Limit sets of unfolding paths in Outer space

Mladen Bestvina, Radhika Gupta, and Jing Tao

We construct an unfolding path in Outer space which does not
converge in the boundary, and instead it accumulates on the entire
1-simplex of projectivized length measures on a non-geometric  arational $\R$-tree $T$.
We also show that $T$ admits exactly two dual ergodic projective

§ INTRODUCTION

For the once-punctured torus, the Thurston compactification of the space by projective measured laminations coincides with the visual
compactification of the hyperbolic plane. In this case, every geodesic
ray has a unique limit point, and the dynamical behavior of the ray in
moduli space is governed by the continued fraction of its limit point.
For hyperbolic surfaces of higher complexity, space with the
metric is no longer negatively curved [35, 36] (or even
Riemannian), and the Thurston boundary is no longer its visual boundary
[26]. More surprisingly, geodesic rays do not always converge
[31, 32].

For hyperbolic surfaces of higher complexity, another interesting
phenomenon is the existence of nontrivial simplices in the Thurston
boundary which correspond to measures on non-uniquely ergodic laminations.
Particularly interesting is the case when the underlying lamination is
minimal and filling, also called arational. Constructions of
non-uniquely ergodic arational laminations have a long history, and
typically used flat structures on surfaces [43, 41, 30, 25]. A topological construction was
introduced in [20]. In [32], Leininger, Lenzhen and Rafi
combined this topological approach with some arithmetic parameters akin to
continued fractions. This allowed them to show that it is possible for
the full simplex of measures on a non-uniquely ergodic arational
lamination to be realized as the limit set of a geodesic ray.

In this paper, we take the above construction into Culler-Vogtmann's
Outer space [18]. A Thurston-type boundary for Outer
space is given by the set of projective classes of minimal, very small
$\free$-trees [15, 1, 14, 23] and the action of
$\Out(\free)$ extends continuously to the compactified space. The
analogue of arational laminations are arational trees; for
example, trees dual to arational laminations on a once-punctured surface
fall into this category. There are other examples, such as trees dual to
minimal laminations on finite 2-complexes that are not surfaces, called
Levitt type; and yet others, called non-geometric, that do
not come from the latter two constructions. The non-uniquely ergodic
phenomenon for laminations has two natural analogues for $\free$-trees:
one in terms of length measures on trees, giving rise to
non-uniquely ergometric trees [21], and the
other in terms of currents, giving non-uniquely ergodic trees, see
[10]. It is an open problem to determine whether these two
notions coincide. An example of a Levitt type non-uniquely ergometric
arational tree, modeled on Keane's construction, was given in

In Outer space, the analogue of metric is the Lipschitz metric and
that of geodesics are folding paths. However, unlike geodesics, a folding path in Outer space has a forward direction,
reflecting the asymmetry of the Lipschitz metric. Even though the
boundary of Outer space is not a visual boundary, a folding path always
converges along its forward direction. Our main result is that this nice
behavior does not persist in the backward direction; in fact, in the
backward direction, folding paths can behave as badly as geodesics. Define an unfolding path in Outer space to be a folding
path with the backward direction. Our main result, as follows, is a
direct analogue of the results of [32].

There exists an unfolding path in Outer space of free group of rank 7
which does not converge to a point in the boundary of Outer space. In
fact, the limit set is a 1-simplex consisting of the full set of length
measures on a non-geometric and arational tree $T$. Moreover, the set
of projective currents dual to $T$ is also a 1-dimensional simplex. In
particular, $T$ is neither uniquely ergometric nor uniquely ergodic.

We use the framework of folding and unfolding sequences. Every such
sequence tracks the combinatorics of an appropriate folding path, resp.  unfolding path, in Outer space.  An infinite folding sequence has a
naturally associated limiting tree in the boundary of Outer space and an
unfolding sequence has a naturally associated algebraic lamination,
called the legal lamination. The graphs in the folding sequence
can be given compatible metrics, which are then  used to parametrize the
different length measures supported on the limiting tree. Compatible edge
thicknesses on the graphs of the unfolding sequence parametrize the
different currents with support contained in the legal lamination. The
latter can then be used to study the currents dual to the trees in the
limit set of the unfolding sequence. See [37] or our
<Ref> for definitions and more precise statements.

Modeling the construction of [32] on a 5-holed sphere, the folding
and unfolding sequences we consider come from explicit sequences of
automorphisms of the free group of rank 7. More explicitly, fix a
non-geometric fully irreducible automorphism on three letters and extend
it to an automorphism $\phi$ of $\free[7]$ by identity on the other four
basis elements. Also let $\rho$ be a finite order automorphism of
$\free[7]$ that rotates the support of $\phi$ off itself. For an integer
$r$, set $\phi_r = \rho \phi^r$.  Given a sequence $(r_i)_{i \ge 1}$ of
positive integers, define a sequence of automorphisms by
\[\Phi_i =
\phi_{r_1} \circ \cdots \circ \phi_{r_i}.\]
From $(\Phi_i)_i$ we get an
unfolding sequence using the train track map induced by $\phi_{r_i}$, and
from $(\Phi_i^{-1})_i$ we get a companion folding sequence. The
parameters $(r_i)_i$ play the role of the continued fraction expansion
for the limiting tree of the folding sequence, and adjusting them
produces different types of trees and behaviors of the unfolding
sequence. In particular, we show that if the sequence $(r_i)_i$ satisfies
certain arithmetic conditions and grows sufficiently fast, then the
limiting tree is arational, non-geometric, non-uniquely ergodic, and
non-uniquely ergometric. Moreover, the limit set of the unfolding
sequence is the full simplex of length measures on the tree. We refer to
<Ref> for the full technical statement.

To see how the parameters $(r_i)_i$ come into play, it is informative to
look at the sequence of free factors $A_i = \Phi_i(A)$, where $A$ is the
support of $\phi$. The $A_i$'s are the projection of the folding sequence
to the free factor complex $\FreeF[7]$. By our construction, $A_i$ and
$A_{i+1}$ are disjoint (meaning $\free[7]=A_i*A_{i+1}*B_i$ for some
$B_i$), but $A_i, A_{i+2}$ are not, and $r_i$ measures the distance
between the projections of $A_{i-2}$ and $A_{i+2}$ to the free factor
complex of $A_i$. Morally, if $r_i$'s are sufficiently large, then
$(A_i)_i$ forms a quasi-geodesic in $\FreeF[7]$. Hence, by
[8, 22], the limiting tree of the folding sequence is
arational. In addition, we show that the tree is non-geometric.
To get two currents on the tree, we take loops in the $A_i$'s, which
correspond to currents on $\free$ and take projective limits of the odd
and even subsequences. Non-unique ergometricity of the tree follows a
similar principle.

Although our construction is general in spirit, the case of rank 7 is
already fairly involved, and some computations used computer assistance.
One issue is that there is no known algorithm to tell if a collection of
free factors has a common complement. This issue appears in the proof of
arationality of the limiting tree that led to the peculiar looking
arithmetic conditions on the parameters; see <Ref>.

§.§.§ Outline

* In <Ref>, we review some background
material, including train track maps, Outer space, currents, length
measures, and arational trees.

* In <Ref> we discuss folding and unfolding
sequences. We relate length measures on a folding sequence with the
length measures on the limiting tree when it is arational. We also
define the legal lamination for an unfolding sequence and state a
result from [37] relating the currents supported on the legal
lamination with those of the unfolding sequence.

* In <Ref>, we discuss our main
construction to generate from a sequence $(r_i)_i$ of positive
integers a sequence of automorphisms of $\free[7]$. The associated
transition matrices for these automorphisms have block shapes which
we use to analyze their asymptotic behavior. From each sequence of
automorphisms and their inverses, we get a folding and unfolding
sequence of graphs of rank 7 induced by their train track maps.

* In <Ref>, we show that under the right
conditions on $(r_i)_i$, the folding sequence converges to a
non-geometric and arational tree $T$ in boundary of Outer space of
rank 7. To show arationality, we project the folding sequence to the
free factor complex and show it is a quasi-geodesic.

* In <Ref>, we study the behavior of the
unfolding sequence. The main result is that if the sequence $(r_i)_i$
grows sufficiently fast, then the legal lamination of the unfolding
sequence supports a 1-simplex of projective currents.

* In <Ref>, we show that if the sequence $(r_i)_i$
grows sufficiently fast, then the limiting tree of the folding
sequence supports a 1-simplex of projective length measures. In
particular, the limiting tree is not uniquely ergometric.

* In <Ref>, we relate the legal lamination
of the unfolding sequence to the dual lamination of the limiting tree of
the folding sequence. This shows the limiting tree is not uniquely

* In <Ref>, we show that the
unfolding sequence limits onto the full simplex of length measures on
the limiting tree of the folding sequence, and thus does not have a
unique limit in the boundary of Outer space.

* In <Ref>, we collect the results to prove
the main theorem.

* In <Ref>, we prove a technical lemma
about convergence of products of matrices.

§.§.§ Acknowledgments

The authors gratefully acknowledge support: M.B. from NSF DMS-1905720,
R.G. from the Sloan Foundation, and J.T. from NSF DMS-1651963.

§ BACKGROUND

Let $\free$ be the free group of rank $\rk$. We review some background on
train track maps, Outer space, laminations, currents, arational trees and the free factor

§.§ Train track maps

We recall some basic definitions from [7]. Identify
$\free$ with $\pi_1(\mathrm{R}_n, \ast)$ where $\mathrm{R}_n$ is a rose
with $\rk$ petals.  A marked graph $G$ is a graph of rank $\rk$,
all of whose vertices have valence at least three, equipped with a
homotopy equivalence $m\from \mathrm{R}_n \to G$ called a marking.

A length vector on $G$ is a vector $\lambda \in \R^{|EG|}$ that
assigns a positive number, i.e. a length, to every edge of $G$. The
volume of $G$ with respect to $\lambda$ is the total length of all the
edges of $G$. This induces a path metric on $G$ where the length of an
edge $e$ is $\lambda(e)$.

A direction $d$ based at a vertex $v \in G$ is an oriented edge of $G$
with initial vertex $v$. A turn is an unordered pair of distinct
directions based at the same vertex. A train track structure on
$G$ is an equivalence relation on the set of directions at each vertex $v
\in G$. The classes of this relation are called gates.  A turn
$(d,d')$ is legal if $d$ and $d'$ do not belong to the same gate,
it is called illegal otherwise.  A path is legal if it only
crosses legal turns.

A map $f \from  G \to G'$ between two graphs is called a morphism
if it is locally injective on open edges and sends vertices to vertices.
If $G$ and $G'$ are metric graphs, then we can homotope $f$ relative to
vertices such that it is linear on edges. Similarly, for an $\R$-tree
$T$, a map $\tilde G\to T$ from the universal cover of $G$ is a morphism
if it is injective on open edges. To a morphism $f\from  G \to G'$ we
associate the transition matrix as follows: Enumerate the
(unoriented) edges $e_1,e_2,\cdots,e_m$ of $G$ and $e_1', e_2', \cdots,
e_n'$ of $G'$ . Then the transition matrix $M$ has size $n\times m$ and
the $ij$-entry is the number of times $f(e_j)$ crosses $e_i'$, i.e. it
is the cardinality of the set $f^{-1}(x)\cap e_j$ for a point $x$ in the
interior of $e_i'$. If $f$ is in addition a homotopy equivalence, then
$f$ is a change-of-marking.

A homotopy equivalence $f\from  G \to G$ induces an outer automorphism of
$\pi_1(G)$ and hence an element $\phi$ of $\Out(\free)$. If $f$ is a
morphism then we say that $f$ is a topological representative of
$\phi$. A topological representative $f \from  G\to G$ induces a
train track structure on $G$ as follows: The map $f$ determines a
map $Df$ on the directions in $G$ by defining $Df(e)$ to be the first
(oriented) edge in the edge path $f(e)$.  We then declare $e_1\sim e_2$
if $(Df)^k(e_1)=(Df)^k(e_2)$ for some $k\geq 1$.

A topological representative $f\from  G \to G$ is called a train
track map if every vertex has at least two gates, and $f$ maps legal
turns to legal turns and legal paths (equivalently, edges) to legal
paths. Equivalently, every positive power $f^k$ is a topological
representative. If $f$ is a train track map with transition matrix $M$,
then the transition matrix of $f^k$ is $M^k$ for every $k\geq 1$. If $M$
is primitive, i.e. $M^k$ has positive entries for some $k \ge 1$,
then Perron-Frobenius theory implies that there is an assignment of
positive lengths to all the edges of $G$ so that $f$ uniformly expands
lengths of legal paths by some factor $\lambda>1$, called the
stretch factor of $f$.

If $\sigma$ is a path (or a circuit) in $G$, we denote by $[\sigma]$ the
reduced path homotopic to $\sigma$ (rel endpoints if $\sigma$ is a path).
A path or circuit $\sigma$ in $G$ is called a periodic Nielsen
path if $[f^k(\sigma)]=\sigma$ for some $k\geq 1$.  If $k=1$, then
$\sigma$ is a Nielsen path.  A Nielsen path is indivisible,
denoted INP, if it cannot be written as a concatenation of nontrivial
Nielsen paths.

The following lemma is an important property of train track maps. For a
very rudimentary form, see <cit.> showing that
INPs have exactly one illegal turn, and for a more involved version see
[5] (some details can also be found in
<cit.>). We will need it
for the proof of <Ref> and include a proof here.

Let $h\from  G \to G$ be a train track map with a primitive transition
matrix. There exists a constant $R >0$ such that for any edge path
$\gamma$, either,

* the number of illegal turns in $[h^{R}(\gamma)]$ is less than
that of $\gamma$, or,

* $\gamma = u_1 v_1 u_2 v_2 \ldots u_n$ where each $u_i$
is a legal subpath, possibly degenerate, and each
$[h^{R}(v_i)]$ is a periodic INP.

Let $\lambda > 1$ be the stretch factor of $h$, and equip $G$ with the
metric so that $h$ uniformly expands the length of every legal path by
$\lambda$. It goes back to the work of Thurston (see [16])
that there is a constant $BCC(h)$, called the bounded cancellation
constant for $h$, such that if $\alpha\beta$ is a reduced edge path
then $[h(\alpha)][h(\beta)]$ have cancellation bounded by $BCC(h)$. The
existence of this constant is really a consequence of the Morse lemma
and the fact that $h$ is a quasi-isometry. Define $C = BCC(h) /

Here is the significance of $C$. To fix ideas let us assume that
$\gamma$ has only one illegal turn, so $\gamma=\alpha\beta$ with both
$\alpha,\beta$ legal. Say $\alpha$ has length $|\alpha|=C+\epsilon>C$.
Then $h(\alpha)$ has length $\lambda |\alpha|$ and after cancellation
with $h(\beta)$ the length is $\geq \lambda |\alpha|
-BCC(h)=|\alpha|+\lambda\epsilon$. Thus assuming $[h^i(\gamma)]$ still
has an illegal turn, the length of the initial subpath to the illegal
turn has length growing exponentially in $i$, assuming it is long

We now prove the lemma for paths $\gamma=\alpha\beta$ with one illegal
turn, and with $\alpha,\beta$ legal. Consider the finite collection of
paths consisting of those with length at most $C$ with both endpoints
at vertices, or with length exactly $C$ with only one endpoint at a
vertex. Let $R$ be a number larger than the square of the size of this
collection. If $[h^i(\gamma)]=\alpha_i\beta_i$ has one illegal turn
(with $\alpha_i,\beta_i$ legal) for $i=1,2,\cdots,R$, then by the
pigeon-hole principle there will be $i<j$ in this range so that the
$C$-neighborhoods of the illegal turns of $[h^i(\gamma)]$ and
$[h^j(\gamma)]$ are the same (if $\alpha_i$ or $\beta_i$ has length
$<C$ this means $\alpha_i=\alpha_j$ or $\beta_i=\beta_j$). We can lift
$h^{j-i}$ and $\gamma$ to the universal cover of the graph and arrange
that (the lift of) $\gamma$ and $[h^{j-i}(\gamma)]$ have the same
illegal turn.  Thus $h^{j-i}$ maps
the terminal $C$-segment of $\alpha_i$ (or $\alpha_i$ itself) over
itself (by the above calculation) and therefore fixes a point in
$\alpha_i$, and similarly for $\beta_i$. The subpath of $[h^i(\gamma)]$
between these fixed points is a periodic INP, proving the lemma in the
case $\gamma$ has one illegal turn.

