Limit sets of unfolding paths in Outer space

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 $\mathbb{R}$-tree T. We also show that T admits exactly two dual ergodic projective currents.


Introduction
For the once-punctured torus, the Thurston compactification of the Teichmüller 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, Teichmüller space with the Teichmüller metric is no longer negatively curved [Mas75,MW95] (or even Riemannian), and the Thurston boundary is no longer its visual boundary [Ker80].More surprisingly, geodesic rays do not always converge [Len08,LLR18].
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 [Vee69,Sat75,KN76,Kea77].A topological construction was introduced in [Gab09].In [LLR18], 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 Teichmüller geodesic ray.
In this paper, we take the above construction into Culler-Vogtmann's Outer space [CV86].A Thurston-type boundary for Outer space is given by the set of projective classes of minimal, very small F n -trees [CM87, BF94, CL95, Hor17] and the action of Out(F n ) 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 F n -trees: one in terms of length measures on trees, giving rise to non-uniquely ergometric trees [Gui00], and the other in terms of currents, giving non-uniquely ergodic trees, see [CHL07].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 [Mar97].
In Outer space, the analogue of Teichmüller metric is the Lipschitz metric and that of Teichmüller geodesics are folding paths.However, unlike Teichmüller 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 Teichmüller 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 [LLR18].
Theorem 1.1.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 [NPR14] or our Section 3 for definitions and more precise statements.
Modeling the construction of [LLR18] 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 φ of F 7 by identity on the other four basis elements.Also let ρ be a finite order automorphism of F 7 that rotates the support of φ off itself.For an integer r, set φ r = ρφ r .Given a sequence (r i ) i≥1 of positive integers, define a sequence of automorphisms by From (Φ i ) i we get an unfolding sequence using the train track map induced by φ ri , and from (Φ −1 i ) 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 Section 10 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 = Φ i (A), where A is the support of φ.The A i 's are the projection of the folding sequence to the free factor complex F F 7 .By our construction, A i and A i+1 are disjoint (meaning F 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 F F 7 .Hence, by [BR15,Ham16], 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 F n 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 Section 5.

Outline
• In Section 2, we review some background material, including train track maps, Outer space, currents, length measures, and arational trees.
• In Section 3 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 [NPR14] relating the currents supported on the legal lamination with those of the unfolding sequence.
• In Section 4, we discuss our main construction to generate from a sequence (r i ) i of positive integers a sequence of automorphisms of F 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 Section 5, 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 Section 6, 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 Section 7, 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 Section 8, 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 ergodic.
• In Section 9, 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 Section 10, we collect the results to prove the main theorem.
• In Section 11, we prove a technical lemma about convergence of products of matrices.

Background
Let F n be the free group of rank n.We review some background on train track maps, Outer space, laminations, currents, arational trees and the free factor complex.

Train track maps
We recall some basic definitions from [BH92].Identify F n with π 1 (R n , * ) where R n is a rose with n petals.A marked graph G is a graph of rank n, all of whose vertices have valence at least three, equipped with a homotopy equivalence m: R n → G called a marking.
A length vector on G is a vector λ ∈ R |EG| that assigns a positive number, i.e. a length, to every edge of G.The volume of G with respect to λ 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 λ(e).
A direction d based at a vertex v ∈ 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 ∈ G.A map f : G → 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 G → T from the universal cover of G is a morphism if it is injective on open edges.To a morphism f : G → G ′ we associate the transition matrix as follows: Enumerate the (unoriented) edges e 1 , e 2 , • • • , e m of G and e ′ 1 , e ′ 2 , • • • , e ′ n of G ′ .Then the transition matrix M has size n × 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) ∩ 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 : G → G induces an outer automorphism of π 1 (G) and hence an element φ of Out(F n ).If f is a morphism then we say that f is a topological representative of φ.A topological representative f : G → 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 ∼ e 2 if (Df ) k (e 1 ) = (Df ) k (e 2 ) for some k ≥ 1.
A topological representative f : G → 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 ≥ 1.If M is primitive, i.e.M k has positive entries for some k ≥ 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 λ > 1, called the stretch factor of f .
If σ is a path (or a circuit) in G, we denote by [σ] the reduced path homotopic to σ (rel 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 [BH92,Lemma 3.4] showing that INPs have exactly one illegal turn, and for a more involved version see [BFH97] (some details can also be found in [KL14, Proposition 3.27, 3.28]).We will need it for the proof of Lemma 4.8 and include a proof here.
Lemma 2.1.Let h: G → G be a train track map with a primitive transition matrix.There exists a constant R > 0 such that for any edge path γ, either, 1. the number of illegal turns in [h R (γ)] is less than that of γ, or, 2. γ = u 1 v 1 u 2 v 2 . . .u n where each u i is a legal subpath, possibly degenerate, and each is a periodic INP.
Proof.Let λ > 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 λ.It goes back to the work of Thurston (see [Coo87]) that there is a constant BCC(h), called the bounded cancellation constant for h, such that if αβ is a reduced edge path then [h(α)][h(β)] 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)/(λ − 1).
Here is the significance of C. To fix ideas let us assume that γ has only one illegal turn, so γ = αβ with both α, β legal.Say α has length |α|= C + ǫ > C. Then h(α) has length λ|α| and after cancellation with h(β) the length is ≥ λ|α|−BCC(h) = |α|+λǫ.Thus assuming [h i (γ)] 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 enough.
We now prove the lemma for paths γ = αβ with one illegal turn, and with α, β 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 (γ)] = α i β i has one illegal turn (with α i , β i legal) for i = 1, 2, • • • , 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 (γ)] and [h j (γ)] are the same (if α i or β i has length < C this means α i = α j or β i = β j ).We can lift h j−i and γ to the universal cover of the graph and arrange that (the lift of) γ and [h j−i (γ)] have the same illegal turn.Thus h j−i maps the terminal C-segment of α i (or α i itself) over itself (by the above calculation) and therefore fixes a point in α i , and similarly for β i .The subpath of [h i (γ)] between these fixed points is a periodic INP, proving the lemma in the case γ has one illegal turn.
The general case is similar.Write γ = γ 1 γ 2 • • • γ s with all γ k legal and with the turn between γ k and γ k+1 illegal.Also assume that [h i (γ)] has the same number of illegal turns for i = 1, s with all γ i k 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 (γ)] and in [h j (γ)] are the same.This gives fixed points of h j−i in γ i k and γ i k+1 and these fixed points split γ into periodic INPs and legal segments, as claimed.
We will use the lemma in the situation that h has no periodic INPs, in which case the conclusion is that whenever γ is not legal, then [h R (γ)] has fewer illegal turns than γ.

