contained in at most one projective line. So $\mathcal{L}(x,z)$ is well-
defined and uniquely determined.

Continuity of $\mathcal{L}(x,z)$ then follows from the continuity of dev,
analogously to the continuity of $\xi^{1}$ in Lemma 5.2. ∎

####  5.1.2. Holonomies of Zero Translation Elements

In this subsection, we examine the consequences of the structure of the
concave regions $B_{\gamma^{\pm}}$ on the holonomy of elements
$\gamma\in\overline{\Gamma}$ of zero translation.

Throughout the subsection, $\gamma\in\overline{\Gamma}$ is a fixed element of
translation $0$ and
$(\gamma^{-},\gamma^{+})\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$ is a fixed-
point pair of $\gamma$. We begin with the general behavior of
$\mathcal{L}(\gamma^{-},\gamma^{+})$:

######  Lemma 5.8.

The restriction of $\mathrm{hol}(\gamma)$ to
$\mathcal{L}(\gamma^{-},\gamma^{+})$ is diagonalizable over $\mathbb{R}$, with
eigenvalues $\lambda_{a}\neq\lambda_{b}$ so $\lambda_{a}\lambda_{b}>0$. The
eigenlines $e_{a}$ and $e_{b}$ corresponding to these eigenvalues are the
endpoints of $\mathcal{S}(\gamma^{-},\gamma^{+})$.

###### Proof.

By our assumption that $\mathcal{S}(\gamma^{-},\gamma^{+})$ is properly convex
and the local injectivity of $\mathrm{dev}$, the two endpoints $e_{a}$ and
$e_{b}$ of $\mathcal{S}(\gamma^{-},\gamma^{+})$ are fixed by
$\mathrm{hol}(\gamma)$, so that
$\mathrm{hol}(\gamma)|_{\mathcal{L}(\gamma^{-},\gamma^{+})}$ is real-
diagonalizable with eigenlines $e_{a}$ and $e_{b}$. Let $\lambda_{a}$ and
$\lambda_{b}$ be the two eigenvalues. We see that $\lambda_{a}\lambda_{b}>0$
as $\mathcal{S}(\gamma^{-},\gamma^{+})$ is connected and setwise fixed by
$\mathrm{hol}(\gamma)$. Now, for a fixed $x\in\widetilde{\partial\Gamma}$ so
that $(\gamma^{-},x,\gamma^{+})$ is positively oriented, we have
$\lim\limits_{n\to\ \infty}\gamma^{N}x=\gamma^{+}$. So from
$\mathrm{hol}$-equivariance of dev, we see $\lambda_{a}\neq\lambda_{b}$. ∎

It is also useful to record the following:

######  Lemma 5.9.

For all $(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$, the line
$\mathcal{L}(x,z)$ is not equal to $\xi^{2}(x)$.

###### Proof.

$\mathcal{S}(x,z)$ is contained in the concave region $B_{x}$ and $\xi^{2}(x)$
is entirely contained in the complement of $B_{x}$. ∎

The relative arrangements of $\mathcal{L}$, $\xi^{1}$, and $\xi^{2}$ are
sufficiently constrained so as to force the existence of three real
eigenspaces of $\mathrm{hol}(\gamma)$:

######  Lemma 5.10.

$\mathrm{hol}(\gamma)$ is diagonalizable over $\mathbb{R}$.

###### Proof.

For the sake of contradiction, suppose that $\mathrm{hol}(\gamma)$ does not
admit three non-colinear fixed points in $\mathbb{RP}^{2}$. By Lemma 5.8, we
know that $\mathcal{L}(\gamma^{-},\gamma^{+})$ already contains two distinct
fixed points of $\mathrm{hol}(\gamma)$. The same holds for
$\mathcal{L}(\gamma^{+},\tau\gamma^{-})$. By Lemma 5.2, we know that
$\xi^{1}(\gamma^{-})\neq\xi^{1}(\gamma^{+})$ are also two distinct fixed
points of $\mathrm{hol}(\gamma)$. So these three pairs of distinct fixed
points must coincide, and
$\mathcal{L}(\gamma^{-},\gamma^{+})=\mathcal{L}(\gamma^{+},\tau\gamma^{-})=\mathrm{Span}\\{\xi^{1}(\gamma^{-}),\xi^{1}(\gamma^{+})\\}.$

By Lemma 5.9, $\mathcal{L}(\gamma^{-},\gamma^{+})\neq\xi^{2}(\gamma^{-})$ and
$\mathcal{L}(\gamma^{+},\tau\gamma^{-})\neq\xi^{2}(\gamma^{+})$. Since
$\mathrm{hol}(\gamma)$ does not admit three distinct fixed points in
$\mathbb{RP}^{2\ast}$ we then have
$\\{\mathcal{L}(\gamma^{-},\gamma^{+}),\xi^{2}(\gamma^{-})\\}=\\{\mathcal{L}(\gamma^{+},\tau\gamma^{-}),\xi^{2}(\gamma^{+})\\}$.
From the first paragraph we have
$\mathcal{L}(\gamma^{-},\gamma^{+})=\mathcal{L}(\gamma^{+},\tau\gamma^{-})$
which then implies that $\xi^{2}(\gamma^{+})=\xi^{2}(\gamma^{-})$.

The intersection of $\mathcal{L}(\gamma^{-},\gamma^{+})$ and
$\xi^{2}(\gamma^{-})$ is a fixed point of $\mathrm{hol}(\gamma)$, and so is
$\xi^{1}(\gamma^{-})$ or $\xi^{1}(\gamma^{+})$. So because
$\xi^{2}(\gamma^{+})=\xi^{2}(\gamma^{-})$, we have $\xi^{1}(x)\in\xi^{2}(x)$
for one $x\in\\{\gamma^{-},\gamma^{+}\\}$.

Now, as $\xi^{1}$ and $\xi^{2}$ are continuous, the property that
$\xi^{1}(x)\in\xi^{2}(x)$ is a closed condition in
$x\in\widetilde{\partial\Gamma}$. As both are
$\mathrm{hol}(\overline{\Gamma})$-equivariant and $\overline{\Gamma}$ acts
minimally on $\widetilde{\partial\Gamma}$, we must have that
$\xi^{1}(x)\in\xi^{2}(x)$ for all $x\in\widetilde{\partial\Gamma}$. So
$\xi^{1}(\gamma^{-})\in\xi^{2}(\gamma^{-})$ and also
$\xi^{1}(\gamma^{+})\in\xi^{2}(\gamma^{+})=\xi^{2}(\gamma^{-})$. This forces
$\mathcal{L}(\gamma^{-},\gamma^{+})$ to be $\xi^{2}(\gamma^{-})$, which is a
contradiction. ∎

######  Definition 5.11.

For $(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$, let
$p_{xz}\in\mathbb{R}\mathbb{P}^{2}$ be given by
$\mathcal{L}(x,z)\cap\xi^{2}(x)$.

Note that the map $(x,z)\mapsto p_{xz}$ is continuous as $\mathcal{L}(x,z)$
and $\xi^{2}(x)$ both vary continuously and are transverse for any
$(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$. We are now prepared to show
that the eigenvalues of $\mathrm{hol}(\gamma)$ are distinct. We retain the
notation from Lemma 5.8 that $e_{a},$ and $e_{b}$ are the eigenlines of
$\mathrm{hol}(\gamma)$ restricted to $\mathcal{L}(\gamma^{-},\gamma^{+})$.

######  Lemma 5.12 (Distinct Eigenvalues).

Let $\gamma\in\overline{\Gamma}$ have translation $0$. Then
$\mathrm{hol}(\gamma)$ has exactly three real eigenspaces, $e_{a},e_{b},$ and
$e_{c}$ whose corresponding eigenvalues have distinct moduli.

###### Proof.

For $\gamma$ of translation $0$, note that because of Lemma 5.10 the property
that $\mathrm{hol}(\gamma)$ has three eigenvalues of distinct modulus is
equivalent to $\mathrm{hol}(\gamma^{2})$ having three distinct eigenvalues,
with no assumption on the modulus. Since for every $\gamma$ of translation
$0$, the square $\gamma^{2}$ has translation zero, it thus suffices to prove
that $\mathrm{hol}(\gamma)$ has exactly three eigenspaces whose eigenvalues
are distinct for every $\gamma$ of translation $0$.

So let $\gamma$ have translation $0$. By Lemma 5.10, there is at least one
eigenline $e_{c}\notin\mathcal{L}(\gamma^{-},\gamma^{+})$, with eigenvalue
$\lambda_{c}$. Then the projective line $L$ defined as the sum of $e_{c}$ and
the eigenspace $e\neq p_{\gamma^{-}\gamma^{+}}$ in
$\mathcal{L}(\gamma^{-},\gamma^{+})$ is not $\xi^{2}(\gamma^{-})$ and
intersects $\xi^{2}(\gamma^{-})$ at a point $q$. By definition,
$p_{\gamma^{-}\gamma^{+}}\in\\{e_{a},e_{b}\\}$. Let us assume
$p_{\gamma^{-}\gamma^{+}}=e_{a}$ so that $L=e_{b}\oplus e_{c}$. The other case
is handled identically.

Let us now show that $e_{a},e_{b},$ and $e_{c}$ are the only eigenlines of
$\mathrm{hol}(\gamma)$, that is that $q$ must be $e_{c}$. Suppose otherwise.
Then Lemma 5.8 forces $\mathrm{hol}(\gamma)$ to fix three points on $L$, and
hence pointwise fix $L$.

We claim that for all $y\in\widetilde{\partial\Gamma}$ such that
$(\gamma^{-},y)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$, the intersection of
$\mathcal{L}(\gamma^{-},y)$ and $L$ is exactly $\\{e_{b}\\}$. Note that this
leads to the desired contradiction. Indeed, this implies that each segment
$\mathcal{S}(\gamma^{-},y)$ cannot meet $L$. Since the concave region
$B_{\gamma^{-}}=\mathrm{pr}_{1}\circ\mathrm{dev}(f_{\gamma^{-}})$ is foliated
by such segments, it follows that it must be disjoint from $L$. This
contradicts the fact that $L$ is not equal to $\xi^{2}(\gamma^{-})$.

To see the claim, for any such $y$, let $p_{y}$ be a point of intersection of
$\mathcal{L}(\gamma^{-},y)$ and $L$. Then
$\lim_{n\to\infty}\mathrm{hol}(\gamma)^{n}p_{y}=p_{y}$. On the other hand, by
continuity of $\mathcal{L}$ and equivariance, we have
$\lim_{n\to\infty}\mathrm{hol}(\gamma)^{n}\mathcal{L}(\gamma^{-},y)=\mathcal{L}(\gamma^{-},\gamma^{+})$.
We conclude that $p_{y}\in\mathcal{L}(\gamma^{-},\gamma^{+})$, and hence that
$p_{y}=e_{b}$. Note that the convergence of
$\mathrm{hol}(\gamma)^{n}\mathcal{L}(\gamma^{-},y)$ to
$\mathcal{L}(\gamma^{-},\gamma^{+})$ also implies
$\mathcal{L}(\gamma^{-},y)\neq L$, and hence $\mathcal{L}(\gamma^{-},y)\cap
L=\\{e_{b}\\}$.

∎

####  5.1.3. Locations of Eigenspaces

By Lemma 5.12, the holonomy of any $\gamma\in\overline{\Gamma}$ of translation
$0$ is _loxodromic_ , i.e. has exactly three eigenvalues of distinct moduli.

Let us introduce new notation that represents the distinct sizes of their
eigenvalues. We shall call these eigenvalues $\lambda_{+},\lambda_{0},$ and
$\lambda_{-}$, and order them by decreasing modulus. We denote the
corresponding eigenlines by $e_{+},e_{0},$ and $e_{-}$, with the obvious
notation. When $\gamma\in\overline{\Gamma}$ is not clear from context, we
write $e_{+}=e_{+}(\gamma),e_{0}=e_{0}(\gamma)$ and $e_{-}=e_{-}(\gamma)$.

We now turn to restricting the locations of eigenspaces.

######  Lemma 5.13.

For all $\gamma\in\overline{\Gamma}$ of translation $0$, we have
$\xi^{1}(\gamma^{+})=e_{+}$.

###### Proof.

Let us first note that the bijection induced on $\\{e_{+},e_{0},e_{-}\\}$ by
interchanging $\gamma$ and $\gamma^{-1}$ has exactly one fixed point, which is
$e_{0}$. So as $\xi^{1}(\gamma^{+})\neq\xi^{1}(\gamma^{-})$ by Lemma 5.2,
$\\{\xi^{1}(\gamma^{+}),\xi^{1}(\gamma^{-})\\}$ is a two-element invariant
subset of $\\{e_{+},e_{0},e_{-}\\}$ and hence $\\{e_{+},e_{-}\\}$.

Now for $\eta$ near $\gamma^{+}$,

$\xi^{1}(\gamma^{+})=\lim_{n\to\infty}\xi^{1}(\gamma^{n}\eta)=\lim_{n\to\infty}\mathrm{hol}(\gamma)^{n}\xi^{1}(\eta)\in\\{e_{0},e_{+}\\}.$

Hence $\xi^{1}(\gamma^{+})=e_{+}$. Here we have used the structure of the
dynamics of $\mathrm{hol}(\gamma)$ acting on $\mathbb{R}\mathbb{P}^{2}$
implied by $\mathrm{hol}(\gamma)$ being real-diagonalizable with three
distinct eigenvalues. ∎

This basic knowledge has two useful consequences that we document. The first
is the following corollary on the central element $\tau\in\overline{\Gamma}$.

