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π−1₍₍

−∞,−3]∪[3,∞))⊂P(),

whereπ:C(S)→Zis the projection to the annulus with core curve v(∗κ)post-composed with a transla- tion such thatπ(b)=0.

Proof. Letκbe any sequence starting and ending atφ0such that every entry in the associated matrixQ_κ is at least 3. This implies that the stable vertex cyclev(∗κ) passes over every band inat least thrice.

Now consider the subsurface projectionπto the annulus with core curvev(∗κ). By post-composing with a translation ofZ, we may assume thatπ(b)=0. By Proposition 12.2, the projection πof every point inC(S) \P() is within distance 3 ofπ(b). This implies that the pre-imageπ−1((−∞,−3)∪(3,∞)) is

In other words, when the vertex cyclev(∗κ) gets nested deep enough inP() a fixed pre-image of the projection tov(∗κ) is contained inP(). It is in this sense that such aκis apush-in sequence. We fix such a sequenceκonce and for all. This achieves a setup to which Theorem 11.6 can be applied.

13. ASINGULAR SET FOR HARMONIC MEASURE.

13.1. Choosing the initial chart. We choose the initial embedding ofφ0inSwith the underlying trackτ0 such that the base-pointb∈C(S) does not belong toP(τ0). The embedding identifies the configuration spaceW0with a chart inP M F. The construction of a singular set will proceed inside this chart.

13.2. Relative probability that the sequenceκ∗n0follows aC-distributed stage. By Lemma 6.7, the relative probability that the push-in sequenceκfollows aC-distributed stage is up to a uniform con- stantc>1, the same as the probability`(PQ_κ(W0)) that an expansion begins withκi.e.,

1

c`(PQκ(W0))<

`(PQ<sub>∗κ</sub>(W0)) `(PQ_(W0)) <

c·`(PQ_κ(W0)).

Sinceκis a priori fixed, whenever the sequenceκ follows aC-distributed stage , the resulting stage

∗κisC0-distributed, for someC0that depends only onCandd. Recall from Section 7 that0denotes the Dehn twist sequence. Denote the matrix associated to the sequence ∗κ∗n0byQ_,n. Again by Lemma 6.7, there exists a constanta1that depends only ona0andC0, such that

1 a1nj < `(PQ<sub></sub>,n(W0)) `(PQ_∗κ(W0))< a1 nj.

Hence the relative probability that the sequenceκ∗n0follows aC-distributed stagesatisfies

(13.3) 1 c a1nj <`(PQ,n(W0)) `(PQ_(W0)) < c a1 nj i.e., the relative probability`(PQ,n(W0))/`(PQ(W0))≈1/nj.

13.4. Construction of the doubly indexed sequence of setsYn(m). By Theorem 6.9, almost every expan- sion becomesC-distributed infinitely often. Hence, for every non-negative integerm, almost every ex- pansion has a stage that is them-th instance ofC-distribution. Whenm=0, we mean the initial stage itself with no splitting whatsoever.

LetSm be the set of stages  that are them-th instances ofC-distribution. For ∈Sm, letY be the set ofxinW0whose expansion begins with  i.e., the subsetY=PQ_(W0). For distinct , ¯ inSm, the setsY_,Y¯have disjoint interiors. By Theorem 6.9, the union over all∈Sm of the setsYis a set of full measure i.e.,

(13.5) X

∈Sm

`(Y_)=1

Follow each  inSm by the sequence κ∗n0, and letQ,n be the matrix associated to ∗κ∗n0. Let Y_,n=PQ,n(W0). Then, the ratio`(Y,n)/`(Y) satisfies Estimate (13.3). LetYn(m)be the union

Y_n(m)= [

∈Sm Y_,n.

Form=0, the setYn(0)is justPQn0(W0) i.e., the set ofxwhose expansion begins withn0. First, we

estimate the Lebesgue measure ofYn(m).

Lemma 13.6. (13.7) 1 c a1nj <`(Yn(m))< c a1 nj

Proof. Write `(Y<sub>n</sub>(m))= X ∈Sm `(Y_,n)= X ∈Sm `(Y<sub></sub>)`(Y,n) `(Y_) .

Estimate (13.3) for the ratio`(Y<sub></sub>,n)/`(Y), and Equation (13.5) finishes the proof. In particular, the above lemma shows that for anym1,m2,

(13.8) `(Y(m1)

n )≈`(Yn(m2))

13.9. Almost independence ofYn(m)for the Lebesgue measure. Let (m1,n1) and (m2,n2) be a pair of

indices withm1<m2. Since the setsYι:ι∈Sm1is a partition of a set of full measure, it is enough to check

that almost independence holds in eachY_ι. For anyι∈Sm1 and∈Sm2, eitherYis contained inYι,n1or

has interior disjoint from it. By Lemma 6.7, givenι∈Sm1, the relative probabilities satisfy

Prob(Y(m1) n1 |ι) ≈ `(Y (0) n1 ) Prob(Y(m2) n2 |ι) ≈ `(Y (m2−m1) n2 ) Prob(Y(m1) n1 ∩Y (m2) n2 |ι) ≈ `(Y (0) n1 ∩Y (m2−m1) n2 )

So it is enough to check that for anym>0, the setsYn(0)1 andY

(m)

n2 are almost independent. Again, the

main point is that for allinSm, the setY is either contained inYn(0)1 or has interior disjoint fromY

(0)

n1 .

LetTm be the subset ofSm consisting of those  for whichY is contained inYn(0)1 . The union over all

∈Tmof the setsYis a set of full measure inYn(0)1 . Hence, using Estimate (13.3) and Equation (13.8), we

get `¡ Yn(0)1 ∩Y (m) n2 ¢ ≈ X ∈Tm `(Y_,n2) ≈ X ∈Tm `(Y<sub>n</sub>(0)<sub>2</sub> )`(Y_) ≈ `(Y<sub>n</sub>(m<sub>2</sub> )) X ∈Jm `(Y_) ≈ `(Y<sub>n</sub>(m<sub>2</sub> ))`(Y_n(0)₁ ) showing almost independence.

13.10. Harmonic measure estimate forYn(m). Recall from Remark 7.3 thatφ0has the property that sim- ple closed curves carried by the underlying train track twist only to the left around the stable vertex cycle v. Letπbe the projection to the annulus with core curvev. Since the twisting is in one direction only, the projection of the set of carried curves is a one-sided interval of the form [M1,∞) or (−∞,−M1]. Reversing

the order onZif necessary, we fix the convention that it is of the form [M1,∞).

For the initial trackτ0, letπbe the projection to the annulus with core curvev(κ). Post-composing by a translation ofZassume thatπ(b)=0. By our chosen convention, there is a positive integerM1>3

such thatπ(P(κ))=[M1,∞).

For∈Sm, letv(∗κ) be the stable vertex cycle of ∗κ. Denote byπ∗κ, the sub-surface projection to the annulus with core curvev(∗κ). Post-composing by a translation, we assume thatπ_∗κ(b)=0.

By Lemma 12.6, the push-in sequenceκensures that the pre-imageπ−1

∗κ([3,∞)∪(−∞,−3]) sits entirely inside the setP(). By our chosen convention,π_∗κ(P(∗κ)) is a one sided interval. We show below that this interval is contained in [M1−3,∞)