Outer space and its boundary
An F n -tree is an R-tree with an isometric action of F n .An F n -tree T has dense orbits if some (every) orbit is dense in T .An F n -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 [CV86].
Unprojectivized Outer space, denoted by cv n , is the set of free, minimal, and simplicial F ntrees.By considering the quotient graphs, cv n is also equivalently the set of marked metric graphs, i.e. the set of triples (G, m, λ), where G is a graph of rank n with all valences at least 3, m: R n → G is a marking, and λ is a positive length vector on G.By [CM87], the map of cv n → R Fn given by T → ( g T ) g∈Fn , where g T is the translation length of g in T , is an inclusion.This endows cv n with a topology.The closure cv n in R Fn is the space of very small F n -trees [BF94,CL95].The boundary ∂cv n = cv n − cv n consists of very small trees that are either not free or not simplicial.
Culler Vogtmann's Outer space, CV n , is the image of cv n in the projective space PR Fn .Elements in CV n can also be described as free, minimal, simplicial F n -trees with unit covolume.Topologically, CV n 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 CV n of CV n in PR Fn is compact and the boundary ∂CV n = CV n − CV n is the projectivization of ∂cv n .
The spaces cv n and CV n and their closures are equipped with a natural (right) action by Out(F n ).That is, for Φ ∈ Out(F n ) and T ∈ cv n the translation length function of T Φ on F n is g T Φ = φ(g) T , where φ is any lift of Φ to Aut(F n ).

Laminations, currents and non-uniquely ergodic trees
In [BFH00], 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 [CHL08a].
Let ∂F n denote the Gromov boundary of F n and let ∆ be the diagonal in which parametrizes the space of unoriented bi-infinite geodesics in a Cayley graph of F n .By an (algebraic) lamination, we mean a non-empty, closed and F n -invariant subset of ∂ 2 F n .
Associated to T ∈ cv n is a dual lamination L(T ), defined as follows in [CHL08b].For ǫ > 0, let so L ǫ (T ) is a lamination and set L(T ) = ǫ>0 L ǫ (T ).Elements of L(T ) are called leaves.For trees in cv n , L(T ) is empty.
A current is an additive, non-negative, F n -invariant function on the set of compact open sets in ∂ 2 F n .Equivalently, it is an F n -invariant Radon measure on the σ-algebra of Borel sets of ∂ 2 F n .Let Curr n denote the space of currents, equipped with the weak* topology.The quotient space of PCurr n of projectivized currents (i.e.homothety classes of nonzero currents) is compact.
For µ ∈ Curr n , let supp(µ) ⊂ ∂ 2 F n denote the support of µ, which is in fact a lamination.For T ∈ cv n and µ ∈ Curr n , if supp(µ) ⊆ L(T ), then we say µ is dual to T .Denote by C(T ) the convex cone of currents dual to T and by PC(T ) the set of projective currents dual to T .PC(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 [CH16], the authors show that if T ∈ ∂cv n has dense orbits, then PC(T ) is the convex hull of at most 3n − 5 projective classes of ergodic currents dual to T .
In [KL09], Kapovich and Lustig established a length pairing, •, • , between cv n and the space of measured currents Curr n .They also showed in [KL10, Theorem 1.1] that for T ∈ cv n and µ ∈ Curr n , T, µ = 0 if and only if µ is dual to T .
Given two trees T and T ′ , we say a map h:

Theorem 2.2 ([CHL07]
).Let T, T ′ ∈ ∂CV n be two trees with dense orbits.The following are equivalent: • There exists an F n -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 [Pau95], a length measure was introduced for R-trees.See [Gui00] for details.
A length measure on an F n -tree T is a collection of finite Borel measures λ I for every compact interval I in T such that if J ⊂ I, then λ J = (λ I )| J .We require the length measure to be invariant under the F n action.The collection of the Lebesgue measures of the intervals of T is F n -invariant, and this will be called the Lebesgue measure of T .A length measure λ is nonatomic or positive if every λ 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 , . . ., g m ∈ F n , such that I ⊂ g i J and g i J ∩ g i+1 J is nondegenerate, then every nonzero length measure is positive (in fact, the condition of mixing [Gui00] suffices).
Let D(T ) be the cone of F n -invariant length measures on T , with projectivization PD(T ), i.e. the homothety classes of F n -invariant length measures on T .PD(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 [Gui00, Corollary 5.2, Lemma 5.3] and D(T ) is naturally a subset of ∂cv n .In fact, Proof.Let λ ∈ D(T ) be a length measure on T .Consider the pseudo-metric d λ on T where d λ (x, y) = λ([x, y]) for x, y ∈ T .In fact, since T is indecomposable, d λ is a metric on T .For the converse, let T ′ ∈ X T .Then the pull back of Lebesgue measure on T ′ under identity map id: T → 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 ∈ cv n and a free factor H of F n , let T H denote the minimal H-invariant subtree of T (this tree is unique unless H fixes an arc).A tree T ∈ ∂cv n is arational if every proper free factor H of F n has a free and simplicial action on T H .By [Rey12] 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 ⊂ ∂CV n denote the set of arational trees with the subspace topology.Using Lemma 2.3, define an equivalence relation ∼ on AT by "forgetting the metric", that is, T ∼ T ′ if T ′ ∈ PD(T ), and endow AT /∼ with the quotient topology.The following lemma is implicit in [Gui00] and we include a proof for completeness.
Proof.If T ∼ T ′ then the identity map id: T → T ′ is an alignment preserving bijection.Therefore, by Theorem 2.2, L(T ) = L(T ′ ).
If L(T ) = L(T ′ ), then by Theorem 2.2 there is an alignment preserving bijection f : T → T ′ .Pulling back the Lebesgue measure on T ′ induces a length measure on T , and the corresponding metric d µ on T is isometric to T ′ , so T ′ ∼ T .
The free factor complex F F n is a simplicial complex whose vertices are given by conjugacy classes of proper free factors of F n and a k-simplex is given by a nested chain 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 [BF14a], the metric space F F n is Gromov hyperbolic.The Gromov boundary of F F n was identified with AT /∼ in [BR15] and [Ham16].
There is a projection map π: CV n → F F n defined as follows [BF14a, Section 3]: for G ∈ CV n , π(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 F Fn (π(G)) ≤ K for some universal K.Note that if G, G ′ belong to the same open simplex of CV n , then π(G) = π(G ′ ), so the projection of a simplex of CV n has uniformly bounded diameter.