######  Corollary 5.14.

$\mathrm{hol}(\tau)\xi^{1}(x)=\xi^{1}(x)$ for all
$x\in\widetilde{\partial\Gamma}$.

###### Proof.

Let $\gamma\in\overline{\Gamma}$ have zero translation and let
$\gamma^{+}\in\widetilde{\partial\Gamma}$ be an attracting fixed-point of
$\gamma$. Because $\mathrm{hol}(\gamma)$ and $\mathrm{hol}(\tau)$ commute and
$\mathrm{hol}(\gamma)$ is loxodromic by Lemma 5.12, the matrices
$\mathrm{hol}(\tau)$ and $\mathrm{hol}(\gamma)$ are simultaneously
diagonalizable. Because $\mathrm{hol}(\gamma)$ is loxodromic,
$\mathrm{hol}(\tau)$ must then fix every eigenspace of $\mathrm{hol}(\gamma)$,
and so must fix $\xi^{1}(\gamma^{+})$ by Lemma 5.13. The claim then follows by
the continuity of $\xi^{1}$ and the density of attracting fixed-points of
zero-translation elements in $\widetilde{\partial\Gamma}$. ∎

The second consequence is a dichotomy for the location of
$p_{\gamma^{-}\gamma^{+}}$:

######  Lemma 5.15 (Dichotomy).

Either $p_{\gamma^{-}\gamma^{+}}=p_{\gamma^{+},\tau\gamma^{-}}=e_{0}$ for all
$\gamma\in\overline{\Gamma}$ of translation $0$, or
$\\{p_{\gamma^{-}\gamma^{+}},p_{\gamma^{+},\tau\gamma^{-}}\\}=\\{e_{+},e_{-}\\}$
for all $\gamma\in\overline{\Gamma}$ of translation $0$.

In the second case, either $p_{xz}=\xi^{1}(x)$ for all
$(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$, or $p_{xz}=\xi^{1}(z)$ for
all $(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$.

The argument boils down to the idea that while $p_{xz}\neq p_{z,\tau x}$ is an
open condition in $(x,z)$, for pole-pair lifts $(\gamma^{-},\gamma^{+})$,
Lemma 5.13 allows an alternative characterization of this condition in terms
of $\xi^{1}$ with closedness properties.

###### Proof.

Suppose that $p_{xz}\neq p_{z,\tau x}$ for some pair
$(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$. We claim that then
$p_{xz}\neq p_{z,\tau x}$ for all
$(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$. By continuity, this
condition is open.

So let $\Omega\subset\widetilde{\partial\Gamma}^{(2)}_{[0]}$ be the nonempty
and open set of pairs $(x,z)$ so that $p_{xz}\neq p_{z,\tau x}$. We note that
by Lemma 5.13 any pole-pair lift $(\gamma^{-},\gamma^{+})\in\Omega$ satisfies
$\\{p_{\gamma^{-}\gamma^{+}},p_{\gamma^{+},\tau\gamma^{-}}\\}=\\{\xi^{1}(\gamma^{+}),\xi^{1}(\gamma^{-})\\}$.
Furthermore, since $\Omega$ is open, standard hyperbolic group theory shows
that such pairs $(\gamma^{-},\gamma^{+})$ are dense in $\Omega$. As $p_{xz}$
and $\xi^{1}$ are both continuous, we see that $\\{p_{xz},p_{z,\tau
x}\\}=\\{\xi^{1}(x),\xi^{1}(z)\\}$ for all $(x,z)$ in the closure
$\overline{\Omega}$ of $\Omega$ in $\widetilde{\partial\Gamma}^{(2)}_{[0]}.$
Lemma 5.2 now shows that for any point $(x,z)\in\overline{\Omega}$ that
$\xi^{1}(x)\neq\xi^{1}(z)$, hence $p_{xz}\neq p_{z,\tau x}$ and hence
$(x,z)\in\Omega$. So $\Omega$ is closed. We conclude that
$\Omega=\widetilde{\partial\Gamma}^{(2)}_{[0]}.$

The last paragraph also shows that for each pair $(x,z)$ we have
$\\{p_{xz},p_{z,\tau x}\\}=\\{\xi^{1}(x),\xi^{1}(z)\\}$, from which it follows
that in this case either $p_{xz}=\xi^{1}(x)$ for all
$(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$ or $p_{xz}=\xi^{1}(z)$ for
all $(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$. ∎

In the following lemma we begin to relate the eigenvalues and eigenspaces of
elements $\gamma$ of translation $0$ with the convex domain $C_{\gamma_{+}}$.

######  Lemma 5.16.

If $g\in{\rm{SL}}_{3}(\mathbb{R})$ is loxodromic with attracting eigenline
$e_{+}$ and repelling eigenline $e_{-}$, and $g$ preserves a properly convex
domain $\Omega$, then $e_{+}$ and $e_{-}$ are in $\partial\Omega$ and their
corresponding eigenvalues have the same sign.

###### Proof.

One sees that $e_{+}$ must at least be in $\overline{\Omega}$ by observing
that as $\Omega$ is open, there is a point in $\Omega$ not contained in the
repelling projective line for $g$, which must limit on $e_{+}$ under iteration
of $g$. Furthermore, $e_{+}$ cannot be contained in $\Omega$, as otherwise
$g$-invariance of $\Omega$ would force $\Omega$ to contain the complement of a
projective line, which is incompatible with proper convexity. The analogues
for $e_{-}$ are symmetric, by considering $g^{-1}$, which shows that $e_{+}$
and $e_{-}$ are in $\partial\Omega$.

Hence a full segment $I$ between $e_{+}$ and $e_{-}$ is contained in
$\overline{\Omega}$. If the eigenvalues of $g$ corresponding to $e_{+}$ and
$e_{-}$ had opposite signs, then $\overline{\Omega}$ would contain a full
projective line, which is impossible. ∎

####  5.1.4. Non-Degeneracy of $\xi^{1}$

We next constrain the images of the maps $\xi^{1}$ and $\xi^{2}$. Our goal is
to prove:

######  Proposition 5.17.

The image of $\xi^{1}$ is not contained in a line.

The following Lemma will force $\xi^{2}$ to be constant in the hypothetical
case where the image of $\xi^{1}$ is contained in a line.

######  Lemma 5.18.

The image of ${{\rm{pr}}_{1}\circ{\rm{dev}}}$ is disjoint from
$\xi^{1}\left(\widetilde{\partial\Gamma}\right)$.

###### Proof.

Suppose for the sake of contradiction that there is an
$x\in\widetilde{\partial\Gamma}$ so $\xi^{1}(x)$ is in the image $U$ of the
developing map. Then by density of pole-pairs, there is a pole-pair
$(\gamma^{-},\gamma^{+})$ so that
$\xi^{1}(\gamma^{-})\in\text{pr}_{1}(\text{dev}(f_{\gamma^{+}}))$. Let $V$ be
a small neighborhood of $\xi^{1}(\gamma^{-})$ contained in
$\text{pr}_{1}(\text{dev}(f_{\gamma^{+}}))$.

Then, since $\mathrm{hol}(\gamma)$ is diagonalizable with distinct real
eigenvalues and $\xi^{1}(\gamma^{-})$ is the eigenline of least modulus by
Lemma 5.12 and Lemma 5.13, we see

$\text{pr}_{1}(\text{dev}(f_{\gamma^{+}}))\supset\bigcup_{n\in\mathbb{N}}\mathrm{hol}(\gamma)^{n}(V)=\mathbb{R}\mathbb{P}^{2}-(e_{0}\oplus
e_{+}).$

Since no concave region contains the complement of a line, this contradicts
the concavity of $\text{dev}(f_{\gamma^{+}})$. ∎

The proof of Proposition 5.17 uses semicontinuity features of the complements
$D_{x}$ of the concave domains $B_{x}$ in $\mathbb{R}\mathbb{P}^{2}$ for
$x\in\widetilde{\partial\Gamma}$. The form of semicontinuity that we have
convenient access to is:

######  Lemma 5.19 (Contraction Control).

For any sequence $x_{n}$ in $\widetilde{\partial\Gamma}$ converging to
$x\in\widetilde{\partial\Gamma}$ the closed set $D_{x}$ contains
$\mathcal{D}_{\\{x_{n}\\}}=\\{p\in\mathbb{R}\mathbb{P}^{2}\mid\text{there
exist infinitely many $n$ so }p\in D_{x_{n}}\\}$.

###### Proof.

If $p\in B_{x}$ then $p=\text{pr}_{1}(\text{dev}(x,y,z))$ for some
$(x,y,z)\in\widetilde{\partial\Gamma^{(3)+}}$. Since dev is a local
homeomorphism, for all $w$ in a neighborhood of $x$ we have $p\in B_{w}$, so
that $p\notin\mathcal{D}_{\\{x_{n}\\}}$. So
$\mathcal{D}_{\\{x_{n}\\}}\subset(\mathbb{R}\mathbb{P}^{2}-B_{x})$, which is
$D_{x}$. ∎

The application of semicontinuity that we shall use is:

######  Lemma 5.20 (Segment Containment).

Let $\gamma,\eta$ be elements of zero translation in $\overline{\Gamma}$ with
fixed-point pairs $(\gamma^{-},\gamma^{+})$ and $(\eta^{-},\eta^{+})$ in
$\widetilde{\partial\Gamma}^{(2)}_{[0]}$, respectively. Suppose that
$\eta^{-}$ is not in the $\langle\tau\rangle$-orbit of $\gamma^{+}$ or
$\gamma^{-}$, that $e_{-}(\eta)\in\partial C_{\gamma^{+}}$, and that
$e_{-}(\eta)\oplus e_{0}(\eta)$ does not support $C_{\gamma^{+}}$ at
$e_{-}(\eta)$. Finally, suppose that $(\gamma^{-},\gamma^{+})$ and
$(\eta^{-},\eta^{+})$ are near to each other in the sense that
$(\eta^{-},\gamma^{+},\eta^{+})$ or $(\eta^{+},\gamma^{+},\tau\eta^{-})$ is a
positively oriented triple in $\widetilde{\partial\Gamma}$.

Then a segment between $e_{-}(\eta)$ and $e_{0}(\eta)$ is contained in
$D_{\eta^{+}}$.

###### Proof.

Since $\mathcal{P}_{\eta}:=e_{-}(\eta)\oplus e_{0}(\eta)$ does not support
$C_{\gamma^{+}}$, there is an open neighborhood
$U\subset\mathcal{P}_{\eta}\cap C_{\gamma^{+}}$ with endpoint $e_{-}(\eta)$.
Apply $\mathrm{hol}(\eta^{n})$. Then $\mathrm{hol}(\eta)^{2n}U$ is a nested
family of open intervals contained in $C_{\eta^{2n}\gamma^{+}}$ limiting on an
open segment between $e_{0}(\eta)$ and $e_{-}(\eta)$. Our hypotheses ensure
that $\eta^{2n}\gamma^{+}$ converges to $\eta^{+}$, and so Lemma 5.19 shows
$D_{\eta^{+}}$ contains an open segment between $e_{0}(\eta)$ and
$e_{-}(\eta)$. ∎

###### Proof of Proposition 5.17.

Suppose for contradiction that $\xi^{1}$ is contained in a projective line
$L$. Note that from local injectivity of $\xi^{1}$ and the dynamics of
$0$-translation elements of $\overline{\Gamma}$ that
$\xi^{1}(\widetilde{\partial\Gamma})=L$. Lemma 5.18 implies for all
$x\in\widetilde{\partial\Gamma}$ that $\xi^{2}(x)=L$.

The argument is now based around leveraging the following claim to show that
the convex domains $C_{x}$ must wildly change in unallowable ways.

######  Claim 5.21.

Under these hypotheses, for all $\gamma\in\overline{\Gamma}$ of zero
translation with pole-pairs
$(\gamma^{-},\gamma^{+})\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$, a closed
segment between $\xi^{1}(\gamma^{+})$ and $\xi^{1}(\gamma^{-})$ is contained
in $\partial C_{\gamma^{+}}$, and all eigenvalues of $\mathrm{hol}(\gamma)$
have the same sign.

###### Proof.

As each $\mathrm{hol}(\gamma)$ for $\gamma\in\overline{\Gamma}-\\{e\\}$ of
translation $0$ is loxodromic and preserves $C_{\gamma^{+}}$ we conclude from
Lemma 5.13 and Lemma 5.16 that $\xi^{1}(\gamma^{+})$ and $\xi^{1}(\gamma^{-})$
are contained in $\partial C_{\gamma^{+}}$. By convexity, a closed segment
between $\xi^{1}(\gamma^{+})$ and $\xi^{1}(\gamma^{-})$ is contained in
$\overline{C_{\gamma^{+}}}$. Since $\xi^{2}(\gamma^{+})=L$ supports
$C_{\gamma^{+}}$, the properly convex domain $C_{\gamma^{+}}$ cannot intersect
such a segment, and the first claim is proved.