The general case is similar. Write
$\gamma=\gamma_1\gamma_2\cdots\gamma_s$ with all $\gamma_k$ legal and
with the turn between $\gamma_k$ and $\gamma_{k+1}$ illegal. Also
assume that $[h^i(\gamma)]$ has the same number of illegal turns for
$i=1,\cdots,R$. We can write
$[h^i(\gamma)]=\gamma_1^i\gamma_2^i\cdots\gamma_s^i$ with all
$\gamma_k^i$ legal and the turns between them illegal. For each illegal
turn corresponding to the pair $(k,k+1)$ there will be $i<j$ in this
range so that the $C$-neighborhoods of the illegal turn in
$[h^i(\gamma)]$ and in $[h^j(\gamma)]$ are the same. This gives fixed
points of $h^{j-i}$ in $\gamma_k^i$ and $\gamma_{k+1}^i$ and these
fixed points split $\gamma$ into periodic INPs and legal segments, as

We will use the lemma in the situation that $h$ has no periodic INPs, in
which case the conclusion is that whenever $\gamma$ is not legal, then
$[h^{R}(\gamma)]$ has fewer illegal turns than $\gamma$.

§.§ Outer space and its boundary

An $\free$-tree is an $\mathbb{R}$-tree with an isometric action of
$\free$. An $\free$-tree $T$ has dense orbits if some (every) orbit is
dense in $T$. An $\free$-tree is called very small if the action
is minimal, arc stabilizers are either trivial or maximal cyclic, and
tripod stabilizers are trivial. We review the definition of Outer space
first introduced in [18].

Unprojectivized Outer space, denoted by $\cv$, is the set of free,
minimal, and simplicial $\free$-trees. By considering the quotient
graphs, $\cv$ is also equivalently the set of marked metric graphs, i.e.  the set of triples $(G,m,\lambda)$, where $G$ is a graph of rank $n$ with
all valences at least $3$, $m\from \mathrm{R}_n \to G$ is a marking, and
$\lambda$ is a positive length vector on $G$. By
[15], the map of $\cv \to \R^{\free}$ given by $T
\mapsto (\norm{g}_T)_{g \in \free}$, where $\norm{g}_T$ is the
translation length of $g$ in $T$, is an inclusion. This endows $\cv$ with
a topology. The closure $\cvclo$ in $\R^{\free}$ is the space of very
small $\free$-trees [1, 14]. The
boundary $\partial \cv = \cvclo-\cv$ consists of very small trees that
are either not free or not simplicial.

Culler Vogtmann's Outer space, $\CV$, is the image of $\cv$ in the
projective space $\P\R^{\free}$. Elements in $\CV$ can also be described
as free, minimal, simplicial $\free$-trees with unit covolume.
Topologically, $\CV$ is a complex made up of simplices with missing
faces, where there is an open simplex for each marked graph $(G,m)$
spanned by positive length vectors on $G$ of unit volume. The closure
$\CVclo$ of $\CV$ in $\P\R^{\free}$ is compact and the boundary $\partial
\CV = \CVclo-\CV$ is the projectivization of $\partial \cvclo$.

The spaces $\cv$ and $\CV$ and their closures are equipped with a natural
(right) action by $\Out(\free)$. That is, for $\Phi \in \Out(\free)$ and $T
\in \cvclo$ the translation length function of $T \Phi$ on $\free$
is $\norm{g}_{T \Phi} = \norm{\phi(g)}_T$, where $\phi$ is any lift of
$\Phi$ to $\Aut(\free)$.

§.§ Laminations, currents and non-uniquely ergodic trees

In [6], Bestvina, Feighn and Handel defined a dynamic
invariant called the attracting lamination associated to a train track
map. In this article, we will consider the more modern definition of a
lamination as given in [11].

Let $\partial \free$ denote the Gromov boundary of $\free$ and let
$\Delta$ be the diagonal in $\partial \free \times \partial \free$. The
double boundary of $\free$ is $\partial^2 \free = (\partial \free
\times \partial \free - \Delta) / \mathbb{Z}_2$, which parametrizes the
space of unoriented bi-infinite geodesics in a Cayley graph of $\free$.
By an (algebraic) lamination, we mean a non-empty, closed and
$\free$-invariant subset of $\partial^2 \free$.

Associated to $T\in \cvclo$ is a dual lamination $L(T)$,
defined as follows in [12]. For $\epsilon > 0$, let

\[
L_{\epsilon}(T) = \overline{\{(g^{-\infty}, g^{\infty}) \quad | \quad
\norm{g}_T < \epsilon, g \in \free \}},
\]

so $L_{\epsilon}(T)$ is a lamination and set $L(T) =
\bigcap_{\epsilon>0}L_{\epsilon}(T)$. Elements of $L(T)$ are called
leaves. For trees in $\cv$, $L(T)$ is empty.

A current is an additive, non-negative, $\free$-invariant function
on the set of compact open sets in $\partial^2 \free$. Equivalently, it
is an $\free$-invariant Radon measure on the $\sigma$-algebra of Borel
sets of $\partial^2 \free$. Let $\Curr$ denote the space of currents,
equipped with the weak* topology. The quotient space of $\PCurr$ of
projectivized currents (i.e. homothety classes of nonzero currents) is

For $\mu \in \Curr$, let $\supp(\mu) \subset \partial^2  \free$ denote
the support of $\mu$, which is in fact a lamination.  For $T \in \cvclo$
and $\mu \in \Curr$, if $\supp(\mu) \subseteq L(T)$, then we say $\mu$ is
dual to $T$. Denote by $\C(T)$ the convex cone of currents dual to
$T$ and by $\P\C(T)$ the set of projective currents dual to $T$.
$\P\C(T)$ is a compact, convex space and its extremal points are called
the ergodic currents dual to $T$. We say $T$ is uniquely
ergodic if there is only one projective class of currents dual to $T$,
and non-uniquely ergodic otherwise. In [9],
the authors show that if $T \in \partial \cv$ has dense orbits, then
$\P\C(T)$ is the convex hull of at most $3\rk-5$ projective classes of
ergodic currents dual to $T$.

In [27], Kapovich and Lustig established a
length pairing, $\la \cdot, \cdot \ra$, between $\cvclo$ and the space
of measured currents $\Curr$. They also showed in <cit.> that for $T \in \cvclo$ and $\mu \in \Curr$, $\la
T, \mu \ra = 0$ if and only if $\mu$ is dual to $T$.

Given two trees $T$ and $T'$, we say a map $h\from  T \to T'$ is
alignment-preserving if whenever $b \in T$ is contained in an arc
$[a,c] \subset T$, then $h(b)$ is contained in the arc $[h(a),h(c)]$.

Let $T, T' \in \partial \CV$ be two trees with dense orbits. The
following are equivalent:

* $L(T) = L(T')$.

* There exists an $\free$-equivariant alignment-preserving
bijection between $T$ and $T'$.

§.§ Length measures and non-uniquely ergometric trees

Since $\R$-trees need not be locally compact, classical measure theory is
not well-suited for them. In [38], a length
measure was introduced for $\R$-trees. See [21] for

A length measure on an $\free$-tree $T$ is a collection of finite
Borel measures $\lambda_I$ for every compact interval $I$ in $T$ such
that if $J \subset I$, then $\lambda_J = (\lambda_I)|_J$. We require the
length measure to be invariant under the $\free$ action. The collection
of the Lebesgue measures of the intervals of $T$ is $\free$-invariant,
and this will be called the Lebesgue measure of $T$. A length measure
$\lambda$ is non-atomic or positive if every $\lambda_I$ is
non-atomic or positive. If every orbit is dense in some segment of $T$,
then $T$ cannot have an invariant measure with atoms. Further, if $T$ is
indecomposable, that is, if for any pair of nondegenerate arcs $I$
and $J$ in $T$, there exist $g_1,\ldots,g_m \in \free$, such that $I
\subset \bigcup g_iJ$ and $g_i J \cap g_{i+1} J$ is nondegenerate, then
every nonzero length measure is positive (in fact, the condition of
mixing [21] suffices).

Let $\D(T)$ be the cone of $\free$-invariant length measures on $T$, with
projectivization $\P\D(T)$, i.e. the homothety classes of
$\free$-invariant length measures on $T$. $\P\D(T)$ is a compact convex
set and we will call its extremal points the ergodic length
measures on $T$. When $T$ has dense orbits there are at most $3n-4$ such
measures for any $T$, see <cit.> and $\D(T)$ is naturally a subset of $\partial
cv_n$. In fact,

If $T \in \cv$ is indecomposable then $\mathcal{D}(T)$ is in one-to-one
correspondence with the set of isometry classes of $\free$-invariant
metrics on $T$, denoted $X_T \subset \cv$.

Let $\lambda \in \mathcal{D}(T)$ be a length measure on $T$. Consider
the pseudo-metric $d_{\lambda}$ on $T$ where $d_{\lambda}(x,y) =
\lambda([x,y])$ for $x, y \in T$. In fact, since $T$ is indecomposable,
$d_{\lambda}$ is a metric on $T$.
For the converse, let $T' \in X_T$. Then the pull back of Lebesgue
measure on $T'$ under identity map $\id \from T \to T'$ gives a
positive length measure on $T$.

We say $T$ is uniquely ergometric if there is only one
projective class of length measures on $T$, which necessarily is the
homothety class of the Lebesgue measure on $T$. It is called
non-uniquely ergometric otherwise.

§.§ Arational trees and the free factor complex

For a tree $T \in \cvclo$ and a free factor $H$ of $\free$, let $T_H$
denote the minimal $H$-invariant subtree of $T$ (this tree is unique
unless $H$ fixes an arc). A tree $T \in \partial \cv$ is arational
if every proper free factor $H$ of $\free$ has a free and simplicial
action on $T_H$.
By [39] every arational tree is free
and indecomposable or it is the dual tree to an arational measured
lamination on a surface with one puncture. The arational trees of the
first kind are either Levitt type or non-geometric.

Let $\AT \subset \partial \CV$ denote the set of arational trees with the
subspace topology. Using <Ref>, define an equivalence relation
$\sim$ on $\AT$ by “forgetting the metric”, that is, $T \sim T'$ if $T'
\in \P \D(T)$, and endow $\AT/\sim$ with the quotient topology. The
following lemma is implicit in [21] and we include a
proof for completeness.

Let $T, T'$ be arational trees. Then $T \sim T'$ if and only if $L(T) =

If $T \sim T'$ then the identity map $\id \from T \to T'$ is an alignment preserving
bijection. Therefore, by <Ref>, $L(T) =

If $L(T) = L(T')$, then by <Ref> there is an alignment
preserving bijection $f \from T \to T'$. Pulling back the Lebesgue
measure on $T'$ induces a length measure on $T$, and the corresponding
metric $d_\mu$ on $T$ is isometric to $T'$, so $T' \sim T$.

The free factor complex $\FreeF$ is a simplicial complex whose
vertices are given by conjugacy classes of proper free factors of $\free$
and a $k$-simplex is given by a nested chain $[A_0] \subset [A_1] \subset
\cdots \subset [A_k]$. When the rank $n=2$ the definition is modified and
an edge connects two conjugacy classes of rank 1 factors if they have
complementary representatives. The free factor complex can be given a
metric as follows: identify each simplex with a standard simplex and
endow the resulting space with path metric. By result of
[3], the metric space $\FreeF$ is Gromov
hyperbolic. The Gromov boundary of $\FreeF$ was identified with
$\AT/\sim$ in [8] and [22].

There is a projection map $\pi\from \CV \to \FreeF$ defined as follows
<cit.>: for $G \in \CV$, $\pi(G)$ is the
collection of free factors given by the fundamental group of proper
subgraphs of $G$ which are not forests. This map is coarsely well
defined, that is, $\diam_{\FreeF}(\pi(G)) \le K$ for some universal $K$.
Note that if $G, G'$ belong to the same open simplex of $\CV$, then
$\pi(G)=\pi(G')$, so the projection of a simplex of $\CV$ has uniformly
bounded diameter.

§ FOLDING AND UNFOLDING SEQUENCES

In this section we introduce (un)folding sequences and review some
work of Namazi-Pettet-Reynolds [37].

A folding/unfolding sequence is a sequence

\[
\begin{tikzcd}
& G_a \arrow[r]
& \cdots \arrow{r}
& G_{-1} \arrow{r}
& G_0 \arrow{r}
& G_1 \arrow{r}
& \cdots \arrow[r]
& G_b
\end{tikzcd}
\]

of graphs, together with maps $f_i\from  G_i \to G_{i+1}$ such that for
any $j \leq i$, $f_{i-1} \circ f_{i-2} \circ \cdots \circ f_j\from  G_j
\to G_i$ is a change-of-marking morphism. Equivalently, a sequence as
above is called a folding/unfolding sequence, if there exists a train
track structure on each $G_i$ and $f_{i-1} \circ f_{i-2} \circ \cdots
\circ f_j$ maps legal paths to legal paths. We allow the sequence to be
infinite in one or both directions. We assume that a marking on $G_0$ has
been specified, so a folding/unfolding sequence determines a sequence of
open simplices in Outer space.

Let $Q_i$ be the transition matrix of $f_{i}$. A length measure
for a folding/unfolding sequence $(G_i)_{a \leq i \leq b}$ is a sequence
$(\lambda_i)_{a \leq i \leq b}$, where $\lambda_i \in \R^{|EG_i|}$ is a
length vector on $G_i$, and for $a \leq i < b$,

\[ \lambda_i  = Q_i^T\lambda_{i+1}.\]

In this way, $f_i$ restricts to an local isometry on every edge of $G_i$.
When $b < \infty$, a length vector on $G_b$ determines a length measure
on the sequence. When the sequence is infinite in the forward direction
we denote by $\D((G_i)_i)$ the space of length measures on $(G_i)_i$, and
$\P\D((G_i)_i)$ its projectivization. Observe that the dimension of
$\D((G_i)_i)$ is bounded by $\liminf_{i \to \infty}|EG_i|$.

A current for a folding/unfolding sequence $(G_i)_{a \leq i \leq
b}$ is a sequence $(\mu_i)_{a \leq i \leq b}$, where $\mu_i \in
\R^{|EG_i|}$ is a length vector on $G_i$ (but thought of as a vector of
thicknesses of edges), and for $a \leq i < b$, we require
$$\mu_{i+1}  =
Q_i \mu_i.$$
Likewise, when the sequence is infinite in the backward
direction, we denote by $\C((G_i)_i)$ the space of currents on $(G_i)_i$,
and $\P\C((G_i)_i)$ its projectivization. The dimension of $\C((G_i)_i)$
is bounded by $\liminf_{i \to - \infty}|EG_i|$.

§.§ Isomorphism between length measures

In this section, we identify the space of length measures on a folding
sequence with that of the limiting tree, when it is an arational tree.