Folding and unfolding sequences
In this section we introduce (un)folding sequences and review some work of Namazi-Pettet-Reynolds [NPR14].
A folding/unfolding sequence is a sequence Equivalently, a sequence as above is called a folding/unfolding sequence, if there exists a train track structure on each G i and 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≤i≤b is a sequence (λ i ) a≤i≤b , where λ i ∈ R |EGi| is a length vector on G i , and for a ≤ i < b, In this way, f i restricts to an local isometry on every edge of G i .When b < ∞, 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 PD((G i ) i ) its projectivization.Observe that the dimension of D( A current for a folding/unfolding sequence (G i ) a≤i≤b is a sequence (µ i ) a≤i≤b , where µ i ∈ R |EGi| is a length vector on G i (but thought of as a vector of thicknesses of edges), and for a ≤ i < b, we require 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 PC((

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 Let Gi be the universal cover of G i , and let fi be a lift of f i .For any positive length measure (λ i ) i ∈ D((G i ) i ), we can realize ( Gi , λi ) i as a sequence in cv n , which can be "filled in" by a folding path in cv n (see [BF14a] for details on folding paths).In particular, ( Gi , λi ) i always converges to a point T ∈ ∂cv n .Furthermore, we have morphisms With respect to the length measure λi , fi and h i restrict to isometries on edges [BR15, Lemma 7.6].Let (U i ) i be the sequence of open simplices CV n associated to the sequence (G i ) i .Recall the projection map π: CV n → F F n is coarsely well-defined on simplices of CV n .We will say the folding sequence (G i ) i converges to an arational tree Proposition 3.1.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 D(T ).
Proof.Fix a positive length measure (λ i ) i ∈ D((G i ) i ) and let T ∈ ∂cv n be the limiting tree of ( Gi , λi ) with corresponding morphism h i : Gi → T .Recall from Section 2.5 that if T is arational, then we can identify D(T ) with the subspace of F n -metrics on T in ∂cv n .We will let λ ∈ D(T ) be a length measure, and T λ its image in ∂cv n . By Conversely, for any positive length measure λ ′ ∈ D(T ), we can use the morphism h i to pull back λ ′ from T to a length measure λ ′ i on Gi .The fact that Remark 3.2.A more general statement of Proposition 3.1 which doesn't involve the assumption that T is arational can be found in [NPR14, Proposition 5.4], 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 [NPR14] first.
Consider an unfolding sequence of marked graphs of rank n Denote the composition Use the marking on G 0 to identify ) is a lamination Λ.We denote by C(Λ) the convex cone of currents supported on Λ, with projectivization PC(Λ).
An invariant sequence of subgraphs is a sequence of nondegenerate (i.e.not forests) proper subgraphs H i ⊂ G i such that f i restricts to a morphism H i → H i−1 .We will need the following theorem from [NPR14].
Theorem 3.3 (Theorem 4.4 [NPR14]).Given an unfolding sequence (G i ) i≥0 without an invariant sequence of subgraphs and with legal lamination Λ, then there is a natural linear isomorphism between C((G i ) i ) and C(Λ).

Main setup
In this section, we will construct an unfolding sequence (τ i ) i and a folding sequence (τ ′ 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.Using θ and ϑ to also denote the corresponding graph maps, and using the convention that a also denotes the initial direction of the oriented edge a, while ā denotes the terminal direction, we have Proof.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 F n [FH18], so one can take a suitable power and check for INPs (though there are more efficient ways, see [Kap19]).Coulbois' train track package [Cou] for the mathematics software system Sage [Sag] computes periodic INPs of train track maps.Now let φ ∈ Aut(F 7 ) be the automorphism:

The automorphisms
and ρ ∈ Aut(F 7 ) be the rotation by 4 clicks: Thus φ is the extension of θ by identity, and ρ rotates the support of φ off itself.

Asymptotics of transition matrices
Let θ, ϑ, φ r , ψ r be the maps defined in the last section.We now analyze the behavior of the transition matrices M r and N r for φ r and ψ r respectively.Lemma 4.5.Let B be the transition matrix for θ, with Perron-Frobenius eigenvalue λ B .There exists a constant κ B > 0 such that if r, s − r → ∞, then where Y is an idempotent matrix of the form Y = u pu qu 0 0 0 0 with u = 0, u 1 , u 2 , u 3 , 0, 0, 0 T and p, q > 0, Proof.There exists a Perron-Frobenius eigenvector x = (x 1 , x 2 , x 3 ) T for B and constants p, q > 0 such that We have The square of the limiting matrix above has a nonzero block where P is of the form (px 1 + qx 2 )P, and zero elsewhere, so we set We have a similar statement for the matrices N r .
Lemma 4.6.Let C be the transition matrix for ϑ = θ −1 , with Perron-Frobenius eigenvalue λ C .There exists a constant κ C > 0 such that if r, s − r → ∞, then where Z is an idempotent matrix of the form T and p, q > 0, Proof.We observe that the matrix N s N r has shape that is the transpose of the matrix in Lemma 4.5, with powers of the PF matrix C forming the nonzero blocks: For future reference, we also record the following.Let P = lim r→∞ B r /λ r B and Lemma 4.7.There are p, q, r, s > 0 such that M ∞ Y = y py qy 0 0 0 0 with y = 0, 0, 0, 0, y 1 , y 2 , y 3 T , (y 1 , y 2 , y 3 ) T is a Perron-Frobenius eigenvector of B; and 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≥1 and the sequence of automorphisms φ ri , with transition matrix M ri and φ −1 ri = ψ ri with transition matrix N ri .Let τ i → τ i−1 (resp.τ ′ i−1 → τ ′ i ) be the train track map induced on the rose by φ ri (resp.ψ ri ) as given by Lemma 4.3 (resp.Lemma 4.4).Thus we have an unfolding sequence and a folding sequence Here, τ 0 is a rose with petals labeled by elements in {a, b, c, d, e, f, g} and hence for i ≥ 1, τ i is a rose labeled by {Φ i (a), . . ., Φ i (g)}.Also, τ ′ 0 is a rose labeled by {a, b, c, d, e, f, g}, so τ ′ i is also a rose labeled by {Φ i (a), . . ., Φ i (g)}.Thus, for every i ≥ 0, τ i and τ ′ 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 Proposition 5.10 to show that the limit tree of the folding sequence is non-geometric.