Lemma 5.16 shows that the eigenvalues of $\mathrm{hol}(\gamma)$ corresponding
to $e_{+}$ and $e_{-}$ have the same sign. Since
$\xi^{2}(\gamma^{+})=\xi^{2}(\gamma^{-})=e_{+}(\gamma)\oplus e_{-}(\gamma)$ in
this situation, Lemma 5.8 shows that the signs of the eigenvalues
corresponding to $e_{0}$ and one of $e_{+},e_{-}$ agree, which finishes the
final claim. ∎

The next step in the argument is to control the location of the middle
eigenspace in an appropriate sense. Let us begin by noting that our hypotheses
force $p_{\gamma^{-}\gamma^{+}}$ to be inside $L$ so that either
$p_{xz}=\xi^{1}(x)$ or $p_{xz}=\xi^{1}(z)$ for all
$(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$ by the Dichotomy Lemma 5.15.
In the first case, $\mathcal{L}(x,z)$ contains $\xi^{1}(x)$ for all
$(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$. In the second case,
$\mathcal{L}(x,z)$ contains $\xi^{1}(z)$ for all
$(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$. In both cases, Lemma 5.9 and
Corollary 5.14 imply that $\mathcal{L}(x,z)$ and $\mathcal{L}(z,\tau x)$ are
transverse for all $(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$.

So for any pair $(x,z)\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$ define
$q_{xz}=\mathcal{L}(x,z)\cap\mathcal{L}(z,\tau x)$. Then $(x,z)\mapsto q_{xz}$
is continuous by construction. Furthermore, we have arranged that
$q_{\gamma^{-}\gamma^{+}}=e_{0}(\gamma)$ when $\gamma$ is a nontrivial element
of zero translation.

We now apply semicontinuity of the complements of the concave domains’
complements $D_{x}$ to force $C_{x}$ to have highly constrained shape.

######  Claim 5.22.

Under these hypotheses, for all pole-pairs
$(\gamma^{-},\gamma^{+})\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$ the closure
of $C_{\gamma^{+}}$ is a triangle with vertices $e_{+}(\gamma),e_{-}(\gamma),$
and $e_{0}(\gamma)$.

###### Proof.

To begin, let $\nu$ be a fixed element of translation $0$. Claim 5.21 shows
that $\partial C_{\nu^{+}}$ contains an open line segment $U$ between
$\xi^{1}(\nu^{+})$ and $\xi^{1}(\nu^{-})$. Since $\xi^{1}$ is a local
homeomorphism with image contained in $L\supset U$, there is an open set
$V\subset\widetilde{\partial\Gamma}$ contained in the interval between
$\nu^{-}$ and $\nu^{+}$ with an endpoint at $\nu^{-}$ so that $\xi^{1}(x)\in
U$ for all $x\in V$. Let us assume that
$(\nu^{-},x)\in\widetilde{\partial\Gamma}^{[2]}_{[0]}$ for all $x\in V$; the
case where $(x,\nu^{-})\in\widetilde{\partial\Gamma}^{[2]}_{[0]}$ is
analogous. In any matter, for any pole-pair
$(\gamma^{-},\gamma^{+})\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$ with both
$\gamma^{-}$ and $\gamma^{+}$ in $V$, the repeller $e_{-}(\gamma^{-})$ is in
$U$ by Lemma 5.13 and $(\gamma^{+},\nu^{+},\tau\gamma^{-})$ is positively
oriented.

Lemma 5.20 then implies that for all such pole-pairs
$(\gamma^{-},\gamma^{+})$, in particular all pole-pairs in an open subset of
$\widetilde{\partial\Gamma}^{(2)}_{[0]}$, a line segment $\ell_{\gamma}^{+}$
in $e_{-}(\gamma)\oplus e_{0}(\gamma)$ is contained in
$\overline{C_{\gamma^{+}}}$. For such $\gamma$, convexity and containment of a
line segment between $e_{+}(\gamma)$ and $e_{-}(\gamma)$ forces
$\overline{C_{\gamma^{+}}}$ to contain a triangle $T_{\gamma^{+}}$ with
vertices $e_{+}(\gamma),e_{0}(\gamma),$ and $e_{-}(\gamma)$.

Because the line spanned by $e_{-}(\gamma)$ and $e_{+}(\gamma)$ is contained
in $\mathbb{R}\mathbb{P}^{2}-C_{\gamma^{+}}$ from the identity
$L=\xi^{2}(\gamma^{+})=\xi^{2}(\gamma^{-})$, it is a straightforward
consequence of the fact that $\mathrm{hol}(\gamma)$ is loxodromic that
$C_{\gamma^{+}}=T_{\gamma^{+}}$. Indeed, observe that for any $p\notin
T_{\gamma^{+}}\cup L$ that at least one of $\mathrm{hol}(\gamma^{n})p$ and
$\mathrm{hol}(\gamma^{-n})p$ converges to $e_{+}(\gamma)$ or $e_{-}(\gamma)$
from the side of $\mathbb{R}\mathbb{P}^{2}-L$ that does not contain
$T_{\gamma^{+}}$ (in any an affine chart containing $T_{\gamma^{+}}$). So if
$p$ were in $C_{\gamma^{+}}$, then $C_{\gamma^{+}}$ would need to contain a
point in $L$ by proper convexity, which is impossible.

We have shown that for all pole-pairs $(\gamma^{-},\gamma^{+})$ in $V\times V$
that $C_{\gamma^{+}}$ is a triangle. Since the collection of pole-pairs of
translation zero elements is $\overline{\Gamma}$-invariant, the topological
transitivity of the action of $\overline{\Gamma}$ on
$\widetilde{\partial\Gamma}^{(2)}_{[0]}$ now forces $C_{\gamma^{+}}$ to be a
triangle for all pole-pairs
$(\gamma^{-},\gamma^{+})\in\widetilde{\partial\Gamma}^{(2)}_{[0]}$. ∎

We are now ready to conclude. By density of pole-pairs, we may produce a
sequence $\gamma_{n}$ of translation zero elements in $\overline{\Gamma}$ so
that:

  1. (1)

$\gamma_{n}^{+}$ converges to $\eta^{+}$ for some $\eta$ of zero translation,
and

  2. (2)

$\gamma_{n}^{-}$ converges to $r$ so that $\xi^{1}(r)$ is not in the boundary
segment of $C_{\eta^{+}}$ between $e_{+}(\eta)$ and $e_{-}(\eta)$.

Then continuity of $(\gamma_{n}^{-},\gamma_{n}^{+})\mapsto
q_{\gamma_{n}^{-}\gamma_{n}^{+}}=e_{0}(\gamma)$ shows that there is a point
$p^{\prime}\notin T_{\eta^{+}}\cup L$ so that infinitely many of the triangles
$T_{\gamma_{n}^{+}}$ contain $p^{\prime}$. Then Lemma 5.19 forces
$p^{\prime}\in C_{\eta^{+}}=T_{\eta^{+}}$, which is a contradiction. ∎

####  5.1.5. Factoring the Holonomy

That the image of $\xi^{1}$ is not contained in a line implies strong
structural constraints on the holonomy and on the maps $\xi^{2}$ and
$\mathcal{L}$, that we document here.

######  Proposition 5.23.

$\mathrm{hol}(\tau)={\rm{Id}}$.

###### Proof.

Since the image of $\xi^{1}$ is not contained in a line, the image of
$\xi^{1}$ contains four points in $\mathbb{R}\mathbb{P}^{2}$ in general
position. As $\tau$ is central, $\mathrm{hol}(\tau)$ must commute with
$\mathrm{hol}(\gamma)$ for all $\gamma\in\overline{\Gamma}$. As each
$\mathrm{hol}(\gamma)$ for $\gamma$ of translation $0$ is diagonalizable with
no repeated eigenvalues, for every $\gamma$ of zero translation,
$\mathrm{hol}(\gamma)$ and $\mathrm{hol}(\tau)$ are simultaneously
diagonalizable.

Since general position is an open condition and attracting fixed-points of
elements of $\overline{\Gamma}$ of translation $0$ are dense in
$\widetilde{\partial\Gamma}$, there are
$\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\in\overline{\Gamma}$ of
translation zero so that
$\xi^{1}(\gamma_{1}^{+}),\xi^{1}(\gamma_{2}^{+}),\xi^{1}(\gamma_{3}^{+}),$ and
$\xi^{1}(\gamma_{4}^{+})$ are in general position. Each of these four points
must be an eigenspace of $\mathrm{hol}(\tau)$ because of simultaneous
diagonalizability of $\mathrm{hol}(\gamma_{i})$ and $\mathrm{hol}(\tau)$
$(i=1,2,3,4)$.

We conclude that $\mathrm{hol}(\tau)$ is a multiple of the identity in
$\text{SL}_{3}(\mathbb{R})$, hence the identity. ∎

####  5.1.6. $\xi$ Maps to Flags

Since $\mathrm{hol}(\tau)=\text{Id}$, the holonomy factors to a homomorphism
$\mathrm{hol}_{*}:\Gamma\to\text{SL}_{3}(\mathbb{R})$, both $\xi^{1}$ and
$\xi^{2}$ descend to maps from $\partial\Gamma$, the developing map dev
descends to a map with domain $\partial\Gamma^{(3)+}$, and $\mathcal{L}$
descends to a map from $\partial\Gamma^{(2)}$. We shall henceforth only work
with these induced maps, and will abuse notation and denote these induced maps
by the same symbols.

######  Lemma 5.24.

$\xi^{2}$ is non-constant.

###### Proof.

Suppose for contradiction that $\xi^{2}$ is constant, and let $L_{0}$ be its
value. We claim that this forces the image of $\xi^{1}$ to be contained in
$L_{0}$.

Note that the condition $\xi^{1}(x)\subset L_{0}$ is closed in
$\partial\Gamma$ by the continuity of $\xi^{1}$. Also, by Lemmas 5.2 and 5.13,
for every $\gamma\in\Gamma-\\{e\\}$ at least one of $\xi^{1}(\gamma^{+})$ and
$\xi^{1}(\gamma^{-})$ is in $\xi^{2}(\gamma^{+})$. Since pole-pairs are dense
in $\partial\Gamma^{2}$, for every $x\in\partial\Gamma$ there is a sequence
$\gamma_{n}\in\Gamma$ so that
$\lim_{n\to\infty}(\gamma^{+}_{n},\gamma^{-}_{n})=(x,x).$ Since at least one
entry $y$ of each pair $(\gamma^{+}_{n},\gamma^{-}_{n})$ has $\xi^{1}(y)\in
L_{0}$, we must have $\xi^{1}(x)\in L_{0}$, as desired.

So the image of $\xi^{1}$ is contained in $L_{0}$, in contradiction to
Proposition 5.17. ∎

We may now deduce that the map $\xi=(\xi^{1},\xi^{2})$ is valued in
$\mathscr{F}_{1,2}$.

######  Lemma 5.25.

$\xi^{1}(x)\in\xi^{2}(x)$ for all $x\in\partial\Gamma$.

###### Proof.

We will show that the collection of elements $x\in\partial\Gamma$ so that
$\xi^{1}(x)\in\xi^{2}(x)$ is thus $\Gamma$-invariant, closed, and nonempty,
hence all of $\partial\Gamma$. Because $\xi^{1}$ and $\xi^{2}$ are continuous
and the action of $\Gamma$ on $\partial\Gamma$ is minimal, it remains to show
that this collection is nonempty.

Now let $x_{0}\in\partial\Gamma$ be fixed and given. Since $\xi^{1}$ is not
contained in a line by Proposition 5.17, there is a point $y\in\partial\Gamma$
so $\xi^{1}(y)\notin\xi^{2}(x_{0})$. From the density of pole-pairs in
$\partial\Gamma^{2}$, there is a $\gamma\in\Gamma-\\{e\\}$ so that
$\xi^{1}(\gamma^{+})$ and $\xi^{1}(\gamma^{-})$ are not contained in
$\xi^{2}(x_{0})$. Then, since $\xi^{2}(x_{0})$ does not contain the repelling
fixed-point $e_{-}(\gamma)$ of $\mathrm{hol}(\gamma)$, we see

$\displaystyle\xi^{2}(\gamma^{+})=\lim_{n\to\
\infty}\xi^{2}(\gamma^{n}x_{0})=\lim_{n\to\infty}\mathrm{hol}(\gamma)^{n}\xi^{2}(x_{0})=e_{+}(\gamma)\oplus
e_{0}(\gamma).$

In particular, $\xi^{1}(\gamma^{+})\in\xi^{2}(\gamma^{+})$, as desired. ∎

####  5.1.7. Hitchin Holonomy

That $\mathrm{hol}_{*}$ is Hitchin is now a fairly straightforward consequence
of what we have proved.

######  Lemma 5.26.

There exists a projective line $L\subset\mathbb{R}\mathbb{P}^{2}$ so that
$\xi^{1}(\partial\Gamma)\cap L=\emptyset$.

###### Proof.

Let $x_{0}\in\partial\Gamma$ be fixed. By Lemma 5.18, the concave region
$B_{x_{0}}=\text{dev}(f_{x_{0}})$ has trivial intersection with
$\xi^{1}(\partial\Gamma)$.

By proper concavity, there is an open set
$U\subset\text{Gr}_{2}(\mathbb{R}^{3})$ of lines so that no $\ell\in U$
intersects the convex domain $C_{x_{0}}$ in
$D_{x_{0}}=\mathbb{R}\mathbb{P}^{2}-B_{x_{0}}$. In particular,

(5.1) $\displaystyle\xi^{1}(\partial\Gamma)\cap\left(\bigcup_{\ell\in
U}\ell\right)\subset\xi^{2}(x_{0}).$

From Proposition 5.17, $\xi^{1}$ maps no nonempty open subset of
$\partial\Gamma$ into a line, so that (5.1) implies for all $\ell\in U$ we
have $\xi^{1}(\partial\Gamma)\cap\ell=\emptyset$. ∎

We then have:

######  Corollary 5.27.

The convex hull $\Omega$ of $\xi^{1}(\partial\Gamma)$ is properly convex and
preserved by $\mathrm{hol}_{*}(\Gamma)$.

