draw[color = red] (0,.5) node{$\times$};
        \draw[color = red] (1,.5) node{$\times$};
        \draw[color = red] (2,.5) node{$\times$};
        \draw[color = red] (1,2.5) node{$\times$};
        \draw[color = red] (2,2.5) node{$\times$};

        \draw[color = red] (0,1) node{$\bullet$};
        \draw[color = red] (1,1) node{$\bullet$};
        \draw[color = red] (2,1) node{$\bullet$};

    \draw (1.5,-1) node{3};

    \draw (.5,.25) node{1};
    \draw (1,.75) node{2};
    \draw (.3,1.25) node{3};
    \draw (1.5,1.25) node{4};
    \draw (1.5,2.25) node{6};
    \draw (2,1.75) node{5};

    \end{scope}

    \begin{scope}[shift = {(4.5,0)}]
    \draw[fill = black, opacity = .1] (1,.5) rectangle (2,1);
    \draw[fill = black, opacity = .1] (1,1.5) rectangle (2,2);

    \draw (0,0) rectangle (2,.5);
    \draw (0,.5) -- (-.5,1) -- (.5,1) -- (1,.5) -- cycle ;
    \draw (1,.5) rectangle (2,1);
    \draw (1,1) rectangle (3,1.5);
    \draw (1,1.5) rectangle (2,2);
    \draw (2,1.5) -- (2.5,2) -- (3.5,2) -- (3,1.5) -- cycle;

    \draw (.5,-.25) node[scale = .7]{B};
    \draw (1.5,2.25) node[scale = .7]{B};
    \draw (1.5,-.25) node[scale = .7]{A};
    \draw (0,1.25) node[scale = .7]{A};
    \draw (2.5,.75) node[scale = .7]{C};
    \draw (3,2.25) node[scale = .7]{C};

    \draw[color = blue] (0,0) node{$\times$};
    \draw[color = blue] (1,0) node{$\times$};
    \draw[color = blue] (2,0) node{$\times$};
    \draw[color = blue] (-.5,1) node{$\times$};
    \draw[color = blue] (.5,1) node{$\times$};
    \draw[color = blue] (1,2) node{$\times$};
    \draw[color = blue] (2,2) node{$\times$};

    \draw[color = blue] (1,1.5) node{$\bullet$};
    \draw[color = blue] (2,1.5) node{$\bullet$};
    \draw[color = blue] (3,1.5) node{$\bullet$};

    \draw[color = red] (0,.5) node{$\bullet$};
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    \draw[color = red] (2,.5) node{$\bullet$};

    \draw[color = red] (1,1) node{$\times$};
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    \draw[color = red] (3,1) node{$\times$};
    \draw[color = red] (2.5,2) node{$\times$};
    \draw[color = red] (3.5,2) node{$\times$};

    \draw (1.5,-1) node{2};

    \draw (1,.25) node{1};
    \draw (.3,.75) node{2};
    \draw (1.5,.75) node{3};
    \draw (2,1.25) node{4};
    \draw (1.5,1.75) node{5};
    \draw (2.8,1.75) node{6};

        \end{scope}

    \begin{scope}[shift = {(8,-4)}]
    \draw[fill = black, opacity = .1] (0,0) rectangle (1,.5);
        \draw[fill = black, opacity = .1] (0,1) rectangle (3,1.5);

        \draw (0,0) rectangle (1,.5);
        \draw (0,.5) rectangle (2,1);
        \draw (0,1) rectangle (3,1.5);
        \draw (0,1.5) rectangle (2,2);
        \draw (2,1.5) -- (2.5,2) -- (3.5,2) -- (3,1.5) -- cycle;
        \draw (1,2) rectangle (2,2.5);

        \draw (.5,-.25) node[scale = .7]{A};
        \draw (.5,2.25) node[scale = .7]{A};
        \draw (1.5,.25) node[scale = .7]{B};
        \draw (3,2.25) node[scale = .7]{B};
        \draw (2.5,.75) node[scale = .7]{C};
        \draw (1.5,2.75) node[scale = .7]{C};

        \draw[color = red] (0,0) node{$\bullet$};
        \draw[color = red] (1,0) node{$\bullet$};
        \draw[color = red] (0,2) node{$\bullet$};
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        \draw[color = red] (2,2) node{$\bullet$};