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 (τ ′ i ) i to the free factor complex F F 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≥1 , recall that Φ i = φ r1 φ r1 • • • φ ri , where φ r = ρφ r .For convenience, also set Φ 0 = id.We have the folding sequence where τ ′ i is a rose labeled by {Φ i (a), • • • , Φ i (g)}, and ψ r = φ −1 r .From the markings, we can associate τ ′ i to an open simplex U i in CV 7 .Consider a sequence of free factors A i ∈ π(U i ), where π: CV 7 → F F 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 Lemma 5.3 which is the main goal of this section.
We now consider the following explicit sequence of free factors.Let A 0 = d, e, f be the free factor in F 7 , and define Note that for any r, s, t > 0, the following holds: Thus, for any sequence (r i ) i , We say two free factors A and A ′ are disjoint if (possibly after conjugating) F n = 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.
Lemma 5.1.For any sequence (r i ) i≥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 disjoint.
Proof.We see from Equation (1) that the statement of the lemma holds for A 0 , A 1 , A 2 and A 3 .Now for i ≥ 1 and k ∈ {1, 2, 3}, by Equation (2), the pair A i , A i+k differs from A 0 , A k by the automorphism Φ i , whence the lemma.
Recall the transition matrix M r for φ r , and the 3 × 3 matrix B whose power B r forms a block of M r .For each i ≥ 1, let M i = M i mod 2. By a simple computation, we see that B 7 = I mod 2. Thus, when i = j mod 7, M i = M j .We have the following lemma.
Proof.For any i ≥ 1 and k ≥ 0, let Abelianizing and reducing mod 2, we have A 0 ≡ V 0 , and B i+k ≡ M i • • • M i+k V 0 .Thus, by Lemma 5.2, the sequence {A 0 , B i , . . ., 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 ≡ i mod 7, so that M ri = M i for all i.Let A i = Φ i A 0 = φ r1 • • • φ ri A 0 .Set Φ 0 = id.For any i ≥ 1, by applying the automorphism Φ −1 i−1 , the sequence of free factors {A i−1 , . . ., A i+107 } is isomorphic to the sequence The latter sequence after abelianization and reducing mod 2 is equivalent to the sequence {A 0 , B i , . . ., B i+107 }.Thus {A i−1 , . . ., A i+107 } cannot be contained in the same factor or be disjoint from a common factor.