Here, the convex hull is taken in any affine chart with the line at infinity
disjoint from $\xi^{1}(\partial\Gamma)$. In order to conclude Hitchin holonomy
from Lemma 5.26 we shall need:

######  Lemma 5.28.

$\mathrm{hol}_{*}$ is injective.

###### Proof.

Lemma 5.2 together with the holonomy factoring through $\Gamma$ gives that
$\xi^{1}:\partial\Gamma\to\mathbb{R}\mathbb{P}^{2}$ is injective. Let
$\gamma\in\Gamma-\\{e\\}$. For any $x\notin\\{\gamma^{-},\gamma^{+}\\}$ from
equivariance of $\xi^{1}$ we have that
$\lim_{n\to\infty}\mathrm{hol}_{*}(\gamma^{n})\xi^{1}(x)=\xi^{1}(\gamma^{+})\neq\xi^{1}(x)$.
We conclude that $\mathrm{hol}(\gamma)\neq\text{Id}$. ∎

From this point it is easy to prove that $\mathrm{hol}_{\ast}$ is Hitchin with
limit map $\xi$. We take advantage of the characterizations of
$\text{Hit}_{3}(S)$ due to Choi-Goldman and Labourie-Guichard.

######  Proposition 5.29.

${\rm{hol}}_{*}$ is Hitchin, with hyperconvex Frenet curve $(\xi^{1},\xi^{2})$
as defined above.

###### Proof.

Since $\mathrm{hol}_{*}$ is injective and $\mathrm{hol}_{*}(\Gamma)$ preserves
a properly convex domain, $\mathrm{hol}_{*}\in\text{Hit}_{3}(S)$ from Choi-
Goldman’s Theorem [15, Theorem A].

It is a proposition of Guichard [32, Proposition 16] that if
$\rho\in\text{Hit}_{n}(S)$ and $\eta:\partial\Gamma\to\mathscr{F}_{n}$ is any
continuous $\rho$-equivariant map, then $\eta$ is the hyperconvex Frenet
curve. ∎

###  5.2. The Leaves

Throughout the following, let $\Omega$ denote the convex region preserved by
$\text{hol}_{*}$. Note that since we have shown $\text{hol}_{*}$ is Hitchin,
$\partial\Omega$ is strictly convex and $C^{1}$, e.g. [6, Théorème 1.1] or as
a standard consequence of [7].

######  Lemma 5.30.

Either $\mathcal{L}(x,z)=\xi^{1}(x)\oplus\xi^{1}(z)$ for all
$(x,z)\in\partial\Gamma^{(2)}$ or $\mathcal{L}(x,z)=\xi^{2}(z)$ for all
$(x,z)\in\partial\Gamma^{(2)}$. In the case where
$\mathcal{L}(x,z)=\xi^{2}(z)$ we have the following alternative:

  1. (1)

For every $(x,z)\in\partial\Gamma^{(2)}$, the line segment $\mathcal{S}(x,z)$
is contained in the connected component of
$\xi^{2}(z)-\\{\xi^{1}(z),\xi^{2}(x)\cap\xi^{2}(z)\\}$ that intersects $w^{2}$
for all $w\in\partial\Gamma$ so that $(x,w,z)$ is positively oriented,

  2. (2)

For every $(x,z)\in\partial\Gamma^{(2)}$, the line segment $\mathcal{S}(x,z)$
is contained in the connected component of
$\xi^{2}(z)-\\{\xi^{1}(z),\xi^{2}(x)\cap\xi^{2}(z)\\}$ that intersects $w^{2}$
for all $w\in\partial\Gamma$ so that $(x,z,w)$ is positively oriented.

###### Proof.

Suppose that there is a pair $(x_{0},z_{0})\in\partial\Gamma^{(2)}$ so that
$\mathcal{L}(x_{0},z_{0})\neq\xi^{2}(z_{0})$. We must prove
$\mathcal{L}(x,z)=\xi^{1}(x)\oplus\xi^{1}(z)$ for all
$(x,z)\in\partial\Gamma^{(2)}$. By continuity of $\mathcal{L}$ and $\xi^{2}$,
this holds on a neighborhood of $(x_{0},z_{0})$, and in particular for a pole-
pair $(\gamma^{-},\gamma^{+})$ for some $\gamma\in\Gamma-\\{e\\}$.

We have shown that $\mathcal{L}(\gamma^{-},\gamma^{+})$ is an invariant
subspace of $\mathrm{hol}_{*}(\gamma)$ other than $\xi^{2}(\gamma^{-})$. So
$\mathcal{L}(\gamma^{-},\gamma^{+})$ is either
$\xi^{2}(\gamma^{+})=e_{+}(\gamma)\oplus e_{0}(\gamma)$ or
$\xi^{1}(\gamma^{+})\oplus\xi^{1}(\gamma^{-})=e_{-}(\gamma)\oplus
e_{+}(\gamma)$, and we have arranged for
$\mathcal{L}(\gamma^{-},\gamma^{+})=\xi^{1}(\gamma^{-})\oplus\xi^{1}(\gamma^{+})$.

Since the lines $\mathcal{L}(x,z)$ are continuous in $(x,z)$, there is a
neighborhood $U$ of $(\gamma^{-},\gamma^{+})$ in $\partial\Gamma$ so that for
all $(x,z)\in U\times U$ the line $\mathcal{L}(x,z)$ is not tangent to
$\partial\Omega$. For all such $(x,z)$, we cannot have
$\mathcal{L}(x,z)=\xi^{2}(z)$, and so must have
$\mathcal{L}(x,z)=\xi^{1}(x)\oplus\xi^{1}(z)$.

Since $\Omega$ is convex with $C^{1}$ boundary
$\partial\Omega=\xi^{1}(\partial\Gamma)$, the condition
$\mathcal{L}(x,z)=\xi^{1}(x)\oplus\xi^{1}(z)$ is closed in
$\partial\Gamma^{(2)}$. From topological transitivity of the action of
$\Gamma$ on $\partial\Gamma^{(2)}$ and closedness of the condition
$\mathcal{L}(x,z)=\xi^{1}(x)\oplus\xi^{1}(z)$, we see that
$\mathcal{L}(x,z)=\xi^{1}(x)\oplus\xi^{1}(z)$ for all
$(x,z)\in\partial\Gamma^{(2)}$.

In the case $\mathcal{L}(x,z)=\xi^{2}(z)$, for a given
$(x,z)\in\partial\Gamma^{(2)}$ exactly one of the conditions (1) or (2) is
satisfied. Which is satisfied is locally constant because of the continuity of
the developing map, which gives the final dichotomy. ∎

The trichotomy of the previous lemma shall eventually correspond to the
choices of developing map between $\Phi_{\mathrm{tr}},\Phi_{\mathrm{tan},+},$
and $\Phi_{\mathrm{tan},-}$. To document this trichotomy and the two connected
components available in
$\xi^{2}(z)-\\{\xi^{1}(z),\xi^{2}(x)\cap\xi^{2}(z)\\}$, we define:

######  Definition 5.31.

The concave foliated flag structure
$(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of transverse type if
$\mathcal{L}(x,z)=\xi^{1}(x)\oplus\xi^{1}(z)$ for all
$(x,z)\in\partial\Gamma^{(2)}$ and
$(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of tangent type if
$\mathcal{L}(x,z)=\xi^{2}(z)$ for all $(x,z)\in\partial\Gamma^{(2)}$.

If $(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of tangent type
and for every $(x,z)\in\partial\Gamma$, the line segment $\mathcal{S}(x,z)$
intersects $\xi^{2}(w)$ for some $w\in\partial\Gamma$ so that $(x,w,z)$ is
positively oriented, we say
$(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of positive tangent
type. If for every $(x,z)\in\partial\Gamma$, the line segment
$\mathcal{S}(x,z)$ intersects $\xi^{2}(w)$ for some $w\in\partial\Gamma$ so
that $(x,z,w)$ is positively oriented we say that
$(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of negative tangent
type.

Lemma 5.30 shows that every concave foliated $\mathscr{F}_{1,2}$ structure on
$\mathrm{T}^{1}S$ is of transverse type or of positive or negative tangent
type. Using this structure, we are now able to constrain exactly what the
shapes of $\mathrm{dev}(f_{x})=B_{x}$ are for $x\in\partial\Gamma$.

######  Proposition 5.32 (Weakly Stable Leaf Characterization).

For all $x\in\partial\Gamma$,
$B_{x}=\mathbb{R}\mathbb{P}^{2}-(\overline{\Omega}\cup\xi^{2}(x))$.

###### Proof.

That $B_{x}$ is contained in
$\mathbb{R}\mathbb{P}^{2}-(\overline{\Omega}\cup\xi^{2}(x))$ follows from
Lemma 5.18 and the definitions of $\Omega$ and $\xi^{2}$.

For the other containment, suppose otherwise. Then there exists a point $p\in
D_{x}\cap(\mathbb{R}\mathbb{P}^{2}-[\overline{\Omega}\cup\xi^{2}(x)])$, where
we maintain the notation that $D_{x}=\mathbb{R}\mathbb{P}^{2}-B_{x}$. We first
consider the case that our concave foliated $\mathscr{F}_{1,2}$ structure is
of transverse type. Then $p\in\mathcal{L}(x,z)$ for some $z\in\partial\Gamma$.
For this pair $(x,z)$, the line segment $\mathcal{S}(x,z)$ is strictly
contained in the line segment $\mathcal{T}(x,z)$ between $\xi^{1}(x)$ and
$\xi^{1}(z)$ complementary to $\Omega$, because the line segments
$(\xi^{1}(x)\oplus\xi^{1}(z))-\Omega$ for $z\neq x$ in $\partial\Gamma$
foliate $\mathbb{R}\mathbb{P}^{2}-(\overline{\Omega}\cup\xi^{2}(x))$. Since
$\xi^{2}$ parameterizes the supporting lines of $\partial\Omega$, there is a
point $w\in\partial\Gamma$ so that $\xi^{2}(w)\cap\mathcal{T}(x,z)$ is not
contained in $\mathcal{S}(x,z)$. We may arrange for $(x,w,z)$ to be positively
oriented.

From the uniform convergence group property of $\Gamma$, there are distinct
$a,b\in\partial\Gamma$ and a sequence $\gamma_{n}\in\Gamma$ so that:

$\displaystyle\lim_{n\to\infty}\gamma_{n}z=a,\qquad\lim_{n\to\infty}\gamma_{n}y=b\qquad(\text{for
all }y\neq z\in\partial\Gamma).$

This sequence $\gamma_{n}$ has the properties that:

  1. (1)

The oriented intervals between $\gamma_{n}x$ and $\gamma_{n}w$ converge to
$\\{b\\}$,

  2. (2)

For all $n$, the line segment $\mathcal{S}(\gamma_{n}x,\gamma_{n}z)$ is
contained in the line segment

$I_{n}=\\{(\xi^{1}(\gamma_{n}x)\oplus\xi^{1}(\gamma_{n}z))\cap\xi^{2}(\gamma_{n}r)\mid(\gamma_{n}x,r,\gamma_{n}w)\text{
is positively oriented}\\}.$

So for all $n$, the complementary closed set $D_{\gamma_{n}x}$ of
$B_{\gamma_{n}x}$ contains the complement of $I_{n}$ in
$\xi^{1}(\gamma_{n}x)\oplus\xi^{1}(\gamma_{n}z)$. Note also that each
$D_{\gamma_{n}x}$ contains the common properly convex domain $\Omega$ and that
$\xi^{1}(\gamma_{n}x)\oplus\xi^{1}(\gamma_{n}z)$ converges to
$\xi^{1}(a)\oplus\xi^{1}(b),$ which is transverse to $\partial\Omega$. So for
any point $p\in\xi^{1}(a)\oplus\xi^{1}(b)$, the closed set $D_{\gamma^{n}x}$
contains $p$ for all sufficiently large $n$. So the Contraction Control Lemma
5.19 forces $D_{b}$ to contain two entire distinct projective lines, namely
$\xi^{2}(b)$ and $\xi^{1}(a)\oplus\xi^{1}(b)$. This contradicts proper
concavity of $D_{b}$. This finishes the the transverse type case.

The tangent type cases are similar, with the difference that the final
contradiction is obtained from forcing $D_{b}$ to contain the segment of
$\mathcal{L}(a,b)$ that must also contain $\mathcal{S}(a,b)$. ∎

We now deduce the exact structure of $\mathcal{S}(x,z)$ in each case on type
of concave flag foliated structure.

######  Corollary 5.33 (Options for $\mathcal{S}$).

We have:

  1. (1)

If $(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of transverse
type, then $\mathcal{S}(x,z)$ is the segment in $\xi^{1}(x)\oplus\xi^{1}(z)$
complementary to $\overline{\Omega}$ for all $(x,z)\in\partial\Gamma$.

  2. (2)

If $(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of positive
tangent type, then $\mathcal{S}(x,z)$ is the line segment in $\xi^{2}(z)$ that
intersects $\xi^{2}(w)$ for all $w\in\partial\Gamma$ so that $(x,w,z)$ is
positively oriented.

  3. (3)

If $(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of negative
tangent type, then $\mathcal{S}(x,z)$ is the line segment in $\xi^{2}(z)$ that
intersects $\xi^{2}(w)$ for all $w\in\partial\Gamma$ so that $(x,z,w)$ is
positively oriented.