Consider a folding sequence of marked graphs of rank $n$

\[
\begin{tikzcd}
G_0 \arrow{r}{f_1}
& G_1 \arrow{r}
& \cdots \cdots \arrow{r}{f_i}
& G_i \arrow{r}{f_{i+1}}
& \cdots \cdots.
\end{tikzcd}
\]

Let $\tilde G_i$ be the universal cover of $G_i$, and let $\tilde f_i$ be
a lift of $f_i$. For any positive length measure $(\lambda_i)_i \in
\D((G_i)_i)$, we can realize $(\tilde G_i, \tilde \lambda_i)_i$ as a
sequence in $\cv$, which can be “filled in” by a folding path in $\cv$
(see [3] for details on folding paths). In
particular, $(\tilde G_i,\tilde \lambda_i)_i$ always converges to a point
$T \in \partial \cv$. Furthermore, we have morphisms $h_i \from  \tilde
G_i \to T$ such that $h_i = h_{i+1} \tilde f_{i+1}$. With respect to the
length measure $\tilde \lambda_i$, $\tilde f_i$ and $h_i$ restrict to
isometries on edges <cit.>.

Let $(U_i)_i$ be the sequence of open simplices $\CV$ associated to the
sequence $(G_i)_i$.  Recall the projection map $\pi \from  \CV \to
\FreeF$ is coarsely well-defined on simplices of $\CV$. We will say the
folding sequence $(G_i)_i$ converges to an arational tree $T$ if
$\pi(U_i)$ converges to $[T] \in \partial \FreeF$.

Suppose a folding sequence $(G_i)_i$ converges to an arational tree
$T$. Then there is a linear isomorphism between $\D((G_i)_i)$ and

Fix a positive length measure $(\lambda_i)_i \in \D((G_i)_i)$ and let
$T \in \partial \cv$ be the limiting tree of $(\tilde
G_i,\tilde\lambda_i)$ with corresponding morphism $h_i \from  \tilde
G_i \to T$. Recall from <Ref> that if $T$ is
arational, then we can identify $\D(T)$ with the subspace of
$\free$-metrics on $T$ in $\partial \cv$. We will let $\lambda \in
\D(T)$ be a length measure, and $T_\lambda$ its image in $\partial
\cv$.

By [8], if $\pi(U_i)$ converges to $[T''] \in \partial
\FreeF$, then for any positive $(\lambda_i')_i \in \D((G_i)_i)$,
$(\tilde{G}_i,\tilde \lambda_i')$ also converges to an arational tree
$T' \in \partial \cv$, such that $[T''] = [T']=[T]$; in other words,
$T' = T_{\lambda'}$ for some $\lambda' \in D(T)$. This gives a linear
map $\D((G_i)_i) \to \D(T)$.

Conversely, for any positive length measure $\lambda' \in \D(T)$, we can
use the morphism $h_i$ to pull back $\lambda'$ from $T$ to a length
measure $\lambda_i'$ on $\tilde{G}_i$. The fact that $h_i = h_{i+1}
\tilde f_{i+1}$ implies $(\lambda_i')_i \in \D((G_i)_i)$. Moreover, the
sequence $(\tilde{G}_i,\tilde \lambda_i')_i$ converges to $T_{\lambda'}
\in \cv$. This gives a linear map $\D(T) \to \D((G_i)_i)$ which is the
inverse of $\D((G_i)_i) \to \D(T)$ defined above. This shows
$\D((G_i)_i) \to \D(T)$ is an isomorphism.

A more general statement of <Ref> which doesn't
involve the assumption that $T$ is arational can be found in
<cit.>, but we will not need such a general
statement here.

§.§ Isomorphism between currents

In this section, we state an analogous result identifying the space of
currents on an unfolding sequence with the space of currents of a legal
lamination associated to a unfolding sequence. We record some definitions
from [37] first.

Consider an unfolding sequence of marked graphs of rank $n$

\[
\begin{tikzcd}
& \cdots \cdots
\arrow{r}{f_{i+1}}
& G_i \arrow{r}{f_i}
& \cdots \cdots \arrow{r}{f_2}
& G_1 \arrow{r}\arrow{r}{f_1}
& G_0.
\end{tikzcd}
\]

Denote the composition $F_i = f_1 \circ \cdots \circ f_i$. Let
$\Omega^L_{\infty}(G_i)$ denote the set of bi-infinite legal paths in
$G_i$. Define the legal lamination of the unfolding sequence
$(G_i)_i$ to be

\[\Lambda = \bigcap_{i} F_i(\Omega^L_{\infty}(G_i)) \subseteq
\Omega_{\infty}^L(G_0).\]
Use the marking on $G_0$ to identify
$\partial^2 \pi_1(G_0)$ with $\partial^2 \free$. The preimage, in
$\partial^2 \free$, of the lift of $\Lambda$ to $\partial^2 \pi_1(G_0)$
is a lamination $\tilde{\Lambda}$. We denote by $\C(\Lambda)$ the convex
cone of currents supported on $\tilde{\Lambda}$, with projectivization

An invariant sequence of subgraphs is a sequence of nondegenerate
(i.e. not forests) proper subgraphs $H_i \subset G_i$ such that $f_i$
restricts to a morphism $H_i \to H_{i-1}$. We will need the following
theorem from [37].

Given an unfolding sequence $(G_i)_{i \geq 0}$ without an invariant
sequence of subgraphs and with legal lamination $\Lambda$, then there
is a natural linear isomorphism between $\C((G_i)_i)$ and $\C(\Lambda)$.

§ MAIN SETUP

In this section, we will construct an unfolding sequence $(\tau_i)_i$ and
a folding sequence $(\tau_i')_i$ in $\CV[7]$ that intersect the same
infinite set of simplices, which we will eventually use to show the
existence of a non-uniquely ergodic and ergometric tree. The construction
is done via a family of outer automorphisms. We will describe these
automorphisms and then analyze the asymptotic behavior of their train
track maps.

§.§ The automorphisms

Let $\free[7]=\la a,b,c,d,e,f,g\ra$. Denote by $\bar{x}$ the inverse of
$x \in \free[7]$. First consider the map induced on the 3-petaled rose by
the automorphism
\[\theta \from  a \mapsto b, b \mapsto c, c \mapsto ca
\in \Aut(\mathbb{F}_3)\]
and the map induced by the inverse automorphism

\[\vartheta \from  a \mapsto \bar{b}c, b \mapsto a, c \mapsto b.\]
$\theta$ and $\vartheta$ to also denote the corresponding graph
maps, and using the convention that $a$ also denotes the initial
direction of the oriented edge $a$, while $\bar{a}$ denotes the
terminal direction, we

\[
D\theta^3 \from
\begin{tikzcd}
a\arrow[r] & c \arrow[out=-120,in=-60,loop] & b\arrow[l] & \bar{a}
\arrow[loop right] & \bar{b} \arrow[loop right]  & \bar{c}
\arrow[loop right] \end{tikzcd}
\]

\[
D\vartheta^3 \from
\begin{tikzcd}
a\arrow[r] & \bar{c} \arrow[loop right] & b\arrow[r] & \bar{a} \arrow[loop right] & c
\arrow[r]  & \bar{b} \arrow[loop right]
\end{tikzcd}
\]

From the structure of the above maps, for $n\equiv 0$ mod 3, $D\theta^n
= D\theta^3$ and $D\vartheta^n = D\vartheta^3$.

The map on the 3-petaled rose labeled $a, b, c$ induced by $\vartheta$
is a train track map with respect to the train track structure with
gates $\{a, \bar{c}\}, \{b, \bar{a}\}, \{c,\bar{b}\}$. Moreover, this
train track map does not have any periodic INPs.

The map on the 3-petaled rose labeled $a, b, c$ induced by $\theta$
is also a train track map with respect to the train track structure with
gates $\{a, b, c\}, \{\bar{a}\}, \{\bar{b}\}, \{\bar{c}\}$ and it has
one periodic Nielsen path (see <cit.>).

The train track structure on the rose induces a metric on the graph
coming from Perron-Frobenius theory. Every INP has length at most twice
the volume of the graph, one illegal turn, and the endpoints are fixed.
Since there are only finitely many fixed points in $G$, it is easy to
enumerate all such paths and check if they are Nielsen. For periodic
INPs one knows that the period is bounded by a function of the rank of
$\free$ [19], so one can take a suitable power and
check for INPs (though there are more efficient ways, see [24]).
Coulbois' train track package [17] for the mathematics
software system Sage [40] computes periodic INPs of train
track maps.

Now let $\phi\in \Aut(\free[7])$ be the automorphism:
b,b\mapsto c,c\mapsto ca,d\mapsto d,e\mapsto e,f\mapsto f,g\mapsto g$$

and $\rho\in \Aut(\mathbb{F}_7)$ be the rotation by 4 clicks:
e, b\mapsto f, c\mapsto g, d\mapsto a, e\mapsto b, f\mapsto c, g\mapsto
Thus $\phi$ is the extension of $\theta$ by identity, and $\rho$
rotates the support of $\phi$ off itself.

For any $r \ge 3$, the map on the $7$-petaled rose induced by $\phi_r
= \rho \phi^r$ is a train track map with respect to the train track
structure with gates
and 8 more gates
consisting of single half edges. The transition matrix $M_r$ has block

\[
\begin{pmatrix}
\end{pmatrix}
\]

where $I$ is the $4\times 4$ identity matrix, and $B$ is the transition
matrix of $\theta$:

\[
\begin{pmatrix}
\end{pmatrix}
\]

By <Ref>, we only have to check the lemma for $\phi_3,
\phi_4, \phi_5$, which can be done by hand or using the train
track package for Sage.

For any $r \ge 3$ and $r \equiv 0$ mod 3, the map on the $7$-petaled rose induced by
$\psi_r = (\rho\phi^r)^{-1}$ is a train track map with respect to the train
track structure with gates

$$\{a,e,\bar{g}\},\{b, \bar{d}\},\{c, \bar{b}\},\{d,
\bar{c}\},\{f,\bar{e}\},\{g,\bar{f}\},\{ \bar{a}\}$$

The transition matrix $N_r$ has block form

\[
\begin{pmatrix}
\end{pmatrix}
\]

where $I$ is the $4\times 4$ identity matrix, and $C$ is the transition
matrix of $\vartheta$:

\[
\begin{pmatrix}
\end{pmatrix}
\]

By <Ref>, we only have to check the lemma for $\psi_3$, which can
be done by hand or using the train track package for Sage.

§.§ Asymptotics of transition matrices

Let $\theta$, $\vartheta$, $\phi_r$, $\psi_r$ be the maps defined in the
last section. We now analyze the behavior of the transition matrices
$M_r$ and $N_r$ for $\phi_r$ and $\psi_r$ respectively.

Let $B$ be the transition matrix for $\theta$, with Perron-Frobenius
eigenvalue $\lambda_B$. There exists a constant $\kappa_B>0$ such
that if $r,s-r\to\infty$, then
$$\frac 1{\kappa_B \lambda_B^s}
M_r M_s\to Y$$
where $Y$ is an idempotent matrix of the form

\[Y = \begin{pmatrix}
u & p u & q u & 0 & 0 & 0 & 0
\end{pmatrix}
\text{ with }
u =
\begin{pmatrix}
0, u_1, u_2, u_3, 0, 0, 0 \end{pmatrix}^T
\text{ and }
p, q > 0,\]

$\begin{pmatrix} u_1,u_2,u_3\end{pmatrix}^T$
is a Perron-Frobenius eigenvector of $B$.

There exists a Perron-Frobenius eigenvector $x=(x_1,x_2,x_3)^T$ for $B$
and constants $p, q > 0$ such that

\[
P = \lim_{s \to \infty} \frac{B^s}{\lambda_B^s} =
\begin{pmatrix} x & px & qx \end{pmatrix}.
\]

We have

\[
\begin{pmatrix}
\begin{matrix}
0 & 0 & 0
\end{matrix}
& \rvline &
\begin{matrix}
0 & 0 & 0
\end{matrix}
& \rvline & 1 \\
\hline
& \rvline & \bigzero & \rvline &
\begin{matrix}
0 \\ 0 \\ 0
\end{matrix}
\\
\hline
\bigzero & \rvline & B^r & \rvline &
\begin{matrix}
0 \\ 0 \\ 0
\end{matrix}
\end{pmatrix}
\quad
\Longrightarrow
\quad
\frac{1}{\lambda_B^s} M_rM_s
\to
\begin{pmatrix}
\begin{matrix}
0 & 0 & 0
\end{matrix}
& \rvline &
\begin{matrix}
0 & 0 & 0
\end{matrix}
& \rvline & 0 \\
\hline
& \rvline & \bigzero & \rvline &
\begin{matrix}
0 \\ 0 \\ 0
\end{matrix}
\\
\hline
\bigzero & \rvline & \bigzero & \rvline &
\begin{matrix}
0 \\ 0 \\ 0
\end{matrix}
\end{pmatrix}
\]

The square of the limiting matrix above has a nonzero block where $P$
is of the form
and zero elsewhere, so we set

\[ \kappa_B=px_1+qx_2 \qquad \text{and} \qquad (u_1,u_2,u_3)^T =
\frac{1}{\kappa_B} (x_1,x_2,x_3)^T. \qedhere \]

We have a similar statement for the matrices $N_r$.

Let $C$ be the transition matrix for $\vartheta = \theta^{-1}$, with
Perron-Frobenius eigenvalue $\lambda_C$. There exists a constant
$\kappa_C>0$ such that if $r, s-r\to\infty$, then

$$\frac 1{\kappa_C \lambda_C^s} N_s N_r \to Z,$$
where $Z$ is an idempotent
matrix of the form

\[
Z =
\begin{pmatrix}
0 & v & p v & q v & 0 & 0 & 0
\end{pmatrix}
\text{ with }
v =
\begin{pmatrix}
v_1 , v_2 , v_3 , 0 , 0 , 0 , 0
\end{pmatrix}^T
\text{ and }
p, q > 0,
\]

and $(v_1,v_2,v_3)^T$ is a Perron-Frobenius eigenvector of $C$.

We observe that the matrix $N_s N_r$ has shape that is the transpose of
the matrix in <Ref>, with powers of the PF matrix $C$
forming the nonzero blocks:

\[
\begin{pmatrix}
\begin{matrix}
\end{matrix}
& \rvline & C^s & \rvline & \bigzero \\
\hline
\begin{matrix}
\end{matrix}
& \rvline & \bigzero & \rvline & C^r \\
\hline
1 & \rvline &
\begin{matrix}
0 & 0 & 0
\end{matrix}
& \rvline &
\begin{matrix}
0 & 0 & 0
\end{matrix}
\end{pmatrix}
\qedhere
\]

For future reference, we also record the following.
Let $P = \lim_{r
\to \infty} B^r/\lambda_B^r$ and $Q = \lim_{r \to \infty}
C^r/\lambda_C^r$. Set

\[
M_\infty =
\lim_{r \to \infty} \frac{M_r}{\lambda^r_B}
\begin{matrix}
\end{matrix}
& \rvline & 0\\
\hline
P & \rvline & 0
\end{pmatrix}
\qquad \text{and} \qquad
N_\infty =
\lim_{r \to \infty} \frac{N_r}{\lambda^r_C}
\begin{matrix}
\end{matrix}
& \rvline & Q\\
\hline
0 & \rvline & 0
\end{pmatrix}
\]

There are $p,q,r,s > 0$ such that

\[
M_\infty Y = \begin{pmatrix}
y & py & qy & 0 & 0 & 0 & 0
\end{pmatrix} \text{ with }
y =
\begin{pmatrix}
0, 0, 0, 0, y_1, y_2, y_3
\end{pmatrix}^T,
\]

$(y_1,y_2,y_3)^T$ is a Perron-Frobenius eigenvector of $B$; and

\[
ZN_\infty = \begin{pmatrix}
0 & 0 & 0 & 0 & z & rz & sz
\end{pmatrix}
\text{ with }
z =
\begin{pmatrix}
z_1, z_2, z_3,  0 , 0 , 0 , 0
\end{pmatrix}^T,
\]

and  $(z_1,z_2,z_3)^T$ is a Perron-Frobeninus eigenvector of $C$.