Subfactor projection
We will now use subfactor projection theory originally introduced in [BF14b] and further developed in [Tay14] 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 ∈ CV n and a rank ≥ 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 ∈ CV(A).Denote by π A (G) := π(A|G) ⊂ F (A) the projection of A|G to F (A).Here CV(A) is the Outer space of the free group A and F (A) is the corresponding free factor complex.
Recall two free factors A and B are disjoint if they are distinct vertex stabilizers of a free splitting of F n .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:  2. If rank(A) ≥ 2, B and C meet A, and B is compatible with C, then 3. If A and B overlap, have rank at least 2, and C meets both, then Theorem 5.5 (Bounded geodesic image theorem [Tay14]).For n ≥ 3, there exists D ′ ≥ 0 such that if A is a free factor with rank(A) ≥ 2 and γ is a geodesic of F F n with each vertex of γ having a well-defined projection to F (A), then diam(π A (γ)) ≤ D ′ .
We now prove the following lemma.
Lemma 5.6.For any K > 0, there exists a constant r = r(K) such that for any sequence (r i ) i≥1 , if r i ≥ r for all i, then the following statements hold: 1.For any j ≥ 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.
2. Let D be the constant of Theorem 5.4.If K > 3D, then for any i < j < k, if j − i ≥ 2 and k − j ≥ 2, the projections of A i and A k to F (A j ) are defined and have distance at least K − 2D.
Proof.Recall for any r, φ r = ρφ r , where φ restricts to a fully irreducible outer automorphism of a, b, c .In particular, φ acts as a loxodromic isometry of the free factor complex F ( a, b, c ), Thus, for any K, there exists r = r(K) such that for all s ≥ r, the distance between φ s ( b, c ) is at least K + 2D away from a, b in F ( a, b, c ).Now consider any sequence (r i ) i with r i ≥ r for all i.By Lemma 5.1 and Theorem 5.4, the projections of A j−2 and A j+2 to F (A j ) are defined.Moreover, by Equation (2), 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 = d, e, f and , f, g ).Note that the rotation ρ sends the free factor a, b, c to A 2 , thus inducing an isometry from F ( a, b, c ) to F (A 2 ).The projection of A 0 to F (A 2 ) is D-close to the factor e, f = ρ( a, b ), and the projection of φ rj−1 (A 3 ) to F (A 2 ) is D-close to the factor ρφ rj−1 ( b, c ).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 ≥ 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 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 Lemma 5.1 we have 4 ≤ 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 ≥ K − 2D > D. Using statement 2 of Theorem 5.4, 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 By the first statement of the lemma, we have d Aj (A j−2 , A j+2 ) ≥ K.We now claim that d Aj (A j+2 , A k ) ≤ D. If k = j + 3, then A j+2 and A k are disjoint, and the claim holds by statement 2 of Theorem 5.4.If k ≥ 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 Aj+2 (A j , A k ) ≥ K − 2D > D. Now, the claim follows by the third statement of Theorem 5.4.By the same argument, we also see that d Aj (A i , A j−2 ) ≤ D. We now conclude d Aj (A i , A k ) ≥ K − 2D by the triangle inequality.
We are now ready to prove the main results of this section.
Proposition 5.7.There exists R > 0 such for any sequence (r i ) i≥1 , if r i ≥ R, and r i ≡ i mod 7, then the sequence (A i ) i≥0 is a quasi-geodesic in F F 7 .
Proof.Let D be the constant of Theorem 5.4 and let D ′ be the constant of Theorem 5.5.Fix K = 4D + D ′ .Let R = r(K) be the constant of Lemma 5.6.Let (r i ) i≥1 be any sequence with r i ≥ R and r i ≡ 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 For every j ∈ {i + 2, . . ., k − 2}, there exists a free factor in γ that is compatible with A j .Indeed, if every free factor in γ meets A j , then by Theorem 5.5, projection of γ to A j will be well-defined and has diameter bounded by D ′ .However, by Lemma 5.6, 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 γ compatible with at least 110 free factors among {A i+2 , . . ., A k−2 }.By Lemma 5.3, it is not possible for B to be compatible with 109 consecutive A j 's.Therefore, it must be possible to find i In particular, π A j ′ (B) is defined.By Lemma 5.6, A i ′ , A k ′ also have well-defined projections to Recall that F F n 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 π(U i ) converges to [T ] ∈ ∂F F n , where U i is the open simplex in in CV n associated to G i .We have the following corollary.
Corollary 5.8.Given any strictly increasing sequence (r i ) i≥1 satisfying r i ≡ i mod 7, the folding sequence (τ ′ 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 [BF95].Definition 5.9 (Geometric tree).[BF94, LP97] Let X be a band complex and T a G = π 1 (X)tree.A resolution f : X → T is exact if for every G-tree T ′ and equivariant factorization 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 [BF94, Proposition 3.6].
Proposition 5.10.For any strictly increasing sequence (r i ) i≥1 , if the corresponding folding sequence (τ ′ i ) i converges to an arational tree T , then T is not geometric.
Proof.Let ψi : τ ′ i−1 → τ ′ i be a lift of the train track map to the universal covers fixing a base vertex.Pick a length measure on (τ ′ i ) i so we get a folding sequence τ ′ −→ • • • in cv 7 that converges to T .Recall that there are morphisms h i : τ ′ i → T such that h i = h i+1 ψi+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 : X → T .We will show that the resolution factors through τ ′ i for sufficiently large i.This will imply T is not geometric.
Let Γ be the underlying real graph of X (disjoint union of metric arcs) with preimage Γ in X.We may assume f embeds the components of Γ.A vertex v of X is either a vertex of Γ or a corner of a band or a 0-cell of X.For every such vertex v choose a point f 0 (v) ∈ τ0 so that f 0 is equivariant and f = h 0 f 0 on the vertices of X.
An edge in X is either a subarc of Γ or a vertical boundary component of a band or a 1-cell in X.Up to the action of F 7 , there are only finitely many edges.Using Lemma 4.8, we can find i > 0 such that for every edge e in X, the edge path in τ ′ i joining the two vertices of ψi i that sends edges to legal paths (or points) and is constant on the leaves.Thus but h i is not an isometry.This shows T is non-geometric.
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 ǫ ri -neighborhood of the 1-simplex [v 567 B , v 234 B ].Moreover, given any ǫ > 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 ǫ-neighborhood of the 1-simplex with endpoints M ij (v 567 B ) and M ij (v 234 B ). Thus each S i is the nested intersection of simplices of dimension ≤ 6 such that for all ǫ > 0 they are eventually all contained in the ǫ-neighborhood of a 1-simplex with definite distance between the endpoints.This proves the Proposition.
We now present a more detailed proof of Proposition 6.1.For a sequence of integers (r i ) i≥1 such that r i , r i+1 − r i → ∞, by Lemma 4.5 (P i ) i converges to an idempotent matrix Y .Let ∆ i = Y − P i and let ||Y || be the operator norm.Lemma 6.2.Let (r i ) i≥1 be a sequence of positive integers such that r i+1 − r i ≥ i.Then there exists an I ≥ 1, such that for all i ≥ I, B be the modulus of the three eigenvalues of B; we have λ B ∼ 1.46 and where the two terms comes from the two blocks in P i .For the last inequality, note that µ < 1 < λ and r i are positive integers.Therefore, µ ri+1 < 1 < λ ri .Now we claim that there exists an I ≥ 1, such that for all i ≥ I, We only need to show that the sequence ri+1−ri i+1 is eventually increasing.Indeed, by assumption, r i+1 − r i ≥ i, so Since i/(i + 1) is an increasing sequence, it follows that our sequence is also increasing.
The following lemma is a consequence of Lemma 4.5 and Lemma 11.1.
Lemma 6.3.Let (r i ) i≥1 be a sequence of positive integers such that r i+1 − r i ≥ i, Y be the idempotent matrix of Lemma 4.5 and M ∞ = lim r→∞ M r /λ r B .Then the following statements hold.
(1) For all i ≥ 1, the sequence of matrices The kernel of Y is a subspace of the kernel of Y i for all i ≥ 1.
(3) For all i ≥ 1, Y i (e 1 ) = 0 with non-negative entries and Y i (e 2 ) and Y i (e 3 ) are positive multiples of Y i (e 1 ).
(4) For all i ≥ 1, M ri Y i+1 (e 1 ) = 0 with non-negative entries, and M ri Y i+1 (e 1 ) and Y i (e 1 ) are not scalar multiples of each other.
(5) Projectively, Proof.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 ≥ I be such that i = i 0 (mod 2) and observe that By assumption By Lemma 6.2, there exists I ≥ 1 such that for all i ≥ I, ∆ i ≤ 1 2•2 i .Also, choose I sufficiently large so that 1 2 I Y ≤ 1/2.Then, by Lemma 11.1, for all i ≥ I, the sequence For (2), it again suffices to show the statement is true for all sufficiently large i, and the statement holds for all i ≥ I by Lemma 11.1.
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 ) = 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 Y i (e 1 ) − Y (e 1 ) ≤ 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 ri Y i+1 (e 1 ) = 0 with non-negative entries since Y i+1 (e 1 ) is so by (3).To see that M ri 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 ∞ = lim r→∞ M r /λ r B .By Lemma 4.5 and Lemma 4.7, Y (e 1 ) and M ∞ Y (e 1 ) are orthogonal.Since r i → ∞, we can make and By Lemma 6.3 (3) -(5), • p i and q i are well defined and distinct. • , and 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, • • • , 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: We may assume that j − 1 = i + 2m, so M ij breaks up into pairs, i.e. for all k, Let ǫ > 0 be arbitrary.Choose δ > 0 such that for any vector u ∈ R 7 + and any v ∈ Y i (e k ), ) ≤ ǫ.Now by Lemma 6.3, we can choose J sufficiently large so that whenever i + 2m ≥ J, then Now we may assume that j − 3 ≥ J.Then, • For k = 1, 2, 3, we have • For k = 7, we have M ij (e 7 ) = M i,j−2 (e 1 ), so [M ij (e 7 )] = [P i • • • P j−3 (e 1 )] is ǫ-close to [p i ] by the same reasoning as the previous bullet point.
We have shown that for any ǫ, the vertices of the simplex S i,j come ǫ-close to p i and q i for all sufficiently large j.Since S i,j+1 ⊂ S i,j and S i = 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 (τ i ) i≥0 where M ri is the transition matrix of the train track map φ ri : τ i → τ i−1 .Let Λ be the legal lamination of (τ i ) i≥0 .Corollary 6.4.If (r i ) i≥1 is a positive sequence with r i+1 − r i ≥ i, then PC(Λ) is a 1-simplex.
Proof.In light of Theorem 3.3, it is enough to show PC((τ i ) i ) is a 1-simplex.For each i ≥ 0, we have a well-defined projection The image of the projection is S i+1 , which is always a 1-simplex by Proposition 6.1.Therefore, PC((τ 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≥1 grows sufficiently fast, then the set of projectivized length measures PD((τ ′ i ) i ) on the folding sequence (τ ′ i ) i ) is a 1-simplex.By Proposition 3.1, if (τ ′ i ) i converges to an arational tree T , then PD(T ) is also a 1-simplex in ∂CV 7 .Recall that N r is a 7 × 7 matrix of the block form 0 C r I 0 where I is the 4 × 4 identity matrix, and C is the transition matrix of ϑ.The transpose of N r has the same shape as M r .Therefore, the same theory from Section 6 holds true.For brevity, we record only the essential statements that will be used later and omit all proofs from this section.
Let λ C be the Perron-Frobenius eigenvalue of C. Let κ C be the constants of Lemma 4.6.Given a sequence (r i ) i , define for each i ≥ 1 Lemma 7.1.Given a sequence (r i ) i≥1 of positive integers such that r i+1 − r i ≥ i.Then for all i ≥ 1, the sequence of matrices k=0 converges to a matrix Z i .Furthermore, lim i→∞ Z i = Z, where Z is the idempotent matrix of Lemma 4.6.