        \draw[color = red] (0,.5) node{$\times$};
        \draw[color = red] (1,.5) node{$\times$};
        \draw[color = red] (2,.5) node{$\times$};
        \draw[color = red] (2.5,2) node{$\times$};
        \draw[color = red] (3.5,2) node{$\times$};

        \draw[color = blue] (0,1) node{$\times$};
        \draw[color = blue] (2,1) node{$\times$};
        \draw[color = blue] (3,1) node{$\times$};
        \draw[color = blue] (1,2.5) node{$\times$};
        \draw[color = blue] (2,2.5) node{$\times$};

        \draw[color = blue] (0,1.5) node{$\bullet$};
        \draw[color = blue] (2,1.5) node{$\bullet$};
        \draw[color = blue] (3,1.5) node{$\bullet$};

        \draw (1.5,-1) node{5};

        \draw (.5,.25) node{1};
        \draw (1,.75) node{2};
        \draw (1.5,1.25) node{3};
        \draw (1,1.75) node{4};
        \draw (1.5,2.25) node{6};
        \draw (2.8,1.75) node{5};

    \end{scope}
    \end{tikzpicture}
\end{center}

We provide a minimal $\M$-isoperiodic deformation of positive degree for each of those decompositions.

\begin{enumerate}
    \item $(c^{-1}_1,0,-c^{-1}_3,-c^{-1}_4,0, c^{-1}_6)$
    \item $(c^{-1}_1,-c^{-1}_2,0, c^{-1}_4,-2 c^{-1}_5,-c^{-1}_6)$
    \item $(c^{-1}_1,0, c^{-1}_3, c^{-1}_4,-2 c^{-1}_5, c^{-1}_6)$
    \item $(0, c^{-1}_2,-c^{-1}_3,0, c^{-1}_5, c^{-1}_6)$
    \item $(c^{-1}_1,0,-c^{-1}_3,0, c^{-1}_5, c^{-1}_6)$
\end{enumerate}

\end{itemize}
\end{proof}

Proposition \ref{pforeigen} together with Theorem \ref{A} and proposition \ref{counterexample} implies the following:

\begin{cor}[Theorem \ref{B}]
Let $\kappa = (2,2)^{odd}, (2,1,1)$ or $(1,1,1,1)$ and let $\M$ be a connected component of $\Omega E_D(\kappa)$. Then either all the leaves of $\FM$ are all closed or all the leaves of $\FM$ are projectively dense. The last case occurs if, and only if, $D$ is not a square.
\end{cor}

\subsection{The hyperelliptic case}

\begin{prop}\label{hyper}
Let $\mathcal{M}$ be a non absolute rank one affine invariant orbifold in $\mathcal{H}(g-1^2)^{hyp}$. Then $\M$ has property $\mathcal{P}$ if and only if it is not arithmetic.
\end{prop}

\begin{proof}
Let $(X,\omega)$ be a translation surface in $\M$, and choose a periodic direction on $X$. Up to a small isoperiodic deformation, we can ensure that the direction is $\FM$-stable (which is equivalent to $\mathcal{F}$-stable here as there are only two singularities). Note that the singularity present on a boundary component of a cylinder is different from the singularity present on the other boundary component. This is due to the fact that the hyperelliptic involution exchanges the singularities of $\omega$ while fixing all the cylinders. A consequence of that is that a minimal $\M$-isoperiodic deformation is supported on all the cylinders and has the form $(\delta_1c_1^{-1},\cdots,\delta_mc_m^{-1})$ with the $\delta_i$ belonging to $\{1,-1\}$. The fact that $\M$ is non arithmetic is equivalent to the fact that this deformation has non vanishing degree by \ref{field}. Therefore $\M$ has property $\mathcal{P}$ if and only if $\M$ is non arithmetic.
\end{proof}

\begin{thm}
Let $\M$ be a non absolute rank 1 affine invariant orbifold contained in $\mathcal{H}(g-1^2)^{hyp}$, and let $(X,\omega)$ be a surface in $\M$. Then the leaf of $\FM_X$ is either closed or projectively dense in $\M$. The latter case occurs if, and only if $\M$ is non arithmetic.
\end{thm}

Paul Apisa has classified the rank one affine invariant orbifolds in the hyperelliptic strata and showed that if $\M$ is a non arithmetic rank $1$ affine invariant orbifold, then it is a translation cover of a surface in $\Omega E_D(1,1)$. See \cite{apisa2017rank}.