§.§ Folding and unfolding sequence

Consider a sequence of positive integers $(r_i)_{i \ge 1}$ and the
sequence of automorphisms $\phi_{r_i}$, with transition matrix $M_{r_i}$
and $\phi_{r_i}^{-1} = \psi_{r_i}$ with transition matrix $N_{r_i}$. Let
$\tau_i \to \tau_{i-1}$ (resp. $\tau_{i-1}' \to \tau_i'$) be the train
track map induced on the rose by $\phi_{r_i}$ (resp. $\psi_{r_i}$) as
given by <Ref> (resp. <Ref>). Thus we have an unfolding sequence

\[
\begin{tikzcd}
\cdots
\arrow[r]
& \tau_{i+1} \arrow[r, "\phi_{r_{i+1}}"]
& \tau_i \arrow[r, "\phi_{r_i}"]
& \tau_{r_{i-1}} \arrow[r]
& \cdots \arrow[r, "\phi_{r_3}"]
& \tau_2 \arrow[r, "\phi_{r_2}"]
& \tau_1 \arrow[r, "\phi_{r_1}"]
& \tau_0,
\end{tikzcd}
\]

and a folding sequence

\[
\begin{tikzcd}
\cdots
& \tau_{i+1}' \arrow[l]
& \tau_i' \arrow{l}[above,pos=.4]{\psi_{r_{i+1}}}
& \tau_{i-1}' \arrow{l}[above,pos=.4]{\psi_{r_i}}
& \cdots \arrow[l]
& \tau_2' \arrow{l}[above,pos=.4]{\psi_{r_3}}
& \tau_1' \arrow{l}[above,pos=.4]{\psi_{r_2}}
& \tau_0' \arrow{l}[above,pos=.4]{\psi_{r_1}}.
\end{tikzcd}
\]

Let $\Phi_i = \phi_{r_1} \circ \ldots \circ \phi_{r_i}$ and $\Phi_i^{-1}
= \Psi_i = \psi_{r_i} \circ \ldots \circ \psi_{r_1}$. Here, $\tau_0$ is a
rose with petals labeled by elements in $\{a,b,c,d,e,f,g\}$ and hence for
$i \ge 1$, $\tau_i$ is a rose labeled by $\{\Phi_i(a), \ldots,
\Phi_i(g)\}$. Also, $\tau_0'$ is a rose labeled by
$\{a,b,c,d,e,f,g\}$, so $\tau_i'$ is also a rose labeled by
$\{\Phi_i(a), \ldots, \Phi_i(g)\}$. Thus, for every $i\geq 0$,
$\tau_i$ and $\tau_i'$ have the same marking but different train track
structures. In other words, they belong to the same simplex in $\CV[7]$.

The next lemma studies the behavior of illegal turns in a path along the
folding sequence. This will be used in the proof of
<Ref> to show that the limit tree of the folding
sequence is non-geometric.

Let $(r_i)_{i \ge 1}$ be strictly increasing such that $r_i \equiv 0$
mod 3 and $r_1 > R$, where $R$ is the constant from <Ref>. Let $(\tau_i')_i$ be the corresponding folding
sequence. Then for any edge path $\beta$ in $\tau'_j$ with at least one
illegal turn, the number of illegal turns in
$[\psi_{r_{j+3}}\psi_{r_{j+2}}\psi_{r_{j+1}}(\beta)]$ is less than the
number of illegal turns in $\beta$.

By <Ref>, the illegal turns in $\tau'_{j}$ are

$$\{a,e\}, \{a,\bar{g}\},\{e,\bar{g}\},
\{b,\bar{d}\},\{c,\bar{b}\},\{d,\bar{c}\},\{f,\bar{e}\},\{g,\bar{f}\} $$

and we have

{a,e}      {d,c̅}

{a,g̅}      {d,c̅}

{b,d̅}      {e,g̅}

{c,b̅}      {f,e̅}

{d,c̅}      {g,f̅}

Thus, for any illegal edge path $\beta \subset \tau_{j}'$, one
of $\beta, \psi_{r_{j+1}}(\beta), \psi_{r_{j+2}}\psi_{r_{j+1}}(\beta)$
has an illegal turn $\{x, y\}$ where $x, y \in \{e, f, g,
\bar{e}, \bar{f}, \bar{g}\}$.

Consider the automorphism $\vartheta$ and corresponding train track map
$h \from  \mathrm{R}_3 \to \mathrm{R}_3$ as in <Ref>. Then $h$ does not have any periodic INPs. Since $R$ is the
constant from <Ref>, we get that
one of $[\psi_{r_{j+1}}(\beta)], [\psi_{r_{j+2}}\psi_{r_{j+1}}(\beta)],
[\psi_{r_{j+3}}\psi_{r_{j+2}}\psi_{r_{j+1}}(\beta)]$ has fewer illegal
turns than $\beta$.

§ LIMITING TREE OF FOLDING SEQUENCE

In this section, we will show that for appropriate choices of $(r_i)_i$,
the projection of the folding sequences $(\tau_i')_i$ to the free factor
complex $\FreeF[7]$ is a quasi-geodesic and hence converges to the
equivalence class of an arational tree. We will also show that this tree
is non-geometric.

§.§ Sequence of free factors

Given a sequence $(r_i)_{i \ge 1}$, recall that $\Phi_i =
\phi_{r_1}\phi_{r_1}\cdots\phi_{r_i}$, where $\phi_r = \rho \phi^r$. For
convenience, also set $\Phi_0 = \id$. We have the folding sequence

\[
\begin{tikzcd}
\cdots
& \tau_{i+1}' \arrow[l]
& \tau_i' \arrow{l}[above,pos=.4]{\psi_{r_{i+1}}}
& \tau_{i-1}' \arrow{l}[above,pos=.4]{\psi_{r_i}}
& \cdots \arrow[l]
& \tau_2' \arrow{l}[above,pos=.4]{\psi_{r_3}}
& \tau_1' \arrow{l}[above,pos=.4]{\psi_{r_2}}
& \tau_0' \arrow{l}[above,pos=.4]{\psi_{r_1}}.
\end{tikzcd}
\]

where $\tau_i'$ is a rose labeled by $\{ \Phi_i(a),\cdots,\Phi_i(g)\}$,
and $\psi_r = \phi_r^{-1}$. From
the markings, we can associate $\tau_i'$ to an open simplex $U_i$ in
$\CV[7]$. Consider a sequence of free factors $A_i \in \pi(U_i)$, where
$\pi \from  \CV[7] \to \FreeF[7]$. For an appropriate sequence of $(r_i)_i$,
we will see that $(A_i)_i$ is a quasi-geodesic (with infinite diameter).
The key will be <Ref> which is the main goal of
this section.

We now consider the following explicit sequence of free factors. Let $A_0
= \la d,e,f\ra$ be the free factor in $\free[7]$, and define
$$ A_i:=
\Phi_i (A_0) = \la \Phi_i(d), \Phi_i(e), \Phi_i(f)\ra.$$
Note that for any
$r,s,t >0$, the following holds:

\begin{equation}\label{eqn:ff1}
\begin{aligned}
A_0 &= \la d, e, f \ra \\
A_1 &= \phi_r(A_0) = \la a, b, c \ra \\
A_2 &= \phi_s\phi_r(A_0) = \la e, f, g \ra \\
A_3 &= \phi_t\phi_s\phi_r(A_0) = \la b,c,d \ra
\end{aligned}
\end{equation}

Thus, for any sequence $(r_i)_i$,

\begin{equation}\label{eqn:ff2}
A_i = \Phi_i(A_0) = \Phi_{i-1}(A_1) = \Phi_{i-2}(A_2) =
\Phi_{i-3}(A_3).
\end{equation}

We say two free factors $A$ and $A'$ are disjoint if (possibly
after conjugating) $\free=A*A'*B$ for a (possibly trivial) free factor
$B$, and $A'$ is compatible with $A$ if it either contains $A$ (up
to conjugation) or is disjoint from $A$.

For any sequence $(r_i)_{i \ge 1}$, if $|i-j|=1$, then $A_i, A_j$ are
disjoint; and if $|i-j| = 2$ or $3$, then they are distinct and not

We see from <Ref> that the statement of the lemma
holds for $A_0, A_1, A_2$ and $A_3$. Now for $i \ge 1$ and $k \in
\{1,2,3\}$, by <Ref>, the pair $A_i, A_{i+k}$
differs from $A_0, A_k$ by the automorphism $\Phi_i$, whence the lemma.

Recall the transition matrix $M_r$ for $\phi_r$, and the $3\times 3$
matrix $B$ whose power $B^r$ forms a block of $M_r$. For each $i \ge 1$,
let $\overline{M}_i = M_i$ mod 2.  By a simple computation, we see that
$B^7 = I$ mod 2. Thus, when $i = j$ mod 7, $\overline{M}_i =
\overline{M}_j$. We have the following lemma.

Let $V_0$ be the 3-dimensional vector
space of $(\mathbb{Z}/2\mathbb{Z})^7$ spanned by the vectors
$(0,0,0,1,0,0,0)^T,(0,0,0,0,1,0,0)^T,(0,0,0,0,0,1,0)^T$. Then for all
$i \ge 0$,

\[
V_0 \bigcup \left( \bigcup_{j=0}^{107} \overline{M}_i
\overline{M}_{i+1}\ldots \overline{M}_{i+j} V_0 \right) =
\]

Since $\overline{M}_i=\overline{M}_j$ whenever $i = j$ mod
7, it is enough to verify the statement for $i \in \{0,\ldots,6\}$. In
these cases, we can check the validity of the statement using Sage
with the following code: language=Python frame=lines

B = matrix(GF(2), [

def M(i):
return block_matrix([
[ matrix(4,3,0) , identity_matrix(4) ],
[ B^i           , matrix(3,4,0)      ]

V0 = set(
for i in (0,1)
for j in (0,1)
for k in (0,1)

for i in range(0,7):
W = set(V0)
P = identity_matrix(7)
for j in range(i,200):
P = P*M(j)
for v in V0:
w = tuple(P*vector(v))
if len(W) >= 2^7:

# Output:
# 0 107
# 1 107
# 2 107
# 3 107
# 4 107
# 5 107
# 6 107

For any sequence $(r_i)_i$, if $r_i \equiv i \mod 7$, then $109$
consecutive $A_i$'s cannot be contained in the same free factor or be
disjoint from a common factor.

For any $i \ge 1$ and $k \ge 0$, let
\[B_{i+k} = \phi_i \phi_{i+1}
\cdots \phi_{i+k} A_0. \]
Abelianizing and reducing mod 2, we have $A_0
\equiv V_0$, and $B_{i+k} \equiv \overline{M}_i \cdots
\overline{M}_{i+k} V_0$. Thus, by <Ref>, the
sequence $\{A_0, B_i, \ldots, B_{i+107}\}$ cannot be contained in the
same free factor or be disjoint from a common factor.

Now consider any sequence $(r_i)_i$ with $r_i \equiv i$ mod $7$, so
that $\overline{M}_{r_i} = \overline{M}_i$ for all $i$. Let $A_i =
\Phi_i A_0 = \phi_{r_1} \cdots \phi_{r_i}A_0$. Set $\Phi_0 = \id$. For
any $i \ge 1$, by applying the automorphism $\Phi_{i-1}^{-1}$, the
sequence of free factors $\{A_{i-1},\ldots,A_{i+107}\}$ is isomorphic to
the sequence
\[\{A_0, \phi_{r_i}A_0, \ldots, \phi_{r_i}\phi_{r_{i+1}}
\cdots \phi_{r_{i+107}}A_0 \}. \]
The latter sequence after
abelianization and reducing mod 2 is equivalent to the sequence
$\{A_0,B_i,\ldots,B_{i+107}\}$. Thus $\{A_{i-1},\ldots,A_{i+107}\}$
cannot be contained in the same factor or be disjoint from a common

§.§ Subfactor projection

We will now use subfactor projection theory originally introduced in
[4] and further developed in [42]
to show that $(A_i)_i$ is  a quasi-geodesic for
appropriate choices of sequence $(r_i)_i$.

We first define subfactor projection and recall the main results about
them. For $G \in \CV$ and a rank $\geq 2$ free factor $A$, let $A|G$
denote the core subgraph of the cover of $G$ corresponding to the
conjugacy class of $A$. Pulling back the metric on $G$, we obtain $A|G
\in \mathrm{CV}(A)$. Denote by $\pi_A(G):= \pi(A|G) \subset \F(A)$ the
projection of $A|G$ to $\F(A)$. Here $\mathrm{CV}(A)$ is the Outer space
of the free group $A$ and $\F(A)$ is the corresponding free factor

Recall two free factors $A$ and $B$ are disjoint if they are
distinct vertex stabilizers of a free splitting of $\free$. If $B$ is not
compatible with $A$, then we say $B$ meets $A$, that is, $B$ and
$A$ are not disjoint and $A$ is not contained in $B$, up to conjugation.
In this case, define the projection of $B$ to $\F(A)$ as follows:

\[\pi_A(B):=\{\pi_A(G) | G \in \CV \text{ and } B|G \subset G \text{ is
embedded }\}\]
If $B$ is compatible with $A$, then define $\pi_A(B)$ to
be empty. If $A$ meets $B$ and $B$ meets $A$, then we say $A$ and $B$

Let $A,B,C$ be free factors of $\free$. There is a constant $D$
depending only on $n$ such that the following statements hold.

* If $\rank(A) \ge 2$, then either $A \subseteq B$ (up to
conjugation), $A$ and $B$ are disjoint, or $\pi_A(B) \subset
\mathcal{F}(A)$ is defined and has diameter $\leq D$.

* If $\rank(A) \ge 2$, $B$ and $C$ meet $A$, and $B$ is
compatible with $C$, then
\[ d_A(B,C) = \diam_{\F(A)}(\pi_A(B) \cup
\pi_A(C)) \le D.\]

* If $A$ and $B$ overlap, have rank at least $2$, and $C$ meets
both, then
\[ \min \{ d_A(B,C), d_B(A,C)\} \le D.\]

For $n \geq 3$, there exists $D'
\geq 0$ such that if $A$ is a free factor with $\text{rank}(A) \geq 2$
and $\gamma$ is a geodesic of $\FreeF$ with each vertex of $\gamma$
having a well-defined projection to $\F(A)$, then
$\text{diam}(\pi_A(\gamma)) \leq D'$.

We now prove the following lemma.

For any $K>0$, there exists a constant $r = r(K)$ such that for any
sequence $(r_i)_{i \ge 1}$, if $r_i \ge r$ for all $i$, then the
following statements hold:

* For any $j \geq 2$, the projections of $A_{j-2}$ and $A_{j+2}$
to the free factor complex $\F(A_j)$ are defined and the distance
between them is at least $K$.

* Let $D$ be the constant of <Ref>. If $K > 3D,$ then for any $i < j <k$, if
$j-i\geq 2$ and $k-j\geq 2$, the projections of $A_i$ and $A_k$ to
$\F(A_j)$ are defined and have distance at least $K-2D$.

Recall for any $r$, $\phi_r = \rho \phi^r$, where $\phi$ restricts to a
fully irreducible outer automorphism of $\la a, b, c\ra$. In
particular, $\phi$ acts as a loxodromic isometry of the free factor
complex $\F(\la a,b,c \ra)$, Thus, for any $K$, there exists $r=r(K)$
such that for all $s \ge r$, the distance between $\phi^s(\la b,c\ra)$
is at least $K+2D$ away from $\la a,b\ra$ in $\F(\la a,b,c\ra)$.