Non-uniquely ergodic tree
In this section we relate the legal lamination Λ associated to the unfolding sequence (τ i ) i defined in Section 6 and the limiting tree T of the folding sequence (τ ′ i ) i defined in Section 5, to show that T is not uniquely ergodic.
Recall the automorphism Φ i = φ r1 • • • • φ ri , with Φ 0 = id.We also use Φ i to denote the induced graph map from τ i to τ 0 .If each τ i and τ ′ 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 Φ i (x) for x ∈ {a, b, c, d, e, f, g} as a word in F 7 = a, b, c, d, e, f, g = π 1 (τ 0 ) = π 1 (τ ′ 0 ).We denote x 0 as above simply by x.Lemma 8.1.If (r i ) i≥1 is positive, then for any length measure Proof.The composition ψ ri ψ ri−1 ψ ri−2 : τ ′ i−3 → τ ′ i has the property that the preimage of every point of τ ′ i consists of at least two (in fact, many more) points of τ ′ i−3 and so the λ i -volume of Lemma 8.2.Suppose (r i ) i≥1 is positive.Let Λ be the legal lamination of the unfolding sequence (τ i ) i .Then every leaf in Λ is obtained as a limit of a sequence {Φ i (w)} i , where w is a legal word in τ 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 ≤ 3.
Proof.Let l be a leaf of Λ realized as a bi-infinite line in τ 0 and let s be any subsegment of l, with combinatorial edge length ℓ s > 0 in τ 0 .By definition, for every i there is a bi-infinite legal path l i in τ i such that l = Φ i (l i ).Let i = i(s) ≥ 0 such that the edge length of x i in τ 0 under the graph map Φ i is ≥ ℓ s for all x ∈ {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 ⊂ Φ i (s i ) (here Φ i is a graph map).Now if s i = x i y i for x, y ∈ {a, b, c, d, e, f, g}, take w = xy.Thus we see that Φ i (w) (here Φ i is an automorphism) covers s in τ 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 τ 0 going to infinity.The fact that legal paths of length ≤ 2 can be closed up to legal loops of length ≤ 3 follows from the description of the train track in Lemma 4.3.
Recall that if (τ ′ i ) i converges to an arational tree T , then we can identify D((τ ′ i ) i ) with D(T ) by Proposition 3.1.Lemma 8.3.Suppose (r i ) i≥1 is positive and that the folding sequence (τ ′ i ) i converges to an arational tree T .Let w be any conjugacy class in F 7 represented by a cyclic word in {a, b, c, d, e, f, g} and their inverses and let λ ∈ D(T ) correspond to a length measure Proof.Under the isomorphism from D((τ ′ i ) i ) → D(T ) that maps (λ i ) i → λ, the sequence (τ ′ i , λ i ) ⊂ cv 7 also converges to (T, λ) ∈ ∂cv 7 .Thus, for any x ∈ F 7 , In fact, the sequence x (τ ′ i ,λi) is monotonically non-increasing.Recall that τ ′ 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 τ ′ 0 .The graph τ ′ i is the rose labeled by {Φ i (a), . . ., Φ i (g)}.Thus, the loop c w in τ ′ i represents the conjugacy class Φ i (w).This shows Φ i (w) (T,λ) ≤ Φ i (w) (τ ′ i ,λi) ≤ w word vol(τ ′ i , λ i ) where w word is the word length of w.By Lemma 8.1 the last term goes to 0.
We now come to the main statement of this section.
Proposition 8.4.Suppose (r i ) i≥1 is positive and that the folding sequence (τ ′ i ) i converges to an arational tree T .Let Λ be the lamination corresponding to the legal lamination Λ of the unfolding sequence(τ i ) i and let L(T ) be the lamination dual to T .Then Λ ⊆ L(T ).In particular, if T is non-geometric, then C(Λ) = C(T ).
Proof.Recall by Lemma 2.4, the lamination dual to an arational tree is independent of the length measure on the tree.So fix an arbitrary length measure λ ∈ 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 Lemma 8.3, for every ǫ > 0, there exists I ǫ > 0 such that for all i ≥ I ǫ , Φ i (w) (T,λ) < ǫ, for every w ∈ W 3 .Then the bi-infinite line By Lemma 8.2, we conclude that Λ ⊆ L(T ).
If T is non-geometric and arational, then it is freely indecomposable by [Rey12].
The following is the consequence of Proposition 8.4 and Corollary 6.4.
Corollary 8.5.For a positive sequence (r i ) i≥1 of integers with r i+1 − r i ≥ i, if the folding sequence (τ ′ i ) i converges to a non-geometric arational tree T , then PC(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≥1 such that r i+1 −r i ≥ i.We will show that the corresponding unfolding sequence (τ i ) i does not converge to a unique point in ∂CV 7 .In fact, we will show in Corollary 9.3 that it converges to a 1-simplex in ∂CV 7 .
Recall the folding and unfolding sequences (τ ′ i ) i and (τ i ) i , respectively, from Section 4.
Here τ i and τ ′ i as marked graphs belong to the same simplex in CV 7 .Also, recall the matrices defined for all i ≥ 0 and the existence of the limiting matrices from Lemma 6.3 and Lemma 7.1 For all even 2m ≥ 0, Similarly, for all odd 2m + 1 ≥ 1, set .
Let ℓ = ℓ 0 ∈ R |Eτ0| be a positive length vector on τ 0 .Then ℓ determines a length vector ℓ i on each τ i given by ℓ Y 2 .Note that both ℓ e and ℓ o are positive vectors.For ℓ e , this follows since ℓ is a positive vector and Y 1 is a non-negative matrix.Similarly, ℓ T M r1 is positive and Y 2 is non-negative, so ℓ o is also positive.
We will show the sequence (τ i , ℓ i ) i ⊂ CV 7 , up to rescaling, does not have a unique limit in ∂CV 7 .We start by showing the even sequence and the odd sequence do converge, up to scaling.More precisely: Lemma 9.1.For any positive length vector ℓ = ℓ 0 on τ 0 , the corresponding even sequence τ 2m , ℓ2m c2m and odd sequence τ 2m+1 , ℓ2m+1 c2m+1 of metric graphs converge to two points T e and T o respectively in ∂CV 7 .In fact, for any conjugacy class x ∈ F 7 , there exists an index i If i x is even, then write i x = 2m x , and set .
If i x is odd, then write i x = 2m x + 1, and set C .First suppose i x is even.Then for all even 2m ≥ i x , we have and for odd 2m + 1 ≥ i x , we have Now suppose i x is odd.Then for all even 2m ≥ i x , we have and for odd 2m + 1 ≥ i x , we have Either way, for any conjugacy class x in F 7 , both x Te = lim 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 PD(T ).
Lemma 9.2.There exist two sequences α i and β i of conjugacy classes of elements of F 7 such that the following holds.For any positive length vector ℓ = ℓ 0 on τ 0 , let T e and T o be the respective limiting trees in ∂CV 7 for τ 2m , ℓ2m Proof.Take the letter e ∈ F 7 and recall the automorphisms Φ i used to define the folding and unfolding sequences.Set x i = Φ i (e).For each i, x i is legal in τ ′ i and is represented by the vector e 5 = (0, 0, 0, 0, 1, 0, 0) T in τ ′ i .Using notation from Lemma 9.1, set c e i = c e xi and c o i = c e xi .Note here i is the smallest index such that x i is legal in τ ′ i .We compare the ratio of c o i and c e i .Since r i+1 − r i → ∞, we have Recall that both ℓ e and ℓ o are positive and by Lemma 7.1 the sequence Z i converges to Z. Since Ze 5 is the zero vector, by continuity of the dot product, Setting α i = x 2i and β i = x 2i+1 finishes the proof.
Corollary 9.3.For a sequence (r i ) i≥1 with r i+1 − r i ≥ i, if the folding sequence (τ ′ i ) i converges to an arational tree T , then for any positive length vector ℓ 0 on τ 0 , the limit set in ∂CV 7 of the rescaled unfolding sequence (τ i , ℓ i ) is always the 1-simplex PD(T ).
Proof.Since the folding (τ i ) ′ i and the unfolding sequence (τ i ) i are equal as marked graphs for all i ≥ 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 F F 7 .Thus, the two limiting trees T e and T o of the even and odd sequences of (τ i , ℓ i ) are length measures on T .
Recall PD(T ) is a 1-simplex by Corollary 7.2.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 ∈ F n , c ′ ≤ x To x Te ≤ 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 ∈ F n , x To x Te ≤ c.In both the cases, we get a contradiction to Lemma 9.2.For any integer r, let φ r = ρφ r .To each sequence (r i ) i≥0 of positive integers, we have an unfolding sequence (τ i ) i with train track map φ ri : τ i → τ i−1 , and a folding sequence (τ ′ i ) i with train track map φ −1 ri : τ ′ i−1 → τ i .By the limit set of the unfolding sequence (τ i ) i in ∂CV n we mean the limit set of (τ i , ℓ i ) with respect to some (any) positive length vector ℓ i on τ i .