\section{Isoperiodic leaves of genus 3 Prym eigenforms}\label{full}

In this section we compute the closure of the isoperiodic leaves or Prym eigenforms of genus 3. We emphasize the fact that Theorem \ref{A} dealt with the subfoliation $\FM$ induced on $\Omega E_D(\kappa)$ while in this section we deal with the full isoperiodic foliation.

\subsection{The case $\mathcal{H}^{odd}(2,2)$}

Let $\M$ be a connected component of $\Omega E_D^{odd}(2,2)$. In the case were there are only two singularities, the dimension of the leaves of $\FM$ is the same as the ones of $\mathcal{F}$. This observation readily implies that for any surface $(X,\omega) \in \M$, there is equality between $\FM_X$ and $\mathcal{F}_X$. Thus the following is only a reformulation of Theorem \ref{A}:

\begin{prop}
Let $(X,\omega)$ be a Prym eigenform in $\mathcal{H}^{odd}(2,2)$, then the leaf $\mathcal{F}_X$ either closed or projectively dense in the connected component of $\Omega E_D^{odd}(2,2)$ in which it belongs. The last case occurs if, and only if, $D$ is not a square.
\end{prop}

\subsection{The case $\mathcal{H}_3(\kappa)$ with $\vert \kappa \vert > 2$}

This section is dedicated to proving Theorem \ref{C}.

\begin{thm}[Theorem C]
Let $(X,\omega)$ be a Prym eigenform in  $\mathcal{H}_3(\kappa)$ with $\vert \kappa \vert >2$, then the leaf $\mathcal{F}_X$ is either closed or projectively dense. The last case occurs if, and only if, $D$ is not a square.
\end{thm}

\begin{proof}
To prove this result, we are going to use theorem \ref{structure} and additional computations to prove that if the leaf of $(X,\omega)$ is not closed, it has to be projectively dense in an affine invariant orbifold $\M$ of rank at least 2. However, the classification of Aulicino and Nguyen of rank 2 affine invariant orbifolds (\cite{aulicino2016rank}) shows that if an affine invariant orbifold is $\mathcal{F}$-saturated, as is $\M$, it cannot be rank 2. Thus the rank is 3, and by \cite{mirzakhani2018full}, it has to be the whole stratum, since the hyperelliptic locus is not saturated by $\mathcal{F}$ neither.

\medskip

If $\M$ is contained in $\mathcal{H}(2,1,1)$, then we saw in the course of the proof of proposition \ref{pforeigen}, that we can assume that $(X,\omega)$ is horitonally periodic with $5$ cylinders, and that $u = (_1,-c^{-1}_2, c^{-1}_3,0,0)$ and $v = (0,0, c^{-1}_3,-c^{-1}_4, c^{-1}_5)$ belong to $ \K(X)$ and have a positive degree, with $c^{-1}_1 = c^{-1}_5$, and $c^{-1}_2 = c^{-1}_4$. The projection of $\K(X)$ on the two first coordinates is a 2-dimensional space, or $c_1$ and $c_2$ would be $\mathbb{Q}$-dependant, and the relation $c_2 = c_1 + c_3$ would imply that the degree of $u$ vanishes. Similarly, the projection of $\K(X)$ on the two last factors is also a 2-dimensional space. In particular, the dimension of $\K(X)$ is at least 4. By lemma \ref{VK}, we have $r+2 \geq 4$, and thus the rank of $\M$ is at least $2$.

\medskip

If $\M$ is contained in $\mathcal{H}(1^4)$, we saw that $(X,\omega)$ can be assumed to be cylinder equivalent to one of the following surfaces:

\begin{center}
    \begin{tikzpicture}

    \begin{scope}

        \draw (0,0) rectangle (2,.5);
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        \draw (1.5,-.75) node[scale = .7]{C};
        \draw (2.5,1.75) node[scale = .7]{C};

        \draw[color = blue] (1,-.5) node{$\bullet$};
        \draw[color = blue] (2,-.5) node{$\bullet$};
        \draw[color = blue] (1,1.5) node{$\bullet$};
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        \draw[color = blue] (3,1.5) node{$\bullet$};

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        \draw[color = blue] (1,0) node{$\times$};
        \draw[color = blue] (2,0) node{$\times$};
        \draw[color = blue] (0,0) node{$\times$};
        \draw[color = blue] (1,2) node{$\times$};
        \draw[color = blue] (2,2) node{$\times$};