Now consider any sequence $(r_i)_i$ with $r_i \ge r$ for all $i$. By
<Ref> and <Ref>, the projections of
$A_{j-2}$ and $A_{j+2}$ to $\F(A_j)$ are defined. Moreover, by
<Ref>, we see that, by applying an automorphism, the distance
between projections of $A_{j-2}$ and $A_{j+2}$ in $\F(A_j)$ is the same
as the distance between the projections of $A_0 = \la d, e, f \ra$ and
$\phi_{r_{j-1}}(A_3) = \la \phi_{r_{j-1}}(b),\phi_{r_{j-1}}(c),a \ra$
to $\F(A_2) = \F(\la e, f, g \ra)$. Note that the rotation $\rho$ sends
the free factor $\la a,b,c\ra$ to $A_2$, thus inducing an isometry from
$\F(\la a,b,c\ra)$ to $\F(A_2)$. The projection of $A_0$ to $\F(A_2)$
is $D$-close to the factor $\la e,f\ra = \rho (\la a,b \ra)$, and the
projection of $\phi_{r_{j-1}}(A_3)$ to $\F(A_2)$ is $D$-close to the
factor $\rho \phi^{r_{j-1}}(\la b,c \ra)$. Thus, the distance in
$\F(A_2)$ of the two projections is at least $K$. This shows the first
statement of the Lemma.

Now fix $K > 3D$ and let $(r_i)_i$ be any sequence with $r_i \ge r(K)$
for all $i$. We will prove the second statement by inducting on $l=k-i$
with the previous statement giving the base case $l = 4$. Suppose we
are given $A_i,A_j,A_k$ with $l=k-i>4$, $j-i,k-j\geq 2$. We first claim
that projections of $A_{j+2},A_{j+3},\cdots, A_k$ to $\F(A_j)$ are
defined, i.e. none of them are equal to or disjoint from $A_j$. For
suppose $A_s$ is the first on the list that is equal to or disjoint
from $A_j$. By <Ref> we have $4\leq s-j < k-i$. By induction,
the projections of both $A_j$ and $A_s$ to $\F(A_{j+2})$ are defined
and the distance between their projections is $\geq K-2D > D$. Using
statement 2 of <Ref>, this implies that $A_s$
and $A_j$ cannot coincide or be disjoint, proving the claim. By the
same argument, we also have that the projections of
$A_i,A_{i+1},\cdots,A_{j-2}$ to $\F(A_j)$ are all defined.

By the first statement of the lemma, we have $d_{A_j}(A_{j-2},A_{j+2})
\ge K$. We now claim that $d_{A_j}(A_{j+2}, A_k) \le D$. If $k=j+3$, then
$A_{j+2}$ and $A_k$ are disjoint, and the claim holds by statement 2 of
<Ref>. If $k \ge j+4$, then applying
induction again to $j$, $j+2$ and $k$, we see that $A_j$ and $A_k$ have
well-defined projections to $\F(A_{j+2})$ and $d_{A_{j+2}}(A_j,A_k) \ge
K-2D > D$. Now, the claim follows by the third statement of
<Ref>. By the same argument, we also see that
$d_{A_j}(A_i,A_{j-2}) \le D$. We now conclude $d_{A_j}(A_i,A_k) \ge
K-2D$ by the triangle inequality.

We are now ready to prove the main results of this section.

There exists $R>0$ such for any sequence $(r_i)_{i \ge 1}$, if $r_i \ge
R$, and $r_i \equiv i$ mod 7, then the sequence $(A_i)_{i \ge 0}$ is a
quasi-geodesic in $\FreeF[7]$.

Let $D$ be the constant of <Ref> and let $D'$
be the constant of <Ref>. Fix $K=4D+D'$. Let $R=r(K)$ be the
constant of <Ref>. Let $(r_i)_{i \ge 1}$ be any
sequence with $r_i \ge R$ and $r_i \equiv i$ mod 7 for all $i$. We will
show that the sequence $(A_i)_i$ goes to infinity with linear speed.
More precisely, we will show that for any $d>0$, if $k-i \ge 110d +4$,
then $d_{\FreeF[7]}(A_i,A_k) \ge d$. Suppose not. Let $\gamma$ be a
geodesic between $A_i$ and $A_k$ of length $< d$.

For every $j \in \{i+2,\ldots,k-2\}$, there exists a free factor in
$\gamma$ that is compatible with $A_j$. Indeed, if every free factor in
$\gamma$ meets $A_j$, then by <Ref>, projection of $\gamma$ to
$A_j$ will be well-defined and has diameter bounded by $D'$. However,
by <Ref>, the projections of $A_i$ and $A_k$ to
$\F(A_j)$ has distance at least $K - 2D > D'$.

By the pigeonhole principle, there exists a vertex $B$ of $\gamma$
compatible with at least 110 free factors among
$\{A_{i+2},\ldots,A_{k-2}\}$. By <Ref>, it is not
possible for $B$ to be compatible with 109 consecutive $A_j$'s.
Therefore, it must be possible to find $i+2 \le i' < j' < k' \le k-2$
with $j' - i' \ge 2$ and $k' - j' \ge 2$, such that $B$ is compatible
with $A_{i'}$ and $A_{k'}$, but $B$ meets $A_{j'}$. In particular,
$\pi_{A_{j'}}(B)$ is defined. By <Ref>, $A_{i'}$,
$A_{k'}$ also have well-defined projections to $\F(A_{j'})$ with
$d_{A_{j'}}(A_{i'},A_{k'}) \ge K-2D > 2D$. On the other hand, since $B$
is compatible with both $A_{i'}$ and $A_{k'}$, we have
$d_{A_{j'}}(A_{i'},B) \le D$ and $d_{A_{j'}}(A_{k'},B)\le D$ by
<Ref>. This is a contradiction, finishing the
proof that $d_{\FreeF[7]}(A_i, A_k) \ge d$ for all $k-i \ge 110d+4$.

Recall that $\FreeF$ is Gromov hyperbolic and that its Gromov boundary is
the space of equivalence class of arational trees. Also recall we say a
folding sequence $(G_i)_i$ converges to an arational tree $T$, if
$\pi(U_i)$ converges to $[T] \in \partial \FreeF$, where $U_i$ is the
open simplex in in $\CV$ associated to $G_i$. We have the following

Given any strictly increasing sequence $(r_i)_{i \geq 1}$ satisfying $r_i
\equiv i \mod 7$, the folding sequence $(\tau_i')_i$ converges to
an arational tree $T$.

§.§ Non-geometric tree

We will now show that the arational tree obtained in the previous
section as the limit of the free factors $(A_i)_i$ is
non-geometric. This section will use the terminology of band
complexes and resolutions, for details see [2].

[1, 33]
Let $X$ be a band complex and $T$ a $G=\pi_1(X)$-tree. A resolution
$f :\widetilde{X} \to T$ is exact if for every $G$-tree
$T'$ and equivariant factorization

\[
\widetilde{X} \stackrel{f'}{\longrightarrow} T'
\stackrel{h}{\longrightarrow} T
\]

of $f$ with $f'$ a surjective resolution it follows that $h$ is an
isometry onto its image. We say $T$ is geometric if every
resolution is exact.

The proof of the following proposition is based on <cit.>.

For any strictly increasing sequence $(r_i)_{i \ge 1}$, if the
corresponding folding sequence $(\tau_i')_i$ converges to an arational
tree $T$, then $T$ is not geometric.

Let $\tilde\psi_i \from \tilde\tau_{i-1}' \to \tilde\tau_i'$ be a lift
of the train track map to the universal covers fixing a base
vertex. Pick a length measure on $(\tau_i')_i$ so we get a folding
sequence $\tilde \tau_0' \stackrel{\tilde \psi_1}{\longrightarrow}
\tilde\tau_1' \stackrel{\tilde \psi_2}{\longrightarrow} \cdots$ in
$\cv[7]$ that converges to $T$.  Recall that there are morphisms
$h_i \from \tilde{\tau}_i' \to T$ such that $h_i = h_{i+1} \tilde
\psi_{i+1}$. Since $T$ is arational, $h_i$'s are not isometries though
they restrict to isometries on edges. Let $X$ be a finite band
complex with resolution $f \from \tilde X \to T$. We will show that
the resolution factors through $\tilde\tau_i'$ for sufficiently
large $i$. This will imply $T$ is not geometric.

Let $\Gamma$ be the underlying real graph of $X$ (disjoint union
of metric arcs) with preimage $\tilde\Gamma$ in $\tilde X$.
We may assume $f$ embeds the components of $\tilde\Gamma$. A vertex $v$
of $\tilde X$ is either a vertex of $\tilde\Gamma$ or a corner of a
band or a 0-cell of $\tilde X$. For every such vertex $v$ choose a
point $f_0(v) \in \tilde\tau_0$ so that $f_0$ is equivariant and $f=h_0
f_0$ on the vertices of $\tilde X$.

An edge in $\tilde X$ is either a subarc of $\tilde\Gamma$ or a
vertical boundary component of a band or a 1-cell in $\tilde X$. Up to
the action of $\free[7]$, there are only finitely many edges. Using
<Ref>, we can find $i>0$ such that for every
edge $e$ in $\tilde X$, the edge path in $\tilde\tau_i'$ joining the
two vertices of $\tilde\psi_i \cdots \tilde\psi_1 f_0(\partial e)$ is
legal. Now extend $\tilde\psi_i \cdots \tilde\psi_1 f_0$ to an
equivariant map $f_i \from \tilde X \to \tilde\tau_i'$ that sends edges
to legal paths (or points) and is constant on the leaves. Thus $f_i$ is
a resolution of $\tilde\tau_i'$.

\[\begin{tikzcd}
&  &
\widetilde{X}\arrow[rd, bend left, "f_i"] \arrow[lld, bend right, "f_0"]
\arrow[rrrd, bend left, "f"]\\
\tilde{\tau}_0' \arrow[r, "\psi_1"]
& \tilde{\tau}_1' \arrow[r, "\psi_2"]
& \cdots \cdots  \arrow[r, "\psi_i"]
& \tilde{\tau}_i' \arrow[r, "\,\, \psi_{i+1}"] \arrow[rr, bend left, "h_i"]
& \cdots
& T
\end{tikzcd}\]

This yields a factorization

\[\begin{tikzcd}
\tilde X \arrow[r, "f_i"]  & \tilde\tau_i' \arrow[r, "h_i"] & T
\end{tikzcd}\]

but $h_i$ is not an isometry. This shows $T$ is non-geometric.

§ NON-UNIQUELY ERGODIC UNFOLDING SEQUENCE

The goal of this section is to show that if a sequence $(r_i)_{i \ge 1}$
grows sufficiently fast, then the set of currents supported on the legal
lamination $\Lambda$ of the unfolding sequence $(\tau_i)_{i \geq 0}$
is a 1-simplex in $\PCurr[7]$.

Recall that $M_r$ is a $7\times 7$ matrix of the block form

\[
\begin{pmatrix}
\end{pmatrix}
\]

where $I$ is the $4\times 4$ identity matrix, and $B$ is the transition
matrix of $\theta$; all that matters is that some positive power of $B$
has all entries positive. Let $\lambda_B$ be the Perron-Frobenius
eigenvalue of $B$. Recall the constant $\kappa_B > 0$ from
<Ref>. Given a sequence $(r_i)_i$, define for each $i \ge 1$

\[
P_i = \frac{1}{\kappa_B \lambda_B^{r_{i+1}}} M_{r_i} M_{r_{i+1}}.
\]

Let $\{e_k:k=1,\ldots,7\}$ be the standard basis for $\R^7$. Denote by
$\mathbb P\R_{\geq 0}^7$ the projectivization of $\R_{\geq 0}^7$, and the
projective class of a vector $v$ by $[v]$. Fix a metric $d$ on
$\P\R_{\geq 0}^7$. We view $M_r$ as a projective transformation
$\P\R_{\geq 0}^7\to \P\R_{\geq 0}^7$. For a sequence $(r_i)_{i \geq  1}$
and for $i<j$ denote by $S_{i,j}\subset\mathbb P \R_{\geq 0}^7$ the image
of the composition
$$M_{ij}:=M_{r_i}M_{r_{i+1}}\cdots M_{r_j}$$
and by
$S_i=\bigcap_{j>i}S_{i,j}$. We denote by $v_B$ a positive
Perron-Frobenius eigenvector of $B$, and by $v_B^{234}$ (resp.
$v_B^{567}$) the vector in $\R^7$ which is $v_B$ in coordinates $2,3,4$
(resp. $5,6,7$) and 0 in all other coordinates. The main result of this
section is the following.

Let $(r_i)_{i \ge 1}$ be a sequence of positive integers with
$r_{i+1}-r_i \ge i$. Then for all $i$ the set $S_i$ is a 1-simplex,
i.e. it is the convex hull of two distinct points $p_i,q_i\in\mathbb P
\R_{> 0}^7$. Moreover, as $i\to\infty$, $\{p_i,q_i\}$ converges (as a
set) to $\{ [v_B^{234}], [v_B^{567}]\}$.

Before we give a technical proof of <Ref>, we will give a
simpler, more intuitive proof where the sequence $r_1<r_2<\cdots$ is
chosen inductively so that $r_1$ is sufficiently large and each $r_i$ is
sufficiently large depending on $r_1,r_2,\cdots,r_{i-1}$. Later, we do a
more careful analysis where we can control the growth of the sequence.

For $\epsilon>0$ we will write $x\overset{\epsilon}=y$ if
$d(x,y)<\epsilon$ in $\P\R_{\geq 0}^7$.
Each $S_{ij}$ is the convex hull of the $M_{ij}$-images of the vectors
$e_i$, $i=1,\cdots,7$. The proof consists of computing these
images using the Perron-Frobenius dynamics.
We first observe that there is a sequence $\epsilon_r\to 0$ such

* $M_r(e_7)=e_4$, $M_r(e_6)=e_3$, $M_r(e_5)=e_2$,

* $M_r(e_i)\overset{\epsilon_r}=v_B^{567}$, $i=1,2,3$,

* $M_r(v_B^{567})=v_B^{234}$,

Next, we consider the composition $M_sM_r$ for $r>>s$. The third
bullet uses uniform continuity of $M_s$ and the assumption that
$r$ is sufficiently large compared to $s$.

* $M_sM_r(e_7)=e_1$,

* $M_sM_r(e_i)\overset{\epsilon_s}=v_B^{567}$, $i=4,5,6$,

* $M_sM_r(e_i)\overset{\epsilon_s}=v_B^{234}$, $i=1,2,3$.

Finally, for $r>>s>>t$ we see similarly:

* $M_tM_sM_r(e_7)\overset{\epsilon_t}=v_B^{567}$,

* $M_tM_sM_r(e_i)\overset{\epsilon_t}=v_B^{234}$,

* $M_tM_sM_r(e_i)\overset{\epsilon_t}=v_B^{567}$,

It follows that if we make suitably large choices for the $r_i$'s, the
set $S_{i,i+3}$ will be contained in the $\epsilon_{r_i}$-neighborhood
of the 1-simplex $[v_B^{567},v_B^{234}]$. Moreover, given any
$\epsilon>0$ and $j>i+3$ we can choose $r_j$ large (depending on
uniform continuity constants of $M_{ij}$) to ensure that
$S_{i,j+3}=M_{ij}(S_{j,j+3})$ is contained in the
$\epsilon$-neighborhood of the 1-simplex with endpoints
$M_{ij}(v_B^{567})$ and $M_{ij}(v_B^{234})$. Thus each $S_i$ is the
nested intersection of simplices of dimension $\leq 6$ such that for
all $\epsilon>0$ they are eventually all contained in the
$\epsilon$-neighborhood of a 1-simplex with definite distance between
the endpoints. This proves the Proposition.

We now present a more detailed proof of <Ref>.
For a sequence of integers $(r_i)_{i\geq 1}$ such that $r_i, r_{i+1}-r_i
\to \infty$, by <Ref> $(P_i)_i$ converges to an idempotent matrix
$Y$.  Let $\Delta_i = Y - P_i$ and let $||Y||$ be the operator norm.

Let $(r_i)_{i\geq 1}$ be a sequence of positive integers such that
$r_{i+1}-r_i \ge i$. Then there exists an $I \geq 1$, such that for all
$i \geq I$, $||\Delta_i|| \leq 1/(2 \cdot 2^i)$.