Conclusion
Main Theorem.Given a strictly increasing sequence (r i ) i≥1 satisfying r i ≡ i mod 7 and r i ≡ 0 mod 3, then the folding sequence (τ ′ 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 ≥ i, then T is both non-uniquely ergometric and non-uniquely ergodic.Both PD(T ) and PC(T ) are 1-dimensional simplices.
Furthermore, the limit set in ∂CV 7 of the unfolding sequence (τ i ) i is always the 1-simplex spanned by the two ergodic metrics on T .
Proof.A sequence as in the statement exists by the Chinese remainder theorem.The first statement follows from Corollary 5.8 and Proposition 5.10.Non-unique ergometricity of T follows from Proposition 3.1 and Corollary 7.2.Non-unique ergodicity of T is Corollary 8.5.Finally, the last statement is Corollary 9.3.
and applying the norm: By adding these for k = 1, 2, • • • , m − 1 and using Σ 1 = ∆ 1 we have So the norms of Σ m Y are bounded by norms of Σ i with i < m.From Equation (4) we also see that the norm of Σ k+1 is bounded by the norms of Σ k Y .Putting this together we have Thus we have an inequality of the form for a = ǫ Y + Y which proves the sequence is Cauchy.
For the second statement, set X k = ∞ i=k (Y + ∆ i ) for k ≥ 1.By the same estimate as above with ǫ replaced with ǫ 2 k−1 , we know that X k exists and By definition, X = (Y + Σ k )X k+1 .Suppose v is a unit vector with Y v = 0. Then Since this is true for all k ≥ 0, letting k → ∞ yields Xv = 0.