###  5.3. Conclusion

We now produce and obstruct the equivalences of concave foliated flag
structures needed to prove Theorem 3.9.

We begin by noting that transverse type and the two tangent types of concave
foliated flag structures are preserved by foliated
$(\mathrm{SL}_{3}(\mathbb{R}),\mathscr{F}_{1,2})$-equivalence.

######  Lemma 5.34.

If $(\mathrm{dev}_{1},\mathrm{hol}_{1},\mathcal{F},\mathcal{G})$ is of
transverse type and
$(\mathrm{dev}_{2},\mathrm{hol}_{2},\mathcal{F},\mathcal{G})$ are of tangent
type, then $(\mathrm{dev}_{1},\mathrm{hol}_{1},\mathcal{F},\mathcal{G})$ and
$(\mathrm{dev}_{2},\mathrm{hol}_{2},\mathcal{F},\mathcal{G})$ are not
foliated-equivalent.

If $(\mathrm{dev}_{1},\mathrm{hol}_{1},\mathcal{F},\mathcal{G})$ is of
positive tangent type and
$(\mathrm{dev}_{2},\mathrm{hol}_{2},\mathcal{F},\mathcal{G})$ is of negative
tangent type, then
$(\mathrm{dev}_{1},\mathrm{hol}_{1},\mathcal{F},\mathcal{G})$ and
$(\mathrm{dev}_{2},\mathrm{hol}_{2},\mathcal{F},\mathcal{G})$ are not
foliated-equivalent.

###### Proof.

If $(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of tangent type,
then $\overline{\mathrm{dev}(g_{xz})}\subset\mathcal{L}(x,z)=\xi^{2}(z)$
($(x,z)\in\partial\Gamma^{(2)}$) does not contain the point $\xi^{1}(x)$ about
which $\mathrm{dev}(g_{xz})$ is a line segment lift. On the other hand, if
$(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ is of transverse type,
then
$\overline{\mathrm{dev}(g_{xz})}\subset\mathcal{L}(x,z)=\xi^{1}(x)\oplus\xi^{1}(z)$
contains the point $\xi^{1}(x)$ about which $\mathrm{dev}(g_{xz})$ is a line
segment lift. These properties are mutually exclusive and preserved by
foliated $(\mathscr{F}_{1,2},\text{SL}_{3}(\mathbb{R}))$-equivalence.

The defining conditions of positive tangent and negative tangent type are also
mutually exclusive and preserved by foliated
$(\mathscr{F}_{1,2},\mathrm{SL}_{3}(\mathbb{R}))$-equivalence. ∎

The conclusion now closely follows Guichard-Wienhard’s method in [33]. In the
preceding we have proved that the holonomy map induces a map
$\mathrm{hol}_{*}:\mathscr{D}_{\mathscr{F}_{1,2}}^{\text{cff}}(\mathrm{T}^{1}S)\to\text{Hit}_{3}(S)$.
We have also shown that
$\mathscr{D}_{\mathscr{F}_{1,2}}^{\text{cff}}(\mathrm{T}^{1}S)$ is a disjoint
union
$\mathcal{C}^{\mathrm{tr}}\sqcup\mathcal{C}^{\mathrm{ta},+}\sqcup\mathcal{C}^{\mathrm{ta},-}$
of transverse, positive tangent, and negative tangent type.

On the other hand, Example 3.5 gives three sections $\sigma_{\mathrm{tr}}$,
$\sigma_{\mathrm{ta},+}$, and $\sigma_{\mathrm{ta},-}$ corresponding to
$\Phi_{\mathrm{tr}},\Phi_{\mathrm{tan},+}$, and $\Phi_{\mathrm{tan},-}$,
respectively, from $\text{Hit}_{3}(S)$ to
$\mathscr{D}_{\mathscr{F}_{1,2}}^{\text{cff}}(\mathrm{T}^{1}S)$. Note that
$\sigma_{\mathrm{ta},+}$ has image in $\mathcal{C}^{\mathrm{ta},+}$,
$\sigma_{\mathrm{ta},+}$ has image in $\mathcal{C}^{\mathrm{ta},-}$, and
$\sigma_{\mathrm{tr}}$ has image in $\mathcal{C}^{\mathrm{tr}}$. To prove
Theorems 3.8 and 3.9 it suffices to show:

######  Lemma 5.35.

The maps $\sigma_{\mathrm{tr}}$ and $\mathrm{hol}_{*}$ restricted to
$\mathcal{C}^{\mathrm{tr}}$ are inverse to each other, the maps
$\sigma_{\mathrm{ta},+}$ and $\mathrm{hol}_{*}$ restricted to
$\mathcal{C}^{\mathrm{ta},+}$ are inverse to each other, and the maps
$\sigma_{\mathrm{ta},-}$ and $\mathrm{hol}_{*}$ restricted to
$\mathcal{C}^{\mathrm{ta},-}$ are inverse to each other.

###### Proof.

We prove the claim for $\sigma_{\mathrm{tr}}$; the other cases are similar.
That $\mathrm{hol}_{*}\circ\sigma_{\mathrm{tr}}=\text{Id}$ is trivial. To
address $\sigma_{\mathrm{tr}}\circ\mathrm{hol}_{*}$, let
$A=(\mathrm{dev},\mathrm{hol},\mathcal{F},\mathcal{G})$ be of transverse type,
and let
$(\mathrm{dev}_{\rho}^{\mathrm{tr}},\mathrm{hol}_{\rho},\mathcal{F},\mathcal{G})$
denote $\sigma_{\mathrm{tr}}(\mathrm{hol}_{*}{A})$. By Remark 3.7, to show
equivalence of $\sigma_{\mathrm{tr}}(\mathrm{hol}_{*}(A))$ and $A$, it
suffices to show that
$(\mathrm{dev}_{\rho}^{\mathrm{tr}})^{-1}\circ\mathrm{dev}$ preserves each
leaf of $\mathcal{G}$. Here, the branch of
$(\mathrm{dev}_{\rho}^{\mathrm{tr}})^{-1}$ is chosen so as to map $g_{xz}$
into $g_{xz}$ for all $(x,z)\in\partial\Gamma^{(2)}$. The claim follows
because both $\mathrm{dev}(g_{xz})$ and
$\mathrm{dev}_{\rho}^{\mathrm{tr}}(g_{xy})$ are the line segment lift
complementary to $\Omega$ in $\xi^{1}(x)\oplus\xi^{1}(z)$ based at
$\xi^{1}(x)$ by Lemma 5.33. ∎

## References

  * [1] Daniele Alessandrini and Brian Collier.  The geometry of maximal components of the $\text{PSp}(4,\mathbb{R})$ character variety.  Geom. Topol., 23(3):1251–1337, 2019.
  * [2] Daniele Alessandrini, Colin Davalo, and Qiongling Li.  Projective structures with (quasi-)Hitchin holonomy, 2021.  arxiv:2110.15407.
  * [3] Daniele Alessandrini, Sara Maloni, Nicolas Tholozan, and Anna Wienhard.  Fiber bundles associated with Anosov representations, 2023.  arxiv:2303.10786.
  * [4] David Baraglia.  $G_{2}$ Geometry and Integrable Systems.  Phd thesis, University of Oxford, June 2010.
  * [5] Thierry Barbot.  Three-dimensional Anosov flag manifolds.  Geom. Topol., 14(1):153–191, 2010.
  * [6] Yves Benoist.  Convexes divisibles. I.  In Algebraic groups and arithmetic, volume 332, pages 339–374. Tata Inst. Fund. Res., Mumbai, 2004.
  * [7] Jean-Paul Benzécri.  Sur les variétés localement affines et localement projectives.  Bull. Soc. Math. France, 88:229–332, 1960.
  * [8] Jonas Beyrer and Fanny Kassel.  $H^{p,q}$-convex cocompactness and higher higher Teichmüller spaces, 2023.  arxiv:2305.15031.
  * [9] Harrison Bray, Richard Canary, Lien-Yung Kao, and Giuseppe Martone.  Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations.  J. Reine Angew. Math., 791:1–51, 2022.
  * [10] Martin Bridgeman, Richard Canary, François Labourie, and Andres Sambarino.  The pressure metric for Anosov representations.  Geom. Funct. Anal., 25:1089–1179, 2015.
  * [11] Jean-Philippe Burelle and Nicolaus Treib.  Schottky groups and maximal representations.  Geometriae Dedicata, 195:215–239, 2018.
  * [12] Jean-Philippe Burelle and Nicolaus Treib.  Schottky presentations of positive representations.  Mathematische Annalen, 382(3-4):1705–1744, 2022.
  * [13] León Carvajales, Xian Dai, Beatrice Pozzetti, and Anna Wienhard.  Thurston’s asymmetric metrics for Anosov representations, 2022.  arxiv:2210.05292.
  * [14] León Carvajales and Florian Stecker.  Anosov representations acting on homogeneous spaces: domains of discontinuity, 2023.  arXiv:2308.08607.
  * [15] Suhyoung Choi and William Goldman.  Convex real projective structures on closed surfaces are closed.  Proc. Amer. Math. Soc., 118(2):657–661, 1993.
  * [16] Brian Collier, Nicolas Tholozan, and Jérémy Toulisse.  The geometry of maximal representations of surface groups into ${\rm SO}_{0}(2,n)$.  Duke Math. J., 168(15):2873–2949, 2019.
  * [17] Jeffrey Danciger, François Guéritaud, and Fanny Kassel.  Convex cocompactness in pseudo-Riemannian hyperbolic spaces.  Geom. Dedicata, 192:87–126, 2018.
  * [18] Jeffrey Danciger, François Guéritaud, and Fanny Kassel.  Convex cocompact actions in real projective geometry.  Ann. Sci. Éc. Norm. Supér., To appear.
  * [19] Colin Davalo.  Geometric structures for maximal representations and pencils.  arxiv:2407.01254.
  * [20] Colin Davalo.  Nearly geodesic immersions and domains of discontinuity, 2023.  arxiv:2303.11260.
  * [21] Colin Davalo and J. Maxwell Riestenberg.  Finite-sided Dirichlet domains and Anosov subgroups, 2024.  arxiv:2402.06408.
  * [22] Benjamin Delarue, Daniel Monclair, and Andrew Sanders.  Locally homogeneous Axiom A flows I: projective Anosov subgroups and exponential mixing, 2024.  arxiv:2403.14257.
  * [23] Subhadip Dey, Dongryul M. Kim, and Hee Oh.  Ahlfors regularity of Patterson-Sullivan measures of Anosov groups and applications, 2024.  arXiv:2401.12398.
  * [24] David Dumas and Andrew Sanders.  Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations.  Geom. Topol., 24(4):1615–1693, 2020.
  * [25] Parker Evans.  Geometric structures for the $G_{2}^{\prime}$ Hitchin component, 2024.  arxiv:2405.04492.
  * [26] Elisha Falbel and Martin Mion-Mouton.  Geometric surgeries of three-dimensional flag structures and non-uniformizable examples, 2024.  arxiv:2406.02053.
  * [27] Simion Filip.  Global properties of some weight 3 variations of Hodge structure.  In European Congress of Mathematics, pages 553–568. EMS Press, Berlin, [2023] ©2023.
  * [28] Vladimir Fock and Alexander Thomas.  Higher complex structures.  Int. Math. Res. Not. IMRN, pages 15873–15893, 2021.
  * [29] William Goldman.  Convex real projective structures on compact surfaces.  J. Differential Geom., 30(3):791–845, 1990.
  * [30] William M. Goldman.  Geometric structures on manifolds, volume 227 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, [2022] ©2022.
  * [31] Olivier Guichard.  Une dualité pour les courbes hyperconvexes.  Geom. Dedicata, 112:141–164, 2005.
  * [32] Olivier Guichard.  Composantes de Hitchin et représentations hyperconvexes de groupes de surface.  J. Differential Geom., 80(3):391–431, 2008.
  * [33] Olivier Guichard and Anna Wienhard.  Convex foliated projective structures and the Hitchin component for ${\rm PSL}_{4}({\bf R})$.  Duke Math. J., 144(3):381–445, 2008.
  * [34] Olivier Guichard and Anna Wienhard.  Anosov representations: domains of discontinuity and applications.  Invent. Math., 190:357–438, 2012.
  * [35] Nigel Hitchin.  Lie groups and Teichmüller space.  Topology, 31(3):449–473, 1992.
  * [36] Michael Kapovitch, Bernhard Leeb, and Joan Porti.  Dynamics on flag manifolds: domains of proper discontinuity and cocompactness.  Geom. Topol., 22:157–234, 2018.
  * [37] Dongryul M. Kim, Hee Oh, and Yahui Wang.  Ergodic dichotomy for subspace flows in higher rank, 2023.  arxiv:2310.19976.
  * [38] Dongryul M. Kim, Hee Oh, and Yahui Wang.  Properly discontinuous actions, growth indicators and conformal measures for transverse subgroups, 2024.  arxiv:2306.06846.
  * [39] Georgios Kydonakis, Charlie Reid, and Alexander Thomas.  Fock bundles and Hitchin components, 2023.  arXiv:2310.04377.
  * [40] François Labourie.  Anosov flows, surface groups and curves in projective space.  Invent. Math., 165(1):51–114, 2006.
  * [41] François Labourie.  Flat projective structures on surfaces and cubic holomorphic differentials.  Pure Appl. Math. Q., 2007.
  * [42] François Labourie.  Cyclic surfaces and Hitchin components in rank 2.  Ann. of Math., 185(1):1–58, 2017.
  * [43] Alexander Livšic.  Cohomology of dynamical systems.  Izv. Akad. Nauk SSSR Ser. Mat., 36:1296–1320, 1972.
  * [44] Vladmir Marković, Nathaniel Sagman, and Peter Smillie.  Unstable minimal surfaces in $\mathbb{R}^{n}$ and in products of hyperbolic surfaces.  Commentarii Mathematici Helvetici, To appear.
  * [45] Filippo Mazzoli and Gabriele Viaggi.  ${\mathrm{S}\mathrm{O}}_{0}(2,n+1)$-maximal representations and hyperbolic surfaces.  to appear in Mem. Amer. Math. Soc., 2022.
  * [46] Filippo Mazzoli and Gabriele Viaggi.  Volume, entropy, and diameter in SO$(p,q+1)$-higher Teichmüller spaces, 2023.  arxiv:2312.17137.
  * [47] Quentin Mérigot and Thierry Barbot.  Anosov AdS representations are quasi-Fuchsian.  Groups, Geometry, and Dynamics, 6(3):441–483, 2012.
  * [48] Geoffrey Mess.  Lorentz spacetimes of constant curvature.  Geom. Dedicata, 126:3–45, 2007.
  * [49] Martin Mion-Mouton.  Geometrical compactifications of geodesic flows and path structures.  Geom. Dedicata, 217(2):Paper No. 18, 43, 2023.
  * [50] Alexander Nolte.  Leaves of foliated projective structures.  Adv. Math., To appear.
  * [51] Alexander Nolte.  Canonical maps from spaces of higher complex structures to Hitchin components, 2022.  arxiv:2204.04732.
  * [52] Alexander Nolte.  Metrics on Hitchin components from Hölder distortion, 2024.  arxiv:2402.16584.
  * [53] Alexander Nolte.  On foliations in $\text{PSL}_{4}(\mathbb{R})$-Teichmüller theory, 2024.  arxiv:2407.02237.
  * [54] Rafael Potrie and Andrés Sambarino.  Eigenvalues and entropy of a Hitchin representation.  Invent. Math., 209(3):885–925, 2017.
  * [55] Beatrice Pozetti and Andrés Sambarino.  Lipschitz limit sets revisited: Hilbert entropy and non-differentiability, 2022.  hal-03898485.
  * [56] Maria Beatrice Pozzetti, Andrés Sambarino, and Anna Wienhard.  Conformality for a robust class of non-conformal attractors.  J. Reine Angew. Math., 774:1–51, 2021.
  * [57] Nathaniel Sagman and Peter Smillie.  Unstable minimal surfaces in symmetric spaces of non-compact type, 2022.  arxiv:2208.04885.
  * [58] Andrés Sambarino.  Quantitative properties of convex representations.  Commentarii Mathematici Helvetici, 89(2):443–488, 2014.
  * [59] Andrés Sambarino.  The orbital counting problem for hyperconvex representations.  Ann. Inst. Fourier (Grenoble), 65(4):1755–1797, 2015.
  * [60] Andrés Sambarino.  A report on an ergodic dichotomy.  Ergodic Theory Dynam. Systems, 44(1):236–289, 2024.
  * [61] Andrea Seppi, Graham Smith, and Jérémy Toulisse.  On complete maximal submanifolds in pseudo-hyperbolic space, 2023.  arxiv:2305.15103.
  * [62] Florian Stecker.  Balanced ideals and domains of discontinuity of Anosov representations.  Geometriae Dedicata, 218(1):9, 2023.
  * [63] Florian Stecker and Nicolaus Treib.  Domains of discontinuity in oriented flag manifolds.  Journal of the London Mathematical Society, 106(2):1380–1442, 2022.
  * [64] Konstantinos Tsouvalas.  The Hölder exponent of Anosov limit maps, 2023.  arxiv:2306.15823.
  * [65] Tengren Zhang and Andrew Zimmer.  Regularity of limit sets of Anosov representations, 2019.  arxiv:1903.11021.
  * [66] Andrew Zimmer.  Projective Anosov representations, convex cocompact actions, and rigidity.  J. Differential Geom., 119(3):513–586, 2021.