        \draw[color = red] (0,.5) node{$\bullet$};
        \draw[color = red] (1,.5) node{$\bullet$};
        \draw[color = red] (2,.5) node{$\bullet$};

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        \draw[color = red] (2,1) node{$\times$};
        \draw[color = red] (3,1) node{$\times$};

        \draw (1.5,-.25) node{1};
        \draw (1,.25) node{2};
        \draw (.3,.75) node{3};
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        \draw (2,1.25) node{5};
        \draw (1.5,1.75) node{6};

        \draw (1.5,-1.5) node{1};

        \end{scope}

        \begin{scope}[shift = {(1,-4)}]
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        \draw (0,0) rectangle (2,.5);
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        \draw (.5,-.25) node[scale = .7]{B};
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We provide a pair of minimal transverse $\M$-isoperiodic deformations for each of the first four decompositions:

\begin{enumerate}
    \item $(c^{-1}_1,0,0,0,-c^{-1}_5, c^{-1}_6)$ and $(0, c^{-1}_2,-c^{-1}_3,-c^{-1}_4,0,0)$
    \item $(c^{-1}_1,-c^{-1}_2,0,0,-c^{-1}_5,0)$ and $(0,0, c^{-1}_3,-c^{-1}_4,0, c^{-1}_6)$
    \item $(c^{-1}_1,0,0,0,-c^{-1}_5, c^{-1}_6)$ and $(0, c^{-1}_2,-c^{-1}_3,-c^{-1}_4,0,0)$
    \item $(c^{-1}_1,0,0,0,-c^{-1}_5,-c^{-1}_6)$ and $(0, c^{-1}_2,-c^{-1}_3, c^{-1}_4,0,0)$
\end{enumerate}

Those minimal deformations have to be a positive degree, and we conclude in these cases using proposition \ref{structure}. Now, if $(X,\omega)$ is cylinder equivalent to the fifth surface, then $\K(X)$ contains the following three vectors $(c^{-1}_1,0,0-c^{-1}_4,0, c^{-1}_6), (0, c^{-1}_2,0,-c^{-1}_4,-c^{-1}_5, c^{-1}_6)$ and $(0,0, c^{-1}_3,-c^{-1}_4,-c^{-1}_5,0)$, and each of them is of positive degree. Using the same argument as in the case where $\M \subset \mathcal{H}(2,1,1)$, we conclude that $k + 3 \geq 5$, and thus the rank of $\M$ has to be at least $2$.