Let $\lambda_B, \mu_B, \mu'_B$ be the modulus of the three
eigenvalues of $B$; we have $\lambda_B \sim 1.46$ and $\mu_B = \mu'_B
\sim 0.826$. Then
\[||\Delta_i|| = || P_i - Y || \leq \text{ max }
\left( \frac{\mu^{r_{i+1}}}{\lambda^{r_{i+1}}},
\frac{\lambda^{r_{i}}}{\lambda^{r_{i+1}}} \right) \leq
\frac{\lambda^{r_{i}}}{\lambda^{r_{i+1}}} \]

where the two terms comes from the two blocks in $P_i$. For the last
inequality, note that $\mu < 1 < \lambda$ and $r_i$ are positive
integers. Therefore,  $\mu^{r_{i+1}} < 1 < \lambda^{r_i}$.

Now we claim that there exists an $I \geq 1$, such that for all $i\geq I$,

\[ \frac{\lambda^{r_{i}}}{\lambda^{r_{i+1}}} \leq \frac{1}{2^{i+1}}
\qquad \text{ equivalently, } \qquad 2 \leq
\lambda^{\frac{r_{i+1}-r_i}{i+1}}\]

We only need to show that the sequence $\frac{r_{i+1}-r_i}{i+1}$ is
eventually increasing. Indeed, by assumption, $r_{i+1}-r_i \ge i$, so

\begin{align*}
i  & \leq r_{i+1}-r_i \\
\frac{ i}{i+1}  & \leq \frac{r_{i+1}-r_i}{i+1}
\end{align*}

Since $i/(i+1)$ is an increasing sequence, it follows that our sequence
is also increasing.

The following lemma is a consequence of <Ref> and

Let $(r_i)_{i \ge 1}$ be a sequence of positive integers such that
$r_{i+1}-r_i \ge i$, $Y$ be the idempotent matrix of <Ref> and
$M_\infty = \lim_{r \to \infty}
M_r/\lambda_B^r$. Then the following statements hold.

* For all $i \ge 1$, the sequence of matrices $\{ P_i P_{i+2}
\cdots P_{i+2k}\}_{k=1}^\infty$ converges to a matrix $Y_i$.
Furthermore, for all sufficiently large $i$,

\[ \norm{Y_i - Y} \le \frac{2}{2^i}\left( \norm{Y}+\norm{Y}^2 \right)\]

* The kernel of $Y$ is a subspace of the kernel of $Y_i$ for all
$i \ge 1$.

* For all $i \ge 1$, $Y_i(e_1) \neq 0$ with non-negative entries and $Y_i(e_2)$
and $Y_i(e_3)$ are positive multiples of $Y_i(e_1)$.

* For all $i \ge 1$, $M_{r_i} Y_{i+1}(e_1) \neq 0$ with non-negative entries,
and $M_{r_i}Y_{i+1}(e_1)$ and $Y_i(e_1)$ are not scalar multiples of
each other.

* Projectively, $[Y_i(e_1)] \to [Y(e_1)]$ and
$[M_{r_i}Y_{i+1}(e_1)] \to [M_\infty Y(e_1)]$ as $i \to \infty$.

For (1), it suffices to show convergence for all $i$ greater than some
$I$. Indeed, if such $I$ exists and $i < I$, then let $i_0 \ge I$ be
such that $i = i_0$ (mod 2) and observe that

\[ \{P_i P_{i+2} \cdots P_{i+2k}\}_{k=\frac{i_0-i}{2}}^\infty =
P_i P_{i+2} \cdots P_{i_0-2} \{P_{i_0}P_{i_0+2} \cdots
By assumption $\{P_{i_0}P_{i_0+2} \cdots
P_{i_0+2k}\}_{k=0}^\infty$ converges. Since matrix multiplication is
continuous, the sequence $\{P_iP_{i+2} \cdots P_{i+2k}\}_{k=0}^\infty$
also converges.

For each $i$, let

\[ \Delta_i = P_i - Y.\]

By <Ref>,
there exists $I \ge 1$ such that for all $i \ge I$, $\norm{\Delta_i} \le
\frac{1}{2 \cdot 2^i}$. Also, choose $I$ sufficiently large so that
$\frac{1}{2^I} \norm{Y} \le 1/2$. Then, by <Ref>,
for all $i \ge I$, the sequence $\{ P_i P_{i+2} \cdots
P_{i+2k}\}_{k=0}^\infty$ converges to some matrix $Y_i$, with

\begin{equation} \label{eqn:bound}
\norm{Y_i - Y} \le \frac{2}{2^i} \left( \norm{Y}+\norm{Y}^2 \right).
\end{equation}

For (2), it again suffices to show the statement is true for all
sufficiently large $i$, and the statement holds for all $i \ge I$ by

For (3), first note that since all the matrices involved are
non-negative, the resulting vectors are all also non-negative. So we
only need to show that they are not the zero vector. It suffices to
check that $Y_i(e_1) \neq 0$ for all sufficiently large $i$, since each
$P_i$ is non-negative and has full rank. For large $i$, the statement
follows because $Y(e_1)$ is not equal to 0 and $\norm{Y_i(e_1) -
Y(e_1)} \le \norm{Y_i-Y}$ can be made arbitrarily small. For the second
statement, we know that $Y(e_2)$ and $Y(e_3)$ are positive multiples of
$Y(e_1)$, so there are $s,t > 0$ such that $se_2 - e_1$ and $te_3 -
e_1$ are in the kernel of $Y$. Then $Y_i(se_2-e_1) = Y_i(te_3-e_1)=0$
for all $i$ by (2).

For (4), $M_{r_i}Y_{i+1}(e_1) \neq 0$ with non-negative entries since
$Y_{i+1}(e_1)$ is so by (3). To see that $M_{r_i}Y_{i+1}(e_1)$ and
$Y_i(e_1)$ are projectively distinct, it is enough to do this for all
sufficiently large $i$. Let $M_\infty = \lim_{r \to \infty}
M_r/\lambda_B^r$. By <Ref> and <Ref>, $Y(e_1)$ and
$M_\infty Y (e_1)$ are orthogonal. Since $r_i \to \infty$, we can make
$\frac{M_{r_i}}{\lambda_B^{r_i}}Y_{i+1}(e_1)$ arbitrarily close to
$M_\infty Y(e_1)$, and $Y_i(e_1)$ close to $Y(e_1)$. This means
$M_{r_i} Y_{i+1}(e_1)$ and $Y_i(e_1)$ are near orthogonal, so they
can't be scalar multiples of each other.

Statement (5) is clear.

By <Ref> and <Ref>,
$[v_B^{234}]=[Y(e_1)]$ and
$[v_B^{567}]=[M_\infty Y(e_1)]$.
Using notation from <Ref>, set
\[p_i = [Y_i(e_1)] \qquad
\text{and} \qquad q_i = [M_{r_i}Y_{i+1}(e_1)].\]
(3) - (5),

* $p_i$ and $q_i$ are well defined and distinct.

* $p_i = [Y_i(e_k)]$, and $q_i = [M_{r_i}Y_{i+1}(e_k)]$, for $k=1,2,3$.

* $p_i \to [v_B^{234}]$ and $q_i \to [v_B^{567}]$.

* $[M_{r_i}(p_{i+1})] = q_i$ and $[M_{r_i}(q_{i+1})] = p_i$.

Our goal is to show $S_i$ is the $1$-simplex spanned by $p_i$ and
$q_i$. To do this, we consider $S_{ij}$, which is the convex hull of
the $M_{ij}$-images of the vectors $e_k$, $k=1,\cdots,7$. That is, we
have to show that $[M_{ij}(e_k)]$ is close to either $p_i$ or $q_i$ for
each $k$. We first observe that for all $r,s > 0$:

* $M_r(e_4)=e_1$, $M_r(e_5)=e_2$, $M_r(e_6)=e_3$,

* $M_rM_s(e_7)=e_1$.

We may assume that $j-1 = i+2m$, so $M_{ij}$ breaks up into pairs,
i.e. for all $k$,

\[ [M_{ij}(e_k)] = [P_i \cdots P_{j-1} (e_k)].\]

Let $\epsilon > 0$ be arbitrary. Choose
$\delta > 0$ such that for any vector $u \in \R_+^7$ and any $v \in
\left\{ Y_i(e_k), \frac{M_{r_i}}{\lambda_B^{r_i}} Y_{i+1}(e_k) :
k=1,2,3\right\}$, if $\norm{u-v} \le \delta$, then $d([u],[v]) \le
\epsilon$. Now by <Ref>, we can choose $J$ sufficiently
large so that whenever $i+2m \ge J$, then

* $\norm{ P_i \cdots P_{i+2m} -Y_i} \le \delta$

* $\norm{ P_{i+1} \cdots P_{i+2m+1} -Y_{i+1}} \le
\frac{\delta}{\norm{M_{r_i}/\lambda_B^{r_i}}}.$

Now we may assume that $j-3 \ge J$. Then,

* For $k=1,2,3$, we have
\[ \norm{P_i \cdots P_{j-1}(e_k) -
Y_i(e_k)} \le
\delta \quad \Longrightarrow \quad d \Big( [ M_{ij} (e_k) ],
p_i \Big) \le
\epsilon. \]

* For $k=7$, we have $M_{ij}(e_7) = M_{i,j-2}(e_1)$, so
$[M_{ij}(e_7)]=[P_i \cdots P_{j-3}(e_1)]$ is $\epsilon$-close to
$[p_i]$ by the same reasoning as the previous bullet point.

* For $k=4,5,6$, $M_{ij}(e_k) = M_{i,j-1}(e_{k-3})$. In this
case, we consider $\frac{M_{r_i}}{\lambda^{r_i}} P_{i+1} \cdots
P_{j-2}(e_{k-3})$ and approximate it by
$\frac{M_{r_i}}{\lambda^{r_i}} Y_{i+1}(e_{k-3})$, as follows:

\begin{align*}
& \norm{\frac{M_{r_i}}{\lambda^{r_i}} P_{i+1} \cdots
P_{j-2}(e_{k-3}) -
\frac{M_{r_i}}{\lambda^{r_i}} Y_{i+1}(e_{k-3})} \\
& \le \norm{\frac{M_{r_i}}{\lambda^{r_i}}} \norm{P_{i+1} \cdots
P_{j-2} - Y_{i+1}}\\
& \le \delta.
\end{align*}

Thus, for $k=4,5,6$, $d \Big( [M_{ij}(e_k)], q_i \Big) \le
\epsilon$.

We have shown that for any $\epsilon$, the vertices of the simplex
$S_{i,j}$ come $\epsilon$-close to $p_i$ and $q_i$ for all sufficiently
large $j$. Since $S_{i,j+1} \subset S_{i,j}$ and $S_i = \bigcap_{j>i}
S_{i,j}$, it follows that $S_i$ must be the $1$-simplex spanned by
$p_i$ and $q_i$. This proves the Proposition.

Recall the unfolding sequence $(\tau_i)_{i \ge 0}$ where $M_{r_i}$ is the
transition matrix of the train track map $\phi_{r_i}: \tau_i \to
\tau_{i-1}$. Let $\Lambda$ be the legal lamination of $(\tau_i)_{i \ge 0}$.

If $(r_i)_{i \ge 1}$ is a positive sequence with  $r_{i+1}-r_i \ge i$,
then $\P\C(\Lambda)$ is a 1-simplex.

In light of <Ref>, it is enough to show
$\P\C((\tau_i)_i)$ is a 1-simplex. For each $i \ge 0$, we have a
well-defined projection
\[ p_i\from \P\C((\tau_i)_i) \to \P\R_+^7
\quad\text{given by}\quad p_i([(\mu_i)_i]) = [\mu_i].\]
The image of
the projection is $S_{i+1}$, which is always a $1$-simplex by
<Ref>. Therefore, $\P\C((\tau_i)_i)$ is a $1$-simplex.

§ NON-UNIQUELY ERGOMETRIC TREE

The goal of this section is to show that if a sequence $(r_i)_{i \ge 1}$
grows sufficiently fast, then the set of projectivized length measures
$\P\D((\tau_i')_i)$ on the folding sequence $(\tau_i')_i)$ is a
1-simplex. By <Ref>, if $(\tau_i')_i$ converges to an
arational tree $T$, then $\P\D(T)$ is also a 1-simplex in

Recall that $N_r$ is a $7\times 7$ matrix of the block form

\[
\begin{pmatrix}
\end{pmatrix}
\]

where $I$ is the $4\times 4$ identity matrix, and $C$ is the transition
matrix of $\vartheta$. The transpose of $N_r$ has the same shape as $M_r$.
Therefore, the same theory from <Ref> holds true.
For brevity, we record only the essential statements that will be used
later and omit all proofs from this section.

Let $\lambda_C$ be the Perron-Frobenius eigenvalue of $C$.   Let
$\kappa_C$ be the constants of <Ref>. Given a sequence
$(r_i)_i$, define for each $i \ge 1$

\[
Q_i = \frac{1}{\kappa_C \lambda_C^{r_{i+1}}} N_{r_{i+1}} N_{r_i}.
\]

Given a sequence $(r_i)_{i \ge 1}$ of positive integers such that
$r_{i+1}-r_i \ge i$. Then for all $i \ge 1$, the sequence of matrices
$\{ Q_{i+2k}\cdots Q_{i+2} Q_i\}_{k=0}^\infty$ converges to a matrix
$Z_i$. Furthermore, $\lim_{i \to \infty} Z_i=Z$, where $Z$ is the
idempotent matrix of <Ref>.

If $(r_i)_{i \ge 1}$ is a positive sequence with  $r_{i+1}-r_i \ge i$.
Then $\P\D((\tau_i')_i)$, and hence $\P\D(T)$, is a 1-simplex.

§ NON-UNIQUELY ERGODIC TREE

In this section we relate the legal lamination $\Lambda$ associated to
the unfolding sequence $(\tau_i)_i$ defined in <Ref> and
the limiting tree $T$ of the folding sequence $(\tau_i')_i$ defined in
<Ref>, to show that $T$ is not uniquely ergodic.

Recall the automorphism $\Phi_i = \phi_{r_1} \circ \cdots \phi_{r_i}$,
with $\Phi_0 = \id$. We also use $\Phi_i$ to denote the induced graph map
from $\tau_i$ to $\tau_0$. If each $\tau_i$ and $\tau_i'$ as a marked
graph is the rose labeled by $\{a_i, b_i, c_i, d_i, e_i, f_i, g_i\}$,
then  $x_i$ is represented by $\Phi_i(x)$ for $x \in \{a,b,c,d,e,f,g\}$
as a word in $\free[7] = \langle a,b,c,d,e,f,g\rangle = \pi_1(\tau_0) =
\pi_1(\tau_0')$. We denote $x_0$ as above simply by $x$.

If $(r_i)_{i \ge 1}$ is positive, then for any length measure
$(\lambda_i)_i \in \D((\tau'_i)_i)$, the $\lambda_i$-volume of
$\tau_i'$ goes to $0$ as $i \to \infty$.

The composition
$\psi_{r_i}\psi_{r_{i-1}}\psi_{r_{i-2}}:\tau_{i-3}'\to\tau_i'$ has
the property that the preimage of every point of $\tau_i'$ consists of
at least two (in fact, many more) points of $\tau_{i-3}'$ and so
the $\lambda_i$-volume of $\tau_i'$ is at most half of the
$\lambda_{i-3}$-volume of $\tau_{i-3}'$.

Suppose $(r_i)_{i \ge 1}$ is positive. Let $\Lambda$ be the legal
lamination of the unfolding sequence $(\tau_i)_i$. Then every leaf in
$\Lambda$ is obtained as a limit of a sequence $\{\Phi_i(w)\}_i$, where
$w$ is a legal word in $\tau_0$ of length at most two in $\{a, b, c, d, e, f, g\}$
and their inverses. Moreover, $w$ can be closed up to a legal loop
which is a cyclic word of length $\leq 3$.