# The Cycle of Dust in the Milky Way: Clues from the High-Redshift and Local
Universe

Eli Dwek  Frédéric Galliano  Anthony Jones

###### Abstract

Models for the evolution of dust can be used to predict global evolutionary
trends of dust abundances with metallicity and examine the relative importance
of dust production and destruction mechanisms. Using these models, we show
that the trend of the abundance of polycyclic aromatic hydrocarbons (PAHs)
with metallicity is the result of the delayed injection of carbon dust that
formed in low mass asymptotic giant branch (AGB) stars into the interstellar
medium. The evolution of dust composition with time will have important
consequences for determining the opacity of galaxies and their reradiated
thermal infrared (IR) emission. Dust evolution models must therefore be an
integral part of population synthesis models, providing a self-consistent link
between the stellar and dust emission components of the spectral energy
distribution (SED) of galaxies. We also use our dust evolution models to
examine the origin of dust at redshifts $>6$, when only supernovae and their
remnants could have been, respectively, their sources of production and
destruction. Our results show that unless an average supernova (or its
progenitor) produces between 0.1 and 1 $M_{\odot}$ of dust, alternative
sources will need to be invoked to account for the massive amount of dust
observed at these redshifts.

Observational Cosmology Lab, Code 665, NASA Goddard Space Flight center,
Greenbelt, MD 20771

Service d’Astrophysique, CEA/Saclay, L’Orme des Merisiers, 91191 Gif-sur-
Yvette, France

Institut d’Astrophysique Spatiale (IAS), Bâtiment 121, Université Paris-Sud 11
and CNRS, F-91405 Orsay, France

##  1\. Historical Overview

Chemical evolution (CE) models follow the formation, destruction, abundance,
and spatial and stellar distribution of elements created during the
nucleosynthesis era of the Big Bang, and the different evolutionary stages in
stars. Models are pitted against a host of observational test, such as the
relative abundances of the various elements and their isotopes in meteorites,
stars, and in the interstellar, intergalactic and intracluster media, the
G-dwarf metallicity distribution, and the age-metallicity relation of the
various systems [e.g. (Matteucci 2001; Pagel 1997)].

CE models provide a natural framework for studying the evolution of dust since
the abundance of the elements locked up in dust must be constrained by the
availability of refractory elements in the interstellar medium (ISM). CE
models need then to be generalized to include processes unique to the
evolution of dust: the condensation efficiency of refractory elements in
stellar ejecta, the destruction of grains in the ISM by expanding supernova
remnants (SNRs), and the growth and coagulation of grains in clouds Dwek
(1998).

The first dust evolution models (Dwek & Scalo 1979, 1980) (hereafter DS)
addressed the origin of the elemental depletion pattern, which was a subject
of considerable debate. Field (1974) showed that the depletion pattern
correlated with condensation temperature, suggesting that it reflects the
condensation efficiency of the elements in their respective sources. Such
causal correlation requires that dust undergoes very little interstellar
processing that can alter the depletion pattern. An equally good correlation
exists between the magnitude of the elements’ depletion and their first
ionization potential (Snow 1975), suggesting that the depletion pattern may
instead be governed by accretion processes in molecular clouds. Pointing out
the intrinsic physical correlation between the condensation temperature, the
first ionization potential, and the threshold for grain destruction by
sputtering, DS suggested that the depletion pattern could reflect the
destruction efficiency of dust in the ISM. It is currently clear that all
three processes play some role in establishing the elemental depletion
pattern, since it depends on the density of the medium in which it is observed
(Savage & Sembach 1996). Globally, their relative importance depends on the
prevalence of and the cycling times between the different phases (hot,
neutral, and molecular) of the ISM.

A more detailed review of dust evolution models is given by Dwek (1998). Since
then, several models have been constructed to follow the evolution of dust in
dwarf galaxies (Lisenfeld & Ferrara 1998), Damped Ly$\alpha$ systems (Kasimova
& Shchekinov 2003), the Milky Way galaxy (Zhukovska et al. 2008), and the
origin of isotopic anomalies in meteorites (Clayton & Nittler 2004; Zinner et
al. 2006a, b).

Early dust evolution models adopted the instantaneous recycling approximation,
which assumes that all the elements and dust are promptly injected back into
the ISM following the formation of their nascent stars. In contrast Dwek
(1998) and Morgan & Edmunds (2003) constructed models that take the finite
main sequence (MS) lifetimes of the stars into account. Silicate dust is
primarily produced in supernovae (SN) that “instantaneously” recycle their
products back to the ISM, whereas carbon dust is mainly produced in low-mass
carbon-rich stars which have significantly longer MS lifetimes. Consequently
these models predicted that the composition of the dust should evolve as a
function of time.

Dust evolution models contain many still uncertain parameters such as the dust
yields in the various sources and the grain destruction efficiency in the ISM.
However, in spite of these uncertainties we will show that they can be very
successful in predicting global evolutionary trends, namely the observed
correlation of the abundance of polycyclic aromatic hydrocarbon (PAH)
molecules with metallicity, and in examining the origin of dust in the high-
redshift universe.

##  2\. Main Ingredients of Dust Evolution Models

###  2.1. The Yield of Dust in Stars

The dust condensation efficiency, and its composition and size distribution
depend very much on the environment in which it is formed. Dust can form in
the quiescent outflows of AGB or Wolf-Rayet stars, or in the explosively
expelled ejecta of SNe and novae. There is substantial observational evidence
for the formation of dust in all these sources, however, their relative
importance as sources of interstellar dust, and the composition and size
distribution of the condensed dust are still uncertain, especially in SNe.
Because of the ease of formation and stability of the CO molecule, dust
sources can be divided into two categories: carbon-rich sources, in which the
C/O abundance ratio is larger than 1. These sources will produce carbon dust;
and oxygen-rich sources with C/O $<$ 1, which will produce silicate-type dust.
Such simple arguments assume that CO formation goes to completion, which may
not always be the case, as suggested by the dynamical condensation models of
Ferrarotti & Gail (2006).

SN explosions mark the death of stars more massive than $\sim 8$ $M_{\odot}$.
Figure 1 depicts the post-explosive composition of a 25 $M_{\odot}$ star
(Woosley 1988). It depicts a typical onion-skin structure in which the
composition of the different layers reflect the pre- and post-explosion
nuclear burning stages of the star. Globally, the ejecta has a C/O ratio $<$
1, and should in principle be only producing silicate dust. However, in spite
of the mixing between the layers caused by Rayleigh-Taylor instabilities in
the ejecta, this mixing is of macroscopic nature and does not occur on an
atomic level. So the layers above above $\sim~{}4.2$ $M_{\odot}$, in which C/O
$>$ 1, will maintain this ratio and produce carbon dust, whereas the inner
layers will produce silicates. If all condensible elements precipitated out of
the ejecta and formed dust, the yield of dust in a typical SN would be about 1
$M_{\odot}$ (Kozasa et al. 1989, 1991).

Figure 1.: The post-explosive composition of a 25 $M_{\odot}$ star (Woosley
1988). A typical SN can potentially produce 1 $M_{\odot}$ of dust.

The yield and composition of dust in lower mass stars depends on the C/O ratio
in their atmosphere during the AGB phase of their evolution. Figure 2 depicts
the C and O yield in stars. Supernovae yields were taken from Woosley & Weaver
(1995) and AGB yields were taken from (Karakas & Lattanzio 2003). Stars with
masses above $\sim 8$ $M_{\odot}$ produce carbon and silicate dust,
irrespective of the global C/O ratio in their ejecta. The mass range of carbon
rich stars depends on the initial stellar metallicity. At zero metallicity
(left panel) the mass range of stars producing carbon dust is between 0.8 and
7.8 $M_{\odot}$. This range narrows significantly to masses between 3.0 and
3.6 $M_{\odot}$ at solar metallicity (right panel). Figure 3 presents a
qualitative depiction of the method we use to calculate the yield of carbon
and silicate dust in AGB stars. A more realistic model calculating the yield
of dust in AGB stars was presented by Ferrarotti & Gail (2006) and Hoefner
(2009, this conference proceedings).

---
Figure 2.: The C and O yields in stars. Stars with masses above $\sim~{}8$
$M_{\odot}$ become SNe and produce both silicate and carbon dust. Stars with
masses below $\sim 8$ $M_{\odot}$ produce either carbon or silicate dust,
depending on the C/O ratio in their ejecta. The light gray area in the
horizontal bar depicts the range of stellar masses in which C/O $>$ 1.
---
Figure 3.: Qualitative depiction of the calculated yield of carbon and
silicate dust in AGB stars. When $C/O>1$ (left panel), the star produces only
carbon dust. The dark shaded area depicts the number of carbon atoms that
condense into dust. When $C/O<1$ (right panel) the star produces silicate
dust, and the dark shaded area depicts the condensing elements.

Figure 4 (left panel) depicts the stellar evolutionary tracks of stars with an
initial solar metallicity. Also shown in the figure is their MS lifetime.
Similar lifetimes are obtained for different initial metallicities (Portinari
et al. 1998). At metallicity of $Z=0$, the first carbon dust producing stars
are about 8 $M_{\odot}$ and will evolve of the MS about 50 Myr after their
formation. At solar metallicities, the production of carbon by AGB stars will
be delayed by about 500 Myr, when $\sim 4$ $M_{\odot}$ stars evolve off the
MS. Since most of the interstellar carbon dust is made in AGB stars, this
delay in its formation can have important observational consequences. Carbon
dust has a significantly higher visual opacity than silicates, so the opacity
of galaxies will change with time, with young systems being more transparent
than older ones.