\end{proof}

\newpage

redpajama_set_name: RedPajamaArXiv


Why do you feel the fear of failing, in the first place, in some situations?
And what are the situations that aren't contaminated by that feeling?
Analyzing that, allows you to see what is missing when the fear of failing kicks in. And then you can move in the right direction to fill in the missing pieces and leave no space for the fear of failure, without the need to inhibit it or overcome it.
Let's say you need to talk to a group of friends about this great event there will be next week and you KNOW they'll love it and they will be grateful that you told them, it is safe to suppose that you'd feel good about it.
Or perhaps you are a bit in a rush, but you know you are a good driver, so you decide to drive a little bit faster to get there on time, no big deal, you've done that many times.
Or maybe you just want a snack, so you decide to stand up, go to the fridge, open it and take that delicious thing you were craving for.
Would you feel the fear of failing in any of these situations? Probably not. You know've never experienced failure in these situations in the past. Or, if you did, you figured out why and how to avoid that. In both cases, You KNOW, that's the point.
But let's get back to the example with the fridge for a minute. Let's say that you live in a house with other people who tend to devour everything in their path like locusts. And one day they cross the line: you open the fridge and your snack is gone! If you forget to take measures, next time you'll be walking there, you might be a bit worried because you suddenly realize that you might not find the snack you were already savoring in your imagination with endorphins happily rushing in your body while you approach the fridge.
And then your snack is not there. Again. But since you have much more important thoughts, tasks and problems that occupy your mind, it is likely that you will forget to take measures again. And again. And again. And again. And every time you will approach the fridge, you'll have that sting of anxiety kicking in (it's not that important, so it will be just a sting). That's how the fear of failing begins. Obviously, this is a trivial example, just to make the demonstration of how the mechanism works easier.
Nonetheless, it could be labeled as failure by our subconscious mind and trigger an unpleasant associative chain of other failures that are much more significant. And the more they accumulate, the more it gets connected to other associations of failures, resulting in a solid mental construction, on a deeper less consciously controlled level.
And let's say that, next time you crave for a snack because, although you're on a diet, you had a very bad day: failures at work, you had a fight with your partner or your child, you have too many things to do and not enough time to do them, you discovered a new scratch on your car, your mom called with annoying questions and you snapped back at her, you even started brooding over your overall results in life and how annoyed or stressed or whatever you've been lately and you were still thinking about that while approaching the fridge for your snack. And, obviously, it wasn't there. Again. The final straw. I'm sure you can imagine how it feels.
So then you take the time to take measures: you buy more snacks, you hide them, you discourage the serial eaters from devouring your precious snacks, you put a locker on the fridge, you find a substitute snack that gives you even higher levels of pleasure and that nobody else likes or whatever. One way or another, you solve it.
So now you know that the situation is solved, that you are organized to prevent it from happening ever again. The walk to the fridge is a path to serenity again. You KNOW it's going to be there or what to do, in case it's not. "Fridge situation" gets detached from the failure mental construction and your subconscious mind re-labels it as something you don't need to worry about anymore. You KNOW you're in control of the situation.
Same mental mechanisms apply to other definitely more important life situations.
So when won't you have the feeling of failure? In other words, what do you need to pay attention to in order to eliminate it without having to overcome or inhibit it?
So how do you get there? How can you shift from failure to success efficiently? How do you figure out things in a way that will make you feel CONFIDENT and NOT feel the fear of failing, at least in a certain sector of your life? Without having to inhibit it or overcome it? And what is the fastest way?
The bad news is that a lot of situations have similarities we tend to not consider as such. So when we fail in one area, the fear of failure will contaminate numerous other areas, making things and life more difficult, full of anxiety and stress.
The good news is that when you single out the real basic mechanism causing the difficulty, it will have positive repercussions in numerous areas of your life. Some clients I worked with stated more than once that this mechanism exceeded their expectations in a significant manner, improving their abilities to deal not only with situations they wanted to solve, but also with some others they weren't even considering. Things just get easier, more natural with the right mental shifts. You don't have to think about certain situations because you see how you can apply new notions over and over again. An "aha moment". You don't have to overcome or inhibit the fear of failure anymore, you just don't feel it anymore.
Let me give you some examples. Let's say that when you were a kid you've never managed to get an agreement with your mother. Now you have a boss who's got some personality traits similar to your mother's and you can't make her really listen to you either. And actually there are a lot of different people who don't really get what you really wanted to say. Sometimes it might appear that they do and that feels good. But then their actions make you realize they didn't get it at all. And that can be really annoying or frustrating. They might seem all different people with different traits, just a few of them have something in common.
What can cause failure and how to solve it?
Although the mechanism causing failure might change from person to person due to numerous factors, the most efficient thing you can do is to learn more of them, expand your horizons and your knowledge. That's what personal development is about and it comes with a lot of unexpected benefits, even if you think that your expectations are already high.
Or, here's another example. Like most of us, your teachers at school started teaching you something without teaching you HOW to LEARN first and WHY you should learn some things, in the first place. And you were right when you had the feeling that some things they taught you, in the way they taught, were outdated even back then. Nobody ever told you HOW to study or maybe they did, but later on, when it was already too late to allow you get the most out of it, when you already had prejudices and your own learning habits. But are they the most efficient learning habits? Are you sure that they didn't interfere with your ability to analyze things objectively? What if there are some misconceptions in something you are sure about? What if there are misconception in the way you see work, relationships, purpose, all those different feelings, especially the ones that could be defined as "emptiness", "anxiety", "yearning", "stress", "depression"? And what if some things you put the same label on were very different? And how can you trust your own judgment if that was the case?
How do you know you're really making progress and not wasting your time?
Another good news is that you can. You can do that through learning or through practice. It doesn't matter if you know how to learn efficiently, in this case. And practice is always one of the most efficient things you can do. If you do both, it is likely that you'll get results faster. Get a mentor, a coach, a trainer, a consultant, a partner that will study and practice with you in order to make it faster. Or get a course.
Whatever you decide to do, focus on feeding your mind with new notions and comparing them to what you already know. Don't just store information, but "make it yours".
It is not a matter of accepting whatever somebody else tells you as undisputed truth that you need to learn by heart. It is a matter of taking it into consideration, browsing through your thoughts, associations and past experiences until it clicks and you get to see it from a different viewpoint. That's the way you "make it yours": you get an "Aha moment" that totally changes something, you get to see what you were doing wrong in a certain situation or even in several areas of your life, you realize how to do it right, you are sure about it and it stops bothering you.
It is not easy, although it is worth it. That's why hiring an expert who has already been there and figured it out, will make it easier and faster, if you choose one wisely.
But whatever path you choose to learn something, if it doesn't click, try to find what you're missing out (if anything) and be aware that, unfortunately, there are a lot of useless and low quality "ultimate guides" and "teachings" around us. Don't waste your time, move on on something that will click. You deserve more.