Let $l$ be a leaf of $\Lambda$ realized as a bi-infinite line in
$\tau_0$ and let $s$ be any subsegment of $l$, with combinatorial edge
length $\ell_s >0$ in $\tau_0$. By definition, for every $i$ there is a
bi-infinite legal path $l_i$ in $\tau_i$ such that $l = \Phi_i(l_i)$.
Let $i = i(s) \geq 0$ such that the edge length of $x_i$ in $\tau_0$
under the graph map $\Phi_i$ is $\geq \ell_s$ for all $x \in \{a, b, c,
d, e, f, g\}$. Thus, there is a segment $s_i$ of $l_i$ of combinatorial
length at most two in $\{a_i, b_i , c_i, d_i, e_i, f_i, g_i\}$, such
that $s \subset \Phi_i(s_i)$ (here $\Phi_i$ is a graph map). Now if
$s_i = x_iy_i$ for $x, y \in \{a,b,c,d,e,f,g\}$, take $w=xy$. Thus we
see that $\Phi_i (w)$ (here $\Phi_i$ is an automorphism) covers $s$ in
$\tau_0$. Since this is true for any segment of $l$, we conclude the
lemma by taking a nested sequence of subsegments of $l$ with edge
length in $\tau_0$ going to infinity. The fact that legal paths of
length $\leq 2$ can be closed up to legal loops of length $\leq 3$
follows from the description of the train track in <Ref>.

Recall that if $(\tau_i')_i$ converges to an arational tree $T$, then we
can identify $\D((\tau_i')_i)$ with $\D(T)$ by

Suppose $(r_i)_{i \ge 1}$ is positive and that the folding sequence
$(\tau_i')_i$ converges to an arational tree $T$. Let $w$ be any
conjugacy class
in $\mathbb F_7$ represented by a cyclic word in $\{a,b,c,d,e,f,g\}$
and their inverses and let $\lambda \in \D(T)$
correspond to a length measure $(\lambda_i)_i \in \D((\tau_i')_i)$.
\[ \lim_{i \to \infty} \norm{\Phi_i(w)}_{(T,\lambda)} = 0. \]

Under the isomorphism from $\D((\tau_i')_i) \to \D(T)$ that maps
$(\lambda_i)_i \mapsto \lambda$, the sequence $(\tau_i',\lambda_i)
\subset \cv[7]$ also converges to $(T,\lambda) \in \partial \cv[7]$.
Thus, for any $x \in \free[7]$,

\[\norm{x}_{(T,\lambda)} = \lim_{i \to \infty}\norm{x}_{(\tau_i',
\lambda_i)}.\]
In fact, the sequence $\norm{x}_{(\tau_i', \lambda_i)}$
is monotonically non-increasing. Recall that $\tau_0'$ as a marked
graph is the rose labeled by $\{a,b,c,d,e,f,g\}$. Represent $w$ by a
loop $c_w$ in $\tau_0'$. The graph $\tau_i'$ is the rose labeled by
$\{\Phi_i(a),\ldots,\Phi_i(g)\}$. Thus, the loop $c_w$ in $\tau_i'$
represents the conjugacy class $\Phi_{i}(w)$. This shows

\[\norm{\Phi_i(w)}_{(T,\lambda)}\leq
\norm{\Phi_i(w)}_{(\tau_i',\lambda_i)}\leq
\norm{w}_{\mbox{word}}\vol(\tau_i',\lambda_i)\]

where $\norm{w}_{\mbox{word}}$ is the word length of $w$. By
<Ref> the last term goes to 0.

We now come to the main statement of this section.

Suppose $(r_i)_{i \ge 1}$ is positive and that the folding sequence
$(\tau_i')_i$ converges to an arational tree $T$. Let $\tilde{\Lambda}$
be the lamination corresponding to the legal lamination $\Lambda$ of
the unfolding sequence$(\tau_i)_i$ and let $L(T)$ be the lamination
dual to $T$. Then $\tilde{\Lambda} \subseteq L(T)$. In particular, if
$T$ is non-geometric, then $\C(\Lambda) = \C(T)$.

Recall by <Ref>, the lamination dual to an arational tree is independent of the
length measure on the tree. So fix an arbitrary length
measure $\lambda \in \D(T)$ on $T$.

Let $W_3$ be the set of legal loops of length at most three in
$\{a, b, c, d, e, f, g\}$ and their inverses. By <Ref>, for every $\epsilon >0$, there exists
$I_\epsilon >0$ such that for all $i \geq I_\epsilon$,
$\norm{\Phi_i(w)_{(T,\lambda)}} < \epsilon$, for every $ w\in
W_3$. Then the bi-infinite line $(\Phi_i(w)^{-\infty},
\Phi_i(w)^{\infty})$ is in $L_{\epsilon}(T)$ for all $i \geq
I_{\epsilon}$. Therefore,
$$\bigcap_{\epsilon > 0}
\overline{\bigcup_{\substack{w \in W_3 \\ i \geq I_{\epsilon}}}
(\Phi_i(w)^{-\infty}, \Phi_i(w)^{\infty})} \subseteq
\bigcap_{\epsilon > 0} L_{\epsilon}(T). $$
By <Ref>, we conclude that $\tilde{\Lambda} \subseteq L(T)$.

If $T$ is non-geometric and arational, then it is freely indecomposable
by [39]. By <cit.>, $\C(\Lambda)
= \C(T)$.

The following is the consequence of <Ref> and <Ref>.

For a positive sequence $(r_i)_{i \ge 1}$ of integers with $r_{i+1}-r_i
\ge i$, if the folding sequence $(\tau_i')_i$ converges to a
non-geometric arational tree $T$, then $\P\C(T)$ is a 1-simplex. In
particular, $T$ is not uniquely ergodic.

§ NON-CONVERGENCE OF UNFOLDING SEQUENCE

In this section, fix a sequence $(r_i)_{i \ge 1}$ such that  $r_{i+1}-r_i
\ge i$. We will show that the corresponding unfolding sequence
$(\tau_i)_i$ does not converge to a unique point in $\partial \CV[7]$. In
fact, we will show in <Ref> that it converges to a
1-simplex in $\partial \CV[7]$.

Recall the folding and unfolding sequences $(\tau_i')_i$ and
$(\tau_i)_i$, respectively, from <Ref>.

\[
\begin{tikzcd}
\cdots
\arrow[r]
& \tau_{i+1} \arrow[r, "M_{r_{i+1}}"]
& \tau_i \arrow[r, "M_{r_i}"]
& \tau_{r_{i-1}} \arrow[r]
& \cdots \arrow[r, "M_{r_3}"]
& \tau_2 \arrow[r, "M_{r_2}"]
& \tau_1 \arrow[r, "M_{r_1}"]
& \tau_0,
\end{tikzcd}
\]

\[
\begin{tikzcd}
\cdots
& \tau_{i+1}' \arrow[l]
& \tau_i' \arrow{l}[above,pos=.4]{N_{r_{i+1}}}
& \tau_{i-1}' \arrow{l}[above,pos=.4]{N_{r_i }}
& \cdots \arrow[l]
& \tau_2' \arrow{l}[above,pos=.4]{N_{r_3}}
& \tau_1' \arrow{l}[above,pos=.4]{N_{r_2}}
& \tau_0' \arrow{l}[above,pos=.4]{N_{r_1}}.
\end{tikzcd}
\]

Here $\tau_i$ and $\tau_i'$ as marked graphs belong to the same
simplex in $\CV[7]$. Also, recall the matrices defined for all $i\ge 0$

\[P_i = \frac{1}{\kappa_B \lambda_B^{r_{i+1} }} M_{r_i} M_{r_{i+1}}
\qquad
\text{and}
\qquad
Q_i = \frac{1}{\kappa_C \lambda_C^{r_{i+1} }} N_{r_{i+1}}N_{r_i},\]

and the existence of the limiting matrices from <Ref> and

\[Y_i = \lim_{k \to \infty}P_iP_{i+2} \cdots P_{i+2k}
\qquad \text{and} \qquad Z_i = \lim_{k \to \infty}Q_{i+2k} \cdots
Q_{i+2} Q_i.
\]

For all even $2m \ge 0$,

\[ c_{2m} = \Big( \kappa_B^m \lambda_B^{r_2} \lambda_B^{r_4} \cdots
\lambda_B^{r_{2m}} \Big) \Big( \kappa_C^m \lambda_C^{r_2} \lambda_C^{r_4}
\cdots \lambda_C^{r_{2m}} \Big).
\]

Similarly, for all odd $2m+1 \ge 1$, set

\[ c_{2m+1} = \Big( \kappa_B^m \lambda_B^{r_1}\lambda_B^{r_3} \cdots
\lambda_B^{r_{2m+1}} \Big) \Big( \kappa_C^m \lambda_C^{r_1}
\lambda_C^{r_3} \cdots \lambda_C^{r_{2m+1}} \Big).
\]

Let $\ell=\ell_0 \in \R^{|E\tau_0|}$ be a positive length vector on
$\tau_0$. Then $\ell$ determines a length vector $\ell_i$ on each
$\tau_i$ given by $\ell_i = M_{r_i}^T \ldots M_{r_1}^T \ell \in
\R^{|E\tau_i|}$. We set $\ell_e^T = \ell^T Y_1$ and $\ell_o^T = \ell^T
\frac{M_{r_1}}{\lambda_B^{r_1}}Y_2$. Note that both $\ell_e$ and $\ell_o$
are positive vectors. For $\ell_e$, this follows since $\ell$ is a
positive vector and $Y_1$ is a non-negative matrix. Similarly, $\ell^T
M_{r_1}$ is positive and $Y_2$ is non-negative, so $\ell_o$ is also

We will show the sequence $(\tau_i, \ell_i)_i \subset \CV[7] $, up to
rescaling, does not have a unique limit in $\partial \CV[7]$. We start by
showing the even sequence and the odd sequence do converge, up to
scaling. More precisely:

For any positive length vector $\ell=\ell_0$ on $\tau_0$, the corresponding
even sequence $\left( \tau_{2m}, \frac{\ell_{2m}}{c_{2m}} \right)$
and odd sequence $\left( \tau_{2m+1}, \frac{\ell_{2m+1}}{c_{2m+1}} \right)$
of metric graphs converge to two points $T_e$ and $T_o$ respectively
in $\partial \CV[7]$. In fact, for any conjugacy class $x \in
\free[7]$, there exists an index $i_x \ge 0$, a vector $v_x \in
\R^{|E\tau_{i_x}'|}$, and matrices $Y_x^e$ and $Y_x^o$, such that

\[
\norm{x}_{T_e} = \ell_e^T Y_x^e v_x
\qquad \text{and} \qquad
\norm{x}_{T_o} = \ell_o^T Y_x^o v_x
\]

Let $x \in \free[7]$ be a cyclically reduced representative of its
conjugacy class. By <Ref>, there exists $i
\geq 0$ such that $x$ is legal in $\tau_{i}'$. Let $i_x$ be the
smallest index among such $i$. Then we can represent $x$ by a vector
$v_x$ in $\R^{|E\tau_{i_x}'|}$ and by the vector
$N_{r_i}\ldots N_{{i_x+1}} v_{x}$ in $\R^{|E\tau_i|}$ for $i
\geq i_x$. Thus, for all $i \ge i_x$, we have

\[
\norm{x}_{(\tau_i,\ell_i)} = \Big( \ell^T  M_{r_1} \cdots
M_{r_i} \Big) \Big( N_{r_i} \cdots N_{{i_x+1}} v_x
\Big).
\]

If $i_x$ is even, then write $i_x = 2m_x$, and set

\[
c_x^e = \kappa_C^{m_x} \lambda_C^{r_2} \lambda_C^{r_4} \cdots
\lambda_C^{r_{i_x}}
\qquad \text{and} \qquad
c_x^o = \kappa_C^{m_x} \lambda_C^{r_1} \lambda_C^{r_3} \cdots
\lambda_C^{r_{i_x-1}}. \]

If $i_x$ is odd, then write $i_x = 2m_x+1$, and set

\[
c_x^e = \kappa_C^{m_x+1} \lambda_C^{r_2} \lambda_C^{r_4}\cdots
\lambda_C^{r_{i_x-1}}
\qquad \text{and} \qquad
c_x^o = \kappa_C^{m_x} \lambda_C^{r_1} \lambda_C^{r_3} \cdots
\lambda_C^{r_{i_x}}. \]

First suppose $i_x$ is even. Then for all even $2m \ge i_x$, we have

\begin{align*}
\norm{x}_{ \left( \tau_{2m}, \frac{\ell_{2m}}{c_{2m}}\right)}
& = \frac{\norm{x}_{(\tau_{2m}, \ell_{2m})}}{c_{2m}} \\[.5em]
& = \frac{\ell^T
\Big(P_1 P_3 \cdots P_{2m-1} \Big) \Big( Q_{2m-1} \cdots
Q_{i_x+3} Q_{i_x+1} \Big) v_x}{c_x^e} \\[.5em]
&\xrightarrow{m \to \infty} \frac{\ell^T Y_1 Z_{i_x+1}
= \ell_e^T \left( \frac{Z_{i_x+1}}{c_x^e}\right) v_x
\end{align*}

and for odd $2m+1 \ge i_x$, we have

\begin{align*}
\norm{x}_{ \left( \tau_{2m+1}, \frac{\ell_{2m+1}}{c_{2m+1}}\right)}
& = \frac{\norm{x}_{(\tau_{2m+1}, \ell_{2m+1})}}{c_{2m+1}} \\[.5em]
& = \frac{\ell^T \frac{M_{r_1}}{\lambda_B^{r_1}} \Big( P_2 P_4 \cdots
P_{2m} \Big) \Big( Q_{2m} \cdots Q_{i_x+4} Q_{i_x+2} \Big) N_{i_x+1}
v_x}{c_x^o \lambda_C^{r_{i_x+1}}} \\[.5em]
& \xrightarrow{m \to \infty}
\frac{\ell^T \frac{M_{r_1}}{ \lambda_B^{r_1}} Y_2 Z_{i_x+2}
N_{{i_x+1}} v_x }{c_x^o}
\ell_o^T \left(\frac{Z_{i_x+2}}{c_x^o}\frac{N_{{i_x+1}}}{
\lambda_C^{i_x+1}} \right) v_x
\end{align*}

Now suppose $i_x$ is odd. Then for all even $2m \ge i_x$, we have

\begin{align*}
\norm{x}_{ \left( \tau_{2m}, \frac{\ell_{2m}}{c_{2m}}\right)}
& = \frac{\ell^T \Big(P_1 P_3\cdots P_{2m-1} \Big) \Big( Q_{2m-1}
\cdots Q_{i_x+3} \Big) N_{r_{i_x+1}} v_x}{c_x^e \lambda_C^{r_{i_x+1}}} \\[1em]
& \xrightarrow{m \to \infty} \ell_e^T \left( \frac{
Z_{i_x+2}}{c_x^e}\frac{N_{r_{i_x+1}}}{\lambda_C^{r_{i_x+1}}} \right) v_x.
\end{align*}

and for odd $2m+1 \ge i_x$, we have

\begin{align*}
\norm{x}_{ \left( \tau_{2m+1}, \frac{\ell_{2m+1}}{c_{2m+1}}\right)}
& = \frac{\ell^T \frac{M_{r_1}}{\lambda_B^{r_1}} \Big( P_2
P_4 \cdots
P_{2m} \Big) \Big( Q_{2m} \cdots Q_{i_x+3} Q_{i_x+1} \Big)
v_x}{c_x^o} \\[.5em]
& \xrightarrow{m \to \infty}
\ell_o^T \left(\frac{Z_{i_x+1}}{c_x^o} \right) v_x.
\end{align*}

Either way, for any conjugacy class $x$ in $\free[7]$, both

\[\norm{x}_{T_e} =
\lim_{m \to \infty} \norm{x}_{ \left( \tau_{2m},
\frac{\ell_{2m}}{c_{2m}}\right)}
\qquad \text{and} \qquad
\norm{x}_{T_o} =
\lim_{m \to \infty}
\norm{x}_{ \left( \tau_{2m+1}, \frac{\ell_{2m+1}}{c_{2m+1}}\right)}
\]

are well-defined and have the desired form.