Figure 4 (right panel) depicts the different evolutionary trends of SN- and
AGB-condensed dust calculated for a CE model with by exponential star
formation rate characterized by a decay time of 6 Gyr, and a Salpeter initial
mass function (Dwek 1998, 2005). The silicate and carbon dust yields were
calculated assuming a condensation efficiency of unity in the ejecta, and
grain destruction was neglected. The model therefore represents an idealized
case, in which grain production is maximized, and grain destruction processes
are totally ignored. Also shown in the figure are the separate contributions
of AGB stars to the abundance of silicate and carbon dust. The onset of the
AGB contribution to the silicate abundance starts at $t\approx 50$ Myr, when
$\sim$ 8 M⊙ stars evolve off the main sequence, whereas AGB stars start to
contribute to the carbon abundance only at $t\approx 500$ Myr, when 4 M⊙ stars
reach the AGB phase. The figure also presents the dust-to-ISM metallicity
ratio, which is almost constant at a value of $\sim$ 0.36.

---
Figure 4.: Left panel: The H-R diagram of stars and their main sequence
lifetime. Right panel: The evolution of silicate (dashed line) and carbon
(solid line) dust from SNe (bold curves) and AGB stars (light curves).

##  3\. The Lifetime of Interstellar Dust

Following their injection into the ISM, the newly-formed dust particles are
subjected to a variety of interstellar processes resulting in the exchange of
elements between the solid and gaseous phases of the ISM, including: (a)
thermal sputtering in high-velocity ($>$200 km s-1) shocks; (b) evaporation
and shattering by grain-grain collisions in lower velocity shocks; and (3)
accretion in dense molecular clouds. Detailed description of the various grain
destruction mechanisms and grain lifetimes in the ISM were presented by (Jones
et al. 1996; Jones 2004). In addition, SN condensates can be destroyed shortly
after their formation by reverse shocks that travel through the expanding
ejecta (Dwek 2005, Bianchi & Schneider 2007, Nozawa et al. 2008).

The most important parameter governing the evolution of the dust is its
lifetime, $\tau_{d}$, against destruction by SNRs. In an interstellar medium
with a uniform dust-to-gas mass ratio, $Z_{d}$, this lifetime is given by
(Dwek & Scalo 1980; McKee 1989):

$\tau_{d}={M_{d}(t)\over\left<m_{d}\right>R_{SN}}={M_{g}(t)\over\left<m_{\rm
ISM}\right>R_{SN}}$ (1)

where $M_{d}$ and $M_{g}$ are, respectively, the total mass of dust and gas in
the galaxy, $\left<m_{d}\right>$ is the total mass of elements that are locked
up in dust and returned by a single SNR back to the gas phase either by
thermal sputtering or evaporative grain-grain collisions. $R_{SN}$ is the SN
rate in the galaxy, so that the product $\left<m_{d}\right>\,R_{SN}$ is the
destruction rate of dust in the ISM. The parameter $\left<m_{\rm
ISM}\right>\equiv\left<m_{d}\right>/Z_{d}$ is the effective ISM mass that is
completely cleared of dust by a single SNR, given by (Dwek et al. 2007):

$\left<m_{\rm ISM}\right>=\int_{v_{0}}^{v_{f}}\ \zeta_{d}(v_{s})\
\left|{dM\over dv_{s}}\right|\ dv_{s}$ (2)

where $\zeta_{d}(v_{s})$ is the fraction of the mass of dust that is destroyed
in an encounter with a shock wave with a velocity $v_{s}$, $(dM/dv_{s})dv_{s}$
is the ISM mass that is swept up by shocks in the [$v_{s},\ v_{s}+dv_{s}$]
velocity range, and $v_{0}$ and $v_{f}$ are the initial and final velocities
of the SNR. Figure 5 depicts the mass fraction of carbon and silicate dust
that is destroyed after being swept up by a shock of velocity $v_{s}$ as a
function of shock velocity. An updated version for the carbon and PAH
destruction efficiency was presented by Jones et al. (2009, this conference
proceedings). For example, in the Milky Way $M_{g}\approx 5\times 10^{9}$
$M_{\odot}$, $R_{SN}\approx 0.03$ yr-1, and $\left<m_{\rm ISM}\right>\approx
300$ $M_{\odot}$, giving a dust lifetime of $\sim 6\times 10^{8}$ yr.

Figure 5.: The mass fraction of carbon and silicate dust that is destroyed as
a function of shock velocity (after Jones et al. 1996).

In addition, SN condensates can be destroyed shortly after their formation by
reverse shocks that travel through their expanding ejecta (Dwek 2004, Nozawa
et al. 2008).

##  4\. PAHs and Silicate Dust as Tracers of AGB- and SN-condensed Dust

An exciting discovery made by spectral and photometric observations of nearby
galaxies with the Infrared Space Observatory (ISO) and Spitzer satellites was
the striking correlation between the strength of their mid-infrared (IR)
aromatic features, commonly attributed to the emission from PAHs, and their
metallicity, depicted in Figure 6 [left panel; see Galliano et al. (2008) for
references]. The figure shows the rise of $F_{8/24}$, the 8 $\mu$m-to-24$\mu$m
band flux ratio with galaxies’ metallicity, and the existence of a metallicity
threshold below which $F_{8/24}$ is equal the flux ratio of the dust continuum
emission. The strength of the aromatic feature is a measure of the PAH
abundance. Since PAHs are predominantly made in C-rich AGB stars, this
correlation provides the first observational evidence for the delayed
injection of AGB condensed dust into the ISM, provided the metallicity is a
measure of the galaxies’ age. The testing of this hypothesis requires first
the determination of the PAH abundance in each galaxy.

---
Figure 6.: Left panel: The observed correlation between the 8-to-24 $\mu$m
bands flux density and metallicity. Right panel: PAH and dust abundances
derived from detailed models of galaxies’ SED versus galaxies’ metallicity.

PAHs are very small macromolecules, typically 50 Å in diameter, that are
stochastically heated by the ambient radiation field. Consequently, only a
fraction of the PAHs are radiating at mid-IR wavelengths at any given time. To
determine the total abundance of PAHs, including those too cold to emit in the
aromatic features requires the determination of the intensity of the
interstellar radiation field (ISRF) to which they are subjected.

Figure 7 depicts the steps used by Galliano et al. (2008) in modeling the
galaxies’ spectral energy distribution (SED). The galaxy used for this
illustrative purpose is the starburst M82. The dust model used in the
calculations is the BARE-GR-S model of Zubko et al. (2004), consisting of
PAHs, and bare silicate and graphite grains with solar abundances constraints.

Figure 7a shows the SED of M82, and its various emission components: stellar
emission at optical, and near-IR wavelengths; the PAH spectrum at mid IR
wavelengths; dust emission from mid- to far-IR wavelengths; and free-free and
synchrotron emission at radio wavelengths.

Fig 7b depicts a fit to the dust spectrum using an ISRF characterized by a
power-law distribution of radiation field intensities. PAH abundance
determined by this method will underestimate the real abundance of PAHs in the
galaxy compared to the method outlined below. Our models use the free-free and
mid-IR emissions to constrain the gas and dust radiation from the gas and dust
from H II regions, and the far-IR and optical emission to constrain the ISRF
that heats the dust in photo-dissociation regions (PDRs). The radio emission
is uniquely decomposed into free-free (dashed) and synchrotron (dotted)
emission components (see Figure 7d).

Massive stars are required to produce the ionizing radiation and the expanding
SN blast waves that generate, respectively, the observed free-free and
synchrotron emission. These stars are produced in an ”instantaneous” burst of
star formation, in contrast to the stars that are continuously created over
the lifetime of the galaxy and mostly contribute to the optical and near-IR
emission (Figure 7c).

The ionizing and a fraction of the non-ionizing radiation emitted by the
starburst component that is absorbed in the H II region is shown as a shaded
area in Figure 7d. This energy is reradiated by the dust and gas, giving rise
to the thermal IR and free-free emission components shown in the figure.

The non-ionizing radiation from the older stellar population and the radiation
escaping the H II regions form the diffuse ISRF that is absorbed by the dust
in photodissociation regions (PDR) (Figure 7e). The shaded area depicts the
fraction of the radiation from the older stellar population that is absorbed
in PDRs. The absorbed radiation is reemitted by the dust, giving rise to the
IR emission (figure 7f).

Figure 7g depicts the different emission components, and Figure 7h shows the
fit of their sum to the SED of M82. The details of the fitting procedure are
described in Galliano et al. (2008).

Using this physical fitting procedure, Galliano et al. (2008) derived the
abundance of PAHs, silicates, and graphite grains in 35 nearby galaxies with
metallicities ranging from 1/50 to 3 times solar. The detailed physical
modeling of their SEDs gives larger PAH abundances compared to models that
employ a template ISRFs to heat the PAHs and the dust. In these models, such
as the one depicted in Fig. 7b, PAHs are subjected to the same intense
radiation field as that required to produce the mid-IR emission from hot dust.
In our model, PAHs are subjected to a weaker radiation field. Since PAHs do
not survive in H II regions, all their emission originates from PDRs, which
are subjected to lower intensity radiation fields than the H II regions. So,
compared to the template ISRF models, a larger amount of PAHs is required to
produce the same PAH spectrum with a weaker ISRF. The resulting PAH abundances
are plotted versus metallicity in Figure 6 (right panel). The figure shows
that the observed trend of increasing 8-to-24 $\mu$m band flux ratio with
metallicity indeed reflects a trend of increasing PAH abundance with
metallicity. The figure also shows the distinct evolutionary trends of PAHs
and the far-IR emitting dust with metallicity.

---
---
Figure 7.:  Construction of the fit to the SED of the starburst galaxy M82.
See text for details.

Figure 8 compares the evolution of the PAHs and dust components derived from
the dust evolution model to the trend of PAH abundances with metallicity. For
the sake of this comparison, the evolution of dust abundances as a function of
time was converted to an evolution as a function of metallicity, using the
age-metallicity relation derived in the model. We emphasize that the
parameters used in the dust evolution model (the star formation rate, the
stellar IMF) were identical to those used in the population synthesis model
that was used to fit the galaxies’ SED. From the dust evolution model we
already derived the two distinct evolutionary trends of SN- and AGB-condensed
dust (see Fig. 6, right panel for the idealized example). The current figure
compares these results with the derived abundance of the PAHs and of the dust
that gives rise to the far-IR emission. The latter is dominated by emission
from silicates, and should therefore follow the trend of the SN condensates,
since most silicate grains are produced in SNe. The figure shows that the
observed far-IR emitting dust falls on the evolutionary track of the
calculated SN-condensed dust, and that the observed PAH abundances fall on the
evolutionary track of the carbon dust that formed in AGB stars. The shaded
regions in the figure represent the range of evolutionary tracks that
correspond to different parameters that determine the star formation rate and
grain destruction efficiencies in the models.

Figure 8.:  Comparison between the metallicity trends of the PAH abundance
derived from the observed SED and those derived from the chemical evolution
model. The shaded area represent the range of model prediction for different
grain destruction and star formation rates. Details of the figure are
described in Galliano et al. (2008).

##  5\. The presence of massive amounts of dust at high redshift

The detection of massive amounts of dust in hyperluminous IR galaxies at
redshifts $z>6$ raises challenging questions about the sources capable of
producing such large amount of dust during the relatively short lifetime of
these galaxies (Maiolino et al. 2006; Beelen et al. 2006; Morgan & Edmunds
2003). For example, the galaxy SDSS J1148+5251 (hereafter J1148+5251) located
at $z=6.4$ was observed at far-IR and submillimeter wavelength (Bertoldi et
al. 2003; Robson et al. 2004; Beelen et al. 2006). The average IR luminosity
of the source is $L_{IR}\sim 2\times 10^{13}$ $L_{\odot}$, and the average
dust mass is $M_{d}\sim 2\times 10^{8}$ $M_{\odot}$. Using the Kennicutt
(1998) relation, one can derive a star formation rate (SFR) of $\sim 3000$
$M_{\odot}$ yr-1 from the observed far-IR luminosity. For comparison, the
Milky Way galaxy is about 10 Gyr old, has an average SFR of $\sim 3$
$M_{\odot}$ yr-1, and contains about $5\times 10^{7}$ $M_{\odot}$ of dust, a
significant fraction of which was produced in AGB stars.

At $z=6.4$ the universe was only 890 Myr old, using standard $\Lambda$CDM
parameters ($\Omega_{m}=0.27$, $\Omega_{\Lambda}=0.73$, and $H_{0}=70$ km s-1
Mpc-1). If J1148+5251 formed at $z=10$ then the galaxy is only 400 Myr old. If
the SFR had occured at a constant rate over the lifetime of the galaxy, its
initial mass should have been about 1012 $M_{\odot}$, which is significantly
larger than the dynamical mass $M_{dyn}\approx 5\times 10^{10}$ $M_{\odot}$ of
the galaxy (Walter et al. 2004). The high observed SFR may therefore represent
a recent burst of star formation that has lasted for only about 20 Myr. The
galaxy J1148+5251 is therefore at most $\sim 400$ Myr old, and probably
significantly younger with an age of only $\sim 20$ Myr. Adopting a current
gas mass of $M_{g}=3\times 10^{10}$ $M_{\odot}$ for this galaxy we get that
the gas mass fraction at 400 Myr is about 0.60. The dust-to-gas mass ratio is
given by $Z_{d}\equiv M_{d}/M_{g}=0.0067$.