Let F 7
= a, b, c, d, e, f, g .Denote by x the inverse of x ∈ F 7 .First consider the map induced on the 3-petaled rose by the automorphism θ: a → b, b → c, c → ca ∈ Aut(F 3 ) and the map induced by the inverse automorphism ϑ: a → bc, b → a, c → b.
Observation 4.1.From the structure of the above maps, for n ≡ 0 mod 3, Dθ n = Dθ 3 and Dϑ n = Dϑ 3 .Lemma 4.2.The map on the 3-petaled rose labeled a, b, c induced by ϑ is a train track map with respect to the train track structure with gates {a, c}, {b, ā}, {c, 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 θ is also a train track map with respect to the train track structure with gates {a, b, c}, {ā}, { b}, {c} and it has one periodic Nielsen path (see[BF94, Example 3.4]).
Lemma 4.3.For any r ≥ 3, the map on the 7-petaled rose induced by φ r = ρφ r is a train track map with respect to the train track structure with gates {a, b, c}, {d, e, f } and 8 more gates consisting of single half edges.The transition matrix M r has block form0 I B r 0where I is the 4 × 4 identity matrix, and B is the transition matrix of θ: By Observation 4.1, we only have to check the lemma for φ 3 , φ 4 , φ 5 , which can be done by hand or using the train track package for Sage.Lemma 4.4.For any r ≥ 3 and r ≡ 0 mod 3, the map on the 7-petaled rose induced by ψ r = (ρφ r ) −1 is a train track map with respect to the train track structure with gates {a, e, ḡ}, {b, d}, {c, b}, {d, c}, {f, ē}, {g, f}, {ā}The transition matrix N r has block form 0 C r I 0 where I is the 4 × 4 identity matrix, and C is the transition matrix of ϑ: By Observation 4.1, we only have to check the lemma for ψ 3 , which can be done by hand or using the train track package for Sage.

π
A (B) := {π A (G)|G ∈ CV n and B|G ⊂ G is embedded } If B is compatible with A, then define π A (B) to be empty.If A meets B and B meets A, then we say A and B overlap.
Theorem 5.4 ([Tay14]).Let A, B, C be free factors of F n .There is a constant D depending only on n such that the following statements hold.1.If rank(A) ≥ 2, then either A ⊆ B (up to conjugation), A and B are disjoint, or π A (B) ⊂ F (A) is defined and has diameter ≤ D.

e 5
e 5 = ℓ T o Ze 5 = 0. Next let N ∞ = lim i→∞ Nr i λ r i C and recall by Lemma 4.7 that the vector ZN ∞ e 5 = (⋆, ⋆, ⋆, 0, 0, 0, 0) is non-negative.Thus there are positive constants A and B such that, lim i→∞ ℓ T e Z 2i+2 N r2i+1 λ r2i+1 C e 5 = ℓ T e ZN ∞ e 5 = A > 0, = ℓ T o ZN ∞ e 5 = B > 0.Combining the above observations and the formulas for length of x i in T e and T o obtained in Lemma 9.1 we get: Recall φ ∈ Aut(F 7 ) is the automorphism:a → b, b → c, c → ca, d → d, e → e, f → f, g → g and ρ ∈ Aut(F 7 ) is the rotation by 4 clicks: a → e, b → f, c → g, d → a, e → b, f → c, g → d.
Let • denote the operator norm.Thus Y ≥ 1 for a nontrivial idempotent matrix Y .Lemma 11.1.Let Y be an idempotent matrix and ∆ i , i ≥ 1, a sequence of matrices with∆ i ≤ ǫ 2 i for some ǫ > 0. Assume also that ǫ Y ≤ 1/2.Then the infinite product ∞ i=1 (Y + ∆ i ) converges to a matrix X with X − Y ≤ 2ǫ Y + Y 2 .Moreover, the kernel of Y is contained in the kernel of X.Proof.WriteY + Σ k = k i=1 (Y + ∆ i ) Then (Y + Σ k )(Y + ∆ k+1 ) = Y + Σ k+1 and since Y 2 = Y it follows that Σ k+1 = Y ∆ k+1 + Σ k (Y + ∆ k+1 )(4)Multiplying on the right by Y and using Y 2 = Y we get

2
and b = ǫ Y .Set c = 2ǫ Y + Y 2 .Then c ≥ ǫ, a ≤ c/2, and b ≤ 1/2 by assumption.Easy induction then shows for all k ≥ 1, Σ k ≤ c.(5)This obtains the inequality X − Y ≤ 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, ∆ i ) − (Y + ∆ k ) By Equation (5), the norm of k−1 i=1 (Y + ∆ i ) = Y + Σ k−1 is bounded by c + Y .Wecan apply the same estimate to the sequence starting with Y + ∆ k and with ǫ replaced with ǫ 2 k−1 to see that m i=k (Y + ∆ i ) − Y ≤ 2ǫ( Y + Y 2 ) 2 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.
Proof.Let x ∈ F 7 be a cyclically reduced representative of its conjugacy class.By Lemma 4.8, there exists i ≥ 0 such that x is legal in τ ′ i .Let i x be the smallest index among such i.Then we can represent x by a vector v x in R |Eτ ′ ix | and by the vector N ri