We now want to show $T_e$ and $T_o$ are not scalar multiples of each
other. In fact, the following lemma will allow us to show that $T_e$ and
$T_o$ are the extreme points of the simplex $\P\D(T)$.

There exist two sequences $\alpha_i$ and $\beta_i$ of conjugacy classes
of elements of $\free[7]$ such that the following holds. For any
positive length vector $\ell=\ell_0$ on $\tau_0$, let $T_e$ and $T_o$
be the respective limiting trees in $\partial \CV[7]$ for $\left(
\tau_{2m}, \frac{\ell_{2m}}{c_{2m}} \right)$ and $\left( \tau_{2m+1},
\frac{\ell_{2m+1}}{c_{2m+1}} \right)$. Then

\[ \frac{||\alpha_i||_{T_o}}{||\alpha_i||_{T_e}} \xrightarrow{i \to
\infty} \infty, \qquad \text{and} \qquad
\frac{||\beta_i||_{T_o}}{||\beta_i||_{T_e}}
\xrightarrow{i \to \infty} 0.\]

Take the letter $e \in \free[7]$ and recall the automorphisms $\Phi_i$
used to define the folding and unfolding sequences. Set $x_i =
\Phi_i(e)$. For each $i$, $x_i$ is legal in $\tau_i'$ and is
represented by the vector $e_5=(0,0,0,0,1,0,0)^T$ in $\tau_i'$.

Using notation from <Ref>, set $c_{i}^e = c_{x_i}^e$ and
$c_{i}^o = c_{x_i}^e$. Note here $i$ is the smallest index such that
$x_i$ is legal in $\tau_i'$. We compare the ratio of $c_i^o$ and
$c_i^e$. Since $r_{i+1} - r_i \to \infty$, we have

\[
\frac{c_{2i}^e}{c_{2i}^o} =
\frac{\lambda_C^{r_2} \cdots \lambda_C^{r_{2i}}}
{\lambda_C^{r_1} \cdots \lambda_C^{r_{2i-1}}}
\xrightarrow{i \to \infty} \infty,
\qquad
\text{while}
\qquad
\frac{c_{2i+1}^e}{c_{2i+1}^o} =
\frac{\kappa_C}{\lambda_C^{r_1}}
\frac{\lambda_C^{r_2} \cdots \lambda_C^{r_{2i}}}
{\lambda_C^{r_3} \cdots \lambda_C^{r_{2i+1}}}
\xrightarrow{i \to \infty} 0.
\]

Recall that both $\ell_e$ and $\ell_o$ are positive and by
<Ref> the sequence $Z_i $ converges to $Z$. Since $Ze_5$
is the zero vector, by continuity of the
dot product,

\[  \lim_{i \to \infty} \ell_e^T Z_{2i+1} e_5 = \ell_e^T Z e_5 = 0
\qquad \text {and} \qquad \lim_{i \to \infty} \ell_o^T Z_{2i+1} e_5 =
\ell_o^T Z e_5 = 0. \]

Next let $N_\infty = \lim_{i \to \infty} \frac{N_{r_i}}{
\lambda_C^{r_i}}$ and recall by  <Ref> that the vector $Z
N_\infty e_5=(\star,\star,\star,0,0,0,0)$ is non-negative. Thus there
are positive constants $A$ and $B$ such that,

\[
\lim_{i \to \infty} \ell_e ^T \left (Z_{2i+2}
\frac{N_{r_{2i+1}}}{\lambda_C^{r_{2i+1}}}\right ) e_5 =
\ell_e^T  ZN_\infty e_5
= A > 0, \]

\[
\lim_{i \to \infty} \ell_o^T \left(Z_{2i+2} \frac{N_{r_{2i+1}}}{
\lambda_C^{r_{2i+1}}}\right) e_5 = \ell_o^T  Z N_\infty e_5
= B > 0.\]

Combining the above observations and the formulas for length of $x_i$
in $T_e$ and $T_o$ obtained in <Ref> we get:

\[
\frac{\norm{x_{2i}}_{T_o}}{\norm{x_{2i}}_{T_e}}
\frac{\ell_o^T \left (Z_{2i+2}
\frac{N_{r_{2i+1}}}{\lambda_C^{r_{2i+1}}}\right ) e_5}{\ell_e^T \left
( Z_{2i+1} \right ) e_5}
\frac{c_{2i}^e}{c_{2i}^o} \quad \xrightarrow{i \to \infty} \quad
\frac{A}{0} \cdot \infty
\]

\[
\frac{\norm{x_{2i+1}}_{T_o}}{\norm{x_{2i+1}}_{T_e}}
\frac{\ell_o^T \left ( Z_{2i+1} \right ) e_5}{\ell_e^T \left
(Z_{2i+2} \frac{N_{r_{2i+1}}}{ \lambda_C^{r_{2i+1}}}\right ) e_5}
\frac{c_{2i+1}^e}{c_{2i+1}^o} \quad \xrightarrow{i \to \infty} \quad
\frac{0}{B} \cdot 0
\]

Setting $\alpha_i = x_{2i}$ and $\beta_i= x_{2i+1}$ finishes the proof.

For a sequence $(r_i)_{i \ge 1}$ with $r_{i+1}-r_i \ge i$, if the
folding sequence $(\tau_i')_i$ converges to an arational tree $T$, then
for any positive length vector $\ell_0$ on $\tau_0$, the limit set in
$\partial \CV[7]$ of the rescaled unfolding sequence $(\tau_i,\ell_i)$
is always the 1-simplex $\P\D(T)$.

Since the folding $(\tau_i)'_i$ and the unfolding sequence $(\tau_i)_i$
are equal as marked graphs for all $i \ge 0$, no matter the metric,
they both visit the same sequence of simplices in $\CV[7]$. In particular,
they both project to the same quasigeodesic in $\FreeF[7]$. Thus, the
two limiting trees $T_e$ and $T_o$ of the even and odd sequences of
$(\tau_i,\ell_i)$ are length measures on $T$.

Recall $\P\D(T)$ is a $1$-simplex by <Ref>. If
neither $T_e$ nor $T_o$ are the extreme points of this simplex, then
there exist constants $c, c'>0$ such that any $x \in \free$,

\[c'\leq \frac{\norm{x}_{T_o}}{\norm{x}_{T_e}} \leq c.\]

On the other hand, if one of them, say $T_o$, is an extreme point but
$T_e$ is not, then we have a constant $c>0$ such that for any $x \in
\free$, $ \frac{\norm{x}_{T_o}}{\norm{x}_{T_e}} \leq c.$ In both the
cases, we get a contradiction to <Ref>.

§ CONCLUSION

Recall $\phi\in \Aut(\free[7])$ is the automorphism:

\[
a\mapsto b,b\mapsto c,c\mapsto ca,d\mapsto d,e\mapsto e,f\mapsto
f,g\mapsto g
\]

and $\rho\in \Aut(\free[7])$ is the rotation by 4 clicks:

\[ a\mapsto e, b\mapsto f, c\mapsto g, d\mapsto a, e\mapsto b, f\mapsto
c, g\mapsto d.\]
For any integer $r$, let $\phi_r = \rho \phi^r$. To each
sequence $(r_i)_{i \ge 0}$ of positive integers, we have an unfolding
sequence $(\tau_i)_i$ with train track map $\phi_{r_i}\from \tau_i \to
\tau_{i-1}$, and a folding sequence $(\tau_i')_i$ with train track map
$\phi_{r_i}^{-1}\from \tau_{i-1}' \to \tau_i$. By the limit set of the
unfolding sequence $(\tau_i)_i$ in $\partial \CV$ we mean the limit set
of $(\tau_i,\ell_i)$ with respect to some (any) positive length vector
$\ell_i$ on $\tau_i$.

Given a strictly increasing sequence $(r_i)_{i \geq 1}$ satisfying $r_i
\equiv i \mod 7$ and $r_i \equiv 0 \mod 3$, then the folding sequence
$(\tau'_i)_i$ converges to a non-geometric arational tree $T$.

If $(r_i)_i$ grows fast enough, that is, if $r_{i+1}-r_i \ge i$, then
$T$ is both non-uniquely ergometric and non-uniquely ergodic. Both
$\P\D(T)$ and $\P\C(T)$ are 1-dimensional simplices.

Furthermore, the limit set in $\partial \CV[7]$ of the unfolding
sequence $(\tau_i)_i$ is always the 1-simplex spanned by the two
ergodic metrics on $T$.

A sequence as in the statement exists by the Chinese remainder theorem.
The first statement follows from <Ref> and
<Ref>. Non-unique ergometricity of $T$
follows from <Ref> and
<Ref>. Non-unique ergodicity of $T$ is
<Ref>. Finally, the last statement is

§ APPENDIX

§.§ Convergence Lemma

Let $\norm{\cdot}$ denote the operator norm. Thus $\norm{Y}\geq 1$ for a
nontrivial idempotent matrix $Y$.

Let $Y$ be an idempotent matrix and $\Delta_i$, $i\geq 1$, a sequence of
matrices with $\norm{\Delta_i}\leq \frac {\epsilon}{2^i}$ for some
$\epsilon>0$. Assume also that $\epsilon \norm{Y}\leq 1/2$. Then the
infinite product
$$\prod_{i=1}^\infty (Y+\Delta_i)$$

converges to a matrix $X$ with $\norm{X-Y}\leq 2\epsilon
\Big(\norm{Y}+\norm{Y}^2 \Big)$. Moreover, the kernel of $Y$ is
contained in the kernel of $X$.

$$Y+\Sigma_k=\prod_{i=1}^k (Y+\Delta_i)$$

Then $(Y+\Sigma_k)(Y+\Delta_{k+1})=Y+\Sigma_{k+1}$ and since $Y^2=Y$
it follows that

\begin{equation}\label{eqn:convergence}
\Sigma_{k+1}=Y\Delta_{k+1}+\Sigma_k(Y+\Delta_{k+1})
\end{equation}

Multiplying on the right by $Y$ and using $Y^2=Y$ we get

and applying the norm:

$$\norm{\Sigma_{k+1}Y}\leq \norm{\Sigma_kY}+\frac
\norm{Y}^2+\norm{\Sigma_k}\frac{\epsilon}{2^{k+1}}\norm{Y}$$

By adding these for $k=1,2,\cdots,m-1$ and using $\Sigma_1=\Delta_1$
we have

\begin{align*}
\norm{\Sigma_mY}
& \leq \norm{\Sigma_1Y}+\epsilon \norm{Y}^2 \left(\frac 14+\cdots +\frac
\frac{\norm{\Sigma_1}}{4}+\cdots+\frac{\norm{\Sigma_{m-1}}}{2^m}\right)
\\
& \leq \epsilon
\Big( \norm{Y}+\norm{Y}^2 \Big) + \epsilon\norm{Y}\left(
\frac{\norm{\Sigma_1}}{4}+\cdots+\frac{\norm{\Sigma_{m-1}}}{2^m} \right)
\end{align*}

So the norms of $\Sigma_mY$ are bounded by norms of $\Sigma_i$ with
$i<m$. From <Ref> we also see that the norm of
$\Sigma_{k+1}$ is bounded by the norms of $\Sigma_kY$. Putting this
together we have

\begin{align*}
\norm{\Sigma_{k+1}}
\norm{Y}\norm{\Delta_{k+1}}+\norm{\Sigma_kY}+\norm{\Sigma_k}\norm{\Delta_{k+1}}\\
\frac{\epsilon}{2^{k+1}} \norm{Y} + \norm{\Sigma_kY} +
\frac{\epsilon}{2^{k+1}} \norm{\Sigma_k}\\
\frac{\epsilon}{2^{k+1}} \norm{Y} + \frac{\epsilon}{2}
\Big( \norm{Y}+\norm{Y}^2 \Big) + \epsilon \norm{Y} \left(
\frac{\norm{\Sigma_1}}{4} + \cdots +
\frac{\norm{\Sigma_{k-1}}}{2^k} \right) + \frac{\epsilon}{2^{k+1}} \norm{\Sigma_k} \\
\epsilon \Big(\norm{Y}+\norm{Y}^2 \Big) + \epsilon\norm{Y} \left(
\frac{\norm{\Sigma_1}}{4}+\cdots+\frac{\norm{\Sigma_{k-1}}}{2^k}
+ \frac{\norm{\Sigma_k}}{2^{k+1}}\right).
\end{align*}

Thus we have an inequality of the form

\[ \norm{\Sigma_{k+1}}\leq
a + b \left( \frac{\norm{\Sigma_1}}{4} +
\cdots + \frac{\norm{\Sigma_k}}{2^{k+1}} \right)
\]

for $a = \epsilon \Big(\norm{Y} + \norm{Y}^2 \Big)$ and $b = \epsilon
\norm{Y}$.

Set $c = 2 \epsilon \Big( \norm{Y}+\norm{Y}^2 \Big)$. Then $c \ge
\epsilon$, $a \le c/2$, and $b \leq 1/2$ by assumption. Easy induction
then shows for all $k \ge 1$,

\begin{equation}\label{eqn:sigma_k}
\norm{\Sigma_k}\leq c.
\end{equation}

This obtains the inequality $\norm{X-Y}\leq c$ from the statement, once
we establish convergence.

To see convergence we argue that the sequence of partial products
forms a Cauchy sequence. For $1<k<m$,

\begin{align*}
\prod_{i=1}^m (Y+\Delta_i) - \prod_{i=1}^k (Y+\Delta_i)=
\prod_{i=1}^{k-1} (Y+\Delta_i) \bigg(\prod_{i=k}^m (Y+\Delta_i) -
\end{align*}

By <Ref>, the norm of $\prod_{i=1}^{k-1}
(Y+\Delta_i) = Y + \Sigma_{k-1}$ is bounded by $c+\norm{Y}$. We can
apply the same estimate to the sequence starting with $Y+\Delta_k$ and
with $\epsilon$ replaced with
$\frac {\epsilon}{2^{k-1}}$ to see that

\[\norm{\prod_{i=k}^m (Y+\Delta_i) - Y} \leq \frac
{2\epsilon(\norm{Y}+\norm{Y}^2) }{2^{k-1}}\leq \frac {c}{2^{k-1}}\]

and so

\[
\norm{\prod_{i=k}^m (Y+\Delta_i) - (Y+\Delta_k)}\leq
\frac{c}{2^{k-1}}+ \frac{1}{2^k}
\]

which proves the sequence is Cauchy.

For the second statement, set $X_k = \prod_{i=k}^\infty (Y+\Delta_i)$
for $k \ge 1$. By the same estimate as above with $\epsilon$ replaced
with $\frac{\epsilon}{2^{k-1}}$, we know that $X_k$ exists and

\[
\norm{X_k - Y} \le \frac{2 \epsilon}{2^{k-1}} \Big( \norm{Y} +
\norm{Y}^2 \Big) = \frac{c}{2^{k-1}}.
\]

By definition, $X = (Y+\Sigma_k)X_{k+1}$. Suppose $v$ is a unit vector
with $Yv = 0$. Then

\begin{align*}
\norm{Xv}
& \le \norm{Y+\Sigma_k} \norm{X_{k+1} v} \\
& = \norm{Y+\Sigma_k} \norm{X_{k+1} v - Yv} \\
& \le \norm{Y+\Sigma_k} \norm{X_{k+1} - Y} \\
& \le (\norm{Y}+c) \frac{c}{2^k}.
\end{align*}

Since this is true for all $k \ge 0$, letting $k \to \infty$ yields
$X v = 0$.

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