A significant fraction of the dust in the Milky Way was produced in AGB stars.
However, these stars are not likely to contribute significantly to the
formation of dust in very young galaxies, since the low mass stars ($M\approx
3$ $M_{\odot}$) that produce most of the dust did not have time to evolve off
the main sequence (Dwek 1998; Morgan & Edmunds 2003; Dwek 2005). In contrast,
core collapse SNe ($M>8$ $M_{\odot}$) and their post-main-sequence progenitors
inject their nucleosynthetic products back into the ISM shortly ($t<20$ Myr)
after their formation, resulting in the rapid enrichment of the interstellar
medium (ISM) with the dust that formed during the mass loss phase prior to the
SN event, or in the explosive SN ejecta. We will hereafter attribute both
contribution to the SN event, since both are return ”promptly” to the ISM. But
can SNe account for the large amount of dust seen in this object? The answer
to this question is complicated by the fact that SNe are also the main source
of grain destruction during the remnant phase of their evolution (Jones et al.
1996; Jones 2004). The problem can therefore only be quantitatively addressed
with CE models for the dust in these systems.

The results of the detailed dust evolution models described summarized in
Figure 8 show that the contribution of AGB stars to metal and dust abundance
can be neglected in galaxies with ages less than about 400 Myr. The equations
for the chemical evolution of the galaxy can then be considerably simplified
using the instantaneous recycling approximation, which assumes that stars
return their ejecta back to the ISM promply after their formation. The
evolution of the dust abundance can then be written in analytical form (Morgan
& Edmunds 2003; Dwek et al. 2007).

In particular, the yield of dust, $Y_{d}$, required to obtain a given dust-to-
gas mass ratio, $Z_{d}$, when the galaxy reaches a given gas mass fraction
$\mu_{g}$, is given by (Dwek et al. 2007):

$Y_{d}=Z_{d}\ \left[{\left<m_{\rm ISM}\right>+R\ m_{\star}\over
1-\mu_{g}^{\nu-1}}\right]$ (3)

where,

$\nu\equiv{\left<m_{\rm ISM}\right>+m_{\star}\over(1-R)\ m_{\star}}\qquad,$
(4)

$R$ is the fraction of the stellar mass that is returned back to the ISM
during the stellar lifetime, $\left<m_{\rm ISM}\right>$ is given by eq. (2),
and $m_{\star}$ is the mass of all stars born per SN event. For example,
$m_{\star}=147$ and $50$ $M_{\odot}$, respectively, for a Salpeter and top-
heavy IMF.

Figure 9 shows how much dust an average SN must produce in order to give rise
to a given dust-to-gas mass ratio, for various grain destruction efficiencies.
The value of $Y_{d}$ was calculated when $\mu_{g}$ reaches a value of 0.60,
the adopted gas mass fraction of J1148+5251 at 400 Myr. Calculations were
performed for two different functional forms of the stellar IMF: a Salpeter
IMF in which $\phi(m)\sim m^{-2.35}$ and $0.1<m$($M_{\odot}$) $<100$; and a
top heavy IMF characterized by the same mass limits but a flatter slope
$\phi(m)\sim m^{-1.50}$. Here, $\phi(m)$ is the number of stars per unit mass
interval, normalized to unity between 0.1 and 100 $M_{\odot}$.

The figure shows that, for example, to produce a value of $Z_{d}=0.0067$ at
$\mu_{g}$=̃ 0.60, a SN must produce about 0.4 (1.2) $M_{\odot}$ of dust for a
top-heavy (Salpeter) IMF, provided the dust is not destroyed in the ISM, that
is, $\left<m_{\rm ISM}\right>$ = 0. Even with modest amount of grain
destruction, $\left<m_{\rm ISM}\right>$ = 100 $M_{\odot}$, the required SN
dust yield is dramatically increased to about $1-2$ $M_{\odot}$, depending on
the IMF. The horizontal line in the figure corresponds to a value of $Y_{d}$ =
0.054 $M_{\odot}$, the largest mass of SN-condensed dust inferred to be
present in a supernova or SNR (Rho et al. 2008; Sugerman et al. 2006).
Contrary to the claim by Rho et al. (2008), this yield is not sufficient to
account for the large amount of dust observed in high redshift galaxies, since
the quoted chemical evolution models of Morgan & Edmunds (2003) do not include
the effect of grain destruction. The figure shows that even without grain
destruction, the largest observed yield can only give rise to a dust-to-gas
mass ratio of $\sim 4\times 10^{-4}$. If the mass of dust in the ejecta of Cas
A represents a typical SN yield, then other processes, such as accretion onto
preexisting grains in molecular clouds is needed to produce the mass of dust
in J1148+5251.

Figure 9.: The IMF-averaged yield of dust by type II supernova,
$\widehat{Y}_{d}$, that is required to account for a given dust-to-gas mass
ratio $Z_{d}$, is presented for different values of $\left<m_{\rm ISM}\right>$
given in units of $M_{\odot}$. Solid and dashed lines correspond to
calculations done for a top-heavy and a Salpeter IMF, respectively. The
horizontal dashed line near the bottom of the figure corresponds a value of
$Y_{d}=0.054$ $M_{\odot}$, the highest inferred yield of dust in supernova
ejecta to date Rho et al. (2008). The vertical dotted line represents the
value of $Z_{d}$ at $\mu_{g}=0.60$. Curves are labeled by $\left<m_{\rm
ISM}\right>$ given in units of $M_{\odot}$. The top two dashed (solid)
horizontal lines represent IMF-averaged theoretical dust yields for a Salpeter
(top-heavy) IMF, assuming 100% condensation efficiency in the SN ejecta.

##  6\. Summary

Dust evolution models have proven to be very successful in predicting global
evolutionary trends in dust abundance and composition, and in analyzing the
origin of dust in the early universe. An important prediction of these models
is that SN- and AGB-condensed dust should follow distinct evolutionary paths
because of the different stellar evolutionary tracks of their progenitor
stars. By analyzing the UV-to-radio SED of 35 nearby galaxies we have
identified silicates and PAHs, respectively, as tracers of SN- and AGB-
condensed dust. Our SED fitting procedure used chemical evolution, dust
evolution, and population synthesis models in a consistent fashion. The models
used the free-free and mid-IR emissions to constrain the gas and dust
radiation from the gas and dust from H II regions, and the far-IR and optical
emission to constrain the ISRF that heats the dust in PDRs. The observed
correlation of the intensity of the mid-IR emission from PAHs with their
metallicity can then be interpreted as the result of stellar evolutionary
effects which cause the delayed injection of carbon dust into the interstellar
medium.

The early universe is a unique environment for studying the role of massive
stars in the formation and destruction of dust. The equations describing their
chemical evolution can be greatly simplified by using the instantaneous
recycling approximation, and by neglecting the delayed contribution of low
mass stars to the metal and dust abundance of the ISM. Neglecting any
accretion of metals onto pre-existing dust in the interstellar medium, the
evolution of the dust is then primarily determined by the condensation
efficiency of refractory elements in the ejecta of Type II supernovae, and the
destruction efficiency of dust by SN blast waves.

We applied our general results to J1148+5251, a dusty, hyperluminous quasar at
redshift $z=6.4$ and found that the formation of a dust mass fraction of
$Z_{d}=0.0067$ in a galaxy with an ISM mass of $3\times 10^{10}$ $M_{\odot}$,
requires an average SN to produce between 0.5 and 1 $M_{\odot}$ of dust if
there was no grain destruction. Such large amount of dust can be produced if
if the condensation efficiency in SNe is about unity. Observationally, the
required dust yield is in excess of the largest amount of dust ($\sim 0.054$
$M_{\odot}$) observed so far to have formed in a SN. This suggests that
accretion in the ISM may play an important role in the growth of dust mass.
For this process to be effective, SNRs must significantly increase, presumably
by non-evaporative grain-grain collisions during the late stages of their
evolution, the number of nucleation centers onto which refractory elements can
condense in molecular clouds.

#### Acknowledgments.

This work was supported by NASA’s LTSA 03-0000-065.

## References

  * Beelen et al. (2006) Beelen, A., Cox, P., Benford, D. J., et al. 2006, ApJ, 642, 694
  * Bertoldi et al. (2003) Bertoldi, F., Carilli, C. L., Cox, P., et al. 2003, A&A, 406, L55
  * Bianchi & Schneider (2007) Bianchi, S. & Schneider, R. 2007, MNRAS, 378, 973
  * Clayton & Nittler (2004) Clayton, D. D. & Nittler, L. R. 2004, ARA&A, 42, 39
  * Dwek (1998) Dwek, E. 1998, ApJ, 501, 643
  * Dwek (2005) Dwek, E. 2005, in American Institute of Physics Conference Series, Vol. 761, The Spectral Energy Distributions of Gas-Rich Galaxies: Confronting Models with Data, ed. C. C. Popescu & R. J. Tuffs, 103–+
  * Dwek et al. (2007) Dwek, E., Galliano, F., & Jones, A. P. 2007, ApJ, 662, 927
  * Dwek & Scalo (1979) Dwek, E. & Scalo, J. M. 1979, ApJ, 233, L81
  * Dwek & Scalo (1980) Dwek, E. & Scalo, J. M. 1980, ApJ, 239, 193
  * Ferrarotti & Gail (2006) Ferrarotti, A. S. & Gail, H.-P. 2006, A&A, 447, 553
  * Field (1974) Field, G. B. 1974, ApJ, 187, 453
  * Galliano et al. (2008) Galliano, F., Dwek, E., & Chanial, P. 2008, ApJ, 672, 214
  * Jones (2004) Jones, A. P. 2004, in ASP Conf. Ser. 309: Astrophysics of Dust, ed. A. N. Witt, G. C. Clayton, & B. T. Draine, 347–+
  * Jones et al. (1996) Jones, A. P., Tielens, A. G. G. M., & Hollenbach, D. J. 1996, ApJ, 469, 740
  * Karakas & Lattanzio (2003) Karakas, A. I. & Lattanzio, J. C. 2003, Publications of the Astronomical Society of Australia, 20, 279
  * Kasimova & Shchekinov (2003) Kasimova, E. R. & Shchekinov, Y. A. 2003, Ap&SS, 284, 433
  * Kennicutt (1998) Kennicutt, Jr., R. C. 1998, ARA&A, 36, 189
  * Kozasa et al. (1989) Kozasa, T., Hasegawa, H., & Nomoto, K. 1989, ApJ, 344, 325
  * Kozasa et al. (1991) Kozasa, T., Hasegawa, H., & Nomoto, K. 1991, A&A, 249, 474
  * Lisenfeld & Ferrara (1998) Lisenfeld, U. & Ferrara, A. 1998, ApJ, 496, 145
  * Maiolino et al. (2006) Maiolino, R., Nagao, T., Marconi, A., et al. 2006, Memorie della Societa Astronomica Italiana, 77, 643
  * Matteucci (2001) Matteucci, F. 2001, The chemical evolution of the Galaxy (The chemical evolution of the Galaxy / by Francesca Matteucci, Astrophysics and space science library, Volume 253, Dordrecht: Kluwer Academic Publishers, ISBN 0-7923-6552-6, 2001, XII + 293 pp.)
  * McKee (1989) McKee, C. 1989, in IAU Symposium, Vol. 135, Interstellar Dust, ed. L. J. Allamandola & A. G. G. M. Tielens, 431–+
  * Morgan & Edmunds (2003) Morgan, H. L. & Edmunds, M. G. 2003, MNRAS, 343, 427
  * Pagel (1997) Pagel, B. E. J. 1997, Nucleosynthesis and Chemical Evolution of Galaxies (Nucleosynthesis and Chemical Evolution of Galaxies, by Bernard E. J. Pagel, pp. 392. ISBN 0521550610. Cambridge, UK: Cambridge University Press, October 1997.)
  * Portinari et al. (1998) Portinari, L., Chiosi, C., & Bressan, A. 1998, A&A, 334, 505
  * Rho et al. (2008) Rho, J., Kozasa, T., Reach, W. T., et al. 2008, ApJ, 673, 271
  * Robson et al. (2004) Robson, I., Priddey, R. S., Isaak, K. G., & McMahon, R. G. 2004, MNRAS, 351, L29
  * Savage & Sembach (1996) Savage, B. D. & Sembach, K. R. 1996, ARA&A, 34, 279
  * Snow (1975) Snow, Jr., T. P. 1975, ApJ, 202, L87
  * Sugerman et al. (2006) Sugerman, B. E. K., Ercolano, B., Barlow, M. J., et al. 2006, Science, 313, 196
  * Walter et al. (2004) Walter, F., Carilli, C., Bertoldi, F., et al. 2004, ApJ, 615, L17
  * Woosley (1988) Woosley, S. E. 1988, ApJ, 330, 218
  * Woosley & Weaver (1995) Woosley, S. E. & Weaver, T. A. 1995, ApJS, 101, 181
  * Zhukovska et al. (2008) Zhukovska, S., Gail, H.-P., & Trieloff, M. 2008, A&A, 479, 453
  * Zinner et al. (2006a) Zinner, E., Nittler, L. R., Alexander, C. M. O. ., & Gallino, R. 2006a, New Astronomy Review, 50, 574
  * Zinner et al. (2006b) Zinner, E., Nittler, L. R., Gallino, R., et al. 2006b, ApJ, 650, 350
  * Zubko et al. (2004) Zubko, V., Dwek, E., & Arendt, R. G. 2004, ApJS, 152, 211