Here a VL, we often hear horror stories of merchants who have overpaid for a complex mess of plug-and-play or custom coded integrations. Usually when that occurs, the merchant will come to VL at their wit's end. Not only have they overpaid for a solution that isn't working, but the whole process has taken an absurd amount of time from signing to implementation.
One merchant we've spoken with, for example, has been waiting for their data integration solution to go live for over 18 months. We see time and time again that merchants are often stuck in the implementation or onboarding phases without a clear go-live data for months; in some cases, the integration solution never goes live. It's clear that some merchants are being talked into wasting valuable time and money on integrations solutions that aren't the right fit — either in the solution proposed itself, or with the partner providing the solution — causing these massive delays.
Implementation and onboarding for a data integration solution should not take 18 months!
With BFCM soon approaching, now is the time to be thinking about your data integration strategy and beginning to automate data in meaningful ways to free you and your resources to grow your business. But if you're going upmarket with a data integration solution that's not out-of-the-box ("plug-and-play"), you should come prepared with knowing how long an upmarket solution should take before your project is live. So we've decided to take a tongue-in-cheek look at all the things that should take merchants longer to accomplish than integrating their technology stack seamlessly with VL OMNI! Read on!
Have you already had your child and still don't have your integration solution up and running? That a great indication you are wasting valuable time and money. It takes a whole 9 months to make a perfect little human! Perfect integrations? Much less.
We've all been there. Has your over-enthusiastic DIY husband or wife promised you that your 80's style kitchen will look like a cover of Style At Home magazine in 8 weeks or less? Depending on the how amateur your spouse or shady contractor is at renovations, it might end up taking much, much longer than that. Even still, your integration project should be completed with much more efficiency (and grace).
Education is key! We all know that. Completing a degree can be one of the greatest accomplishments of someone's life. All those hours in the library (or let's be real – the bar) studying for endless exams and thesis projects is time well wasted! But your integrations should definitely not take that long.
Game of Thrones fans — we feel you! We are over here still waiting for season 8! (Ugh.) Other great shows can take just as long to come back when their creators decide to take their own sweet time (or axe the show all together)! God bless Torrentz? What's for sure is your integrations projects should never keep you waiting this long.
Your wedding day! The most beautiful day of your life right? Too bad none of your guests know the mental anguish of planning a wedding. The dress, flowers, the food and god-forbid unhappy in-laws? (That last one can hold up plans for weeks!) But your integration projects should never be as much trouble as your in-laws or delay your plans for this long.
In reality, creating and implementing an integration solution that is both agile, flexible and scalable with VL OMNI can take only take a couple of weeks.* The overall benefit and ROI of these integrations should be immediately evident to your business. For merchants, our biggest advice is having a complete data integration strategy realized before implementing a solution to avoid heading down the wrong path, partnering with a solution that may not fit your goals, or ultimately getting stuck in an "it's almost ready!" loop for weeks (or months).
Here at VL OMNI, we have a tried, tested and true onboarding process that allows the merchant to be fully involved in the implementation and kickoff. That's the difference a great data integration provider like VL OMNI makes — one that looks at your business holistically, understands your business' needs deeply, and deploys a solution to match your needs.
*This is an average time length for implementing VL OMNI projects for merchants who are responsive to our requests and have fully bought-in to co-ownership of their data integration project. VL OMNI also has many merchants with more complex projects, or with multiple third parties involved — all of which can slow down our implementation timelines. But not to fret! Even the most complex of projects only take an average of 1-2 months for VL OMNI to implement.