The goal of this section is to provide another approach to compute the spectra of the adjacency operators∆ξon the Schreier graphsΓξof spinal groups withm= 1using a renormalization approach, in Theorem 4.2.8. Even though for∆ξit is a particular case of Theorem 4.1.10, the strategy is more elegant, and the result is valid not only for the graphsΓξ,ξ ∈XN, but also for thed-ended graphsΓπfrom Theorem 3.7.3, so for all graphs in the space of Schreier
graphsGG
ω,XN. This method was developed by Quint in [60] for a graph related with the Pascal graph. In that case, the renormalization maps are an example of a more general construction studied in [48] called the para-line graph. Nevertheless, such construction is not suitable for our case, so we use a different approach to define the renormalization maps.
LetGω be a spinal group withd ≥3, m = 1andω ∈ Ωd,1. Letξ ∈ XNandΓξ be its Schreier graph with respect to the spinal generating setS, and let us write`2ξ =`2(Γξ).
Recall that the shift mapσ :XN→XNremoves the first digit of any point inXN. This shift induces an operatorΠ∗ :`2σξ →`2ξ, defined asΠ∗f(η) =f(ση). Conversely, we consider as well the operatorΠ :`2ξ →`2σξ, defined asΠf(η) =P
i∈Xf(iη). As it turns out,Πand Π∗are adjoint operators, and we have the relationΠΠ∗=d.
Geometrically, we can think ofΓξas an inflated version ofΓσξ, where every vertex has been replaced by a graph ondvertices. Iff ∈`2σξassigns a certain value to a vertex inΓσξ, thenΠ∗f assigns the same value to alldvertices replacing it. Conversely, iff ∈`2ξ, then Πf assigns to a vertex the sum of the values off at thedcorresponding vertices inΓξ. See Figure 4.3 for a description ofΠ∗.
α
βj γj
7−→Π∗ Γσξ
α α
α α
α βj
βj βj
βj
βj γj
γj γj
γj γj Γξ
Figure 4.3: The operatorΠ∗ copies the value of the function at each vertexpinΓσξto the correspondingdverticesip,i∈X, inΓξ.
In a similar way, we may define the renormalization mapsΠ∗andΠforΓπas well. Let Gω be a spinal group withd ≥ 3, m = 1andω ∈ Ωd,1. Letπ ∈ Epi(B, A) occurring infinitely often inω. Thed-ended Schreier graphsΓπ from Theorem 3.7.3 correspond to Schreier graphs of neighborhoods of the point(d−1)N∈XN(see Section 3.7).
Recall that the graphsΓπhave vertex setCof((d−1)N)×X, and can be decomposed asdcopies ofΓ(d−1)N joined by a copy ofΛπ. Let`2π =`2(Γπ)and∆π be the adjacency operator onΓπ. We can extend the renormalization mapsΠ∗ andΠto the graphsΓπ in a natural way asΠ∗,Π :`2π →`2π asΠ∗f(η, i) =f(ση, i)andΠf(η, i) =P
j∈Xf(jη, i).
These operatorsΠ∗andΠare the renormalization maps that we will use in order to relate
∆ξwith∆σξand∆π with itself in order to find their spectra.
Lemma 4.2.1. LetGbe the quadratic polynomialG(x) =x2−2(d−2)x−2(d−1). We
This shows the first relation while the second one is its dual, as∆ξ,∆σξ are self-adjoint operators.
Remark 4.2.2. The mapGarising in Lemma 4.2.1 is the same as the mapGdefined in Remark 4.1.11 relating the mapsψandFfrom Theorem 4.1.10 form= 1. For spinal groups withm≥2, there does not exist any quadratic map playing the role ofGin Lemma 4.2.1, which limits this approach to spinal groups withm= 1.
We will now use the equalities in Lemma 4.2.1 to find a relation between the spectrum of
∆σξ(∆π) with that of∆ξ|H (∆π|H), the restriction of∆ξ(∆π) to a subspaceHof`2ξ(`2π).
In order to do that, we will need a general result from functional analysis. We only include its statement (Lemma 4.2.3), for its proof, see [60].
Lemma 4.2.3. Let H be a Hilbert space and T a self-adjoint bounded operator of H.
Let q ∈ R[x]be a quadratic polynomial. Let K ⊂ H be a closed subspace such that q(T)K ⊂ K andK and T K generateH. Thenq(sp(T)) = sp(q(T)|K). Moreover, if T−1K∩K= 0, thensp(T) =q−1(sp(q(T)|K)).
LetH =hΠ∗`2σξ,∆ξΠ∗`2σξi(alternativelyH=hΠ∗`2π,∆πΠ∗`2πi) be the subspace of`2ξ (`2π) generated by the images of the operatorsΠ∗and∆ξΠ∗(∆πΠ∗). Now we are ready to establish the relation betweensp(∆ξ|H)andsp(∆σξ)(sp(∆π|H)andsp(∆π)).
Proposition 4.2.4. For everyξ∈XN,His invariant under∆ξ(∆π). Moreover, sp(∆ξ|H) =G−1(sp(∆σξ)),
sp(∆π|H) =G−1(sp(∆π)).
Proof. We provide again the proof only for∆ξ, as for∆π it is analogous. For anyf ∈`2σξ,
∆ξΠ∗f ∈Hby definition and, by Lemma 4.2.1,
∆2ξΠ∗f = Π∗∆σξf+ 2(d−2)∆ξΠ∗f+ 2(d−1)Π∗f ∈H.
ThereforeHis invariant under∆ξ. LetK be the image ofΠ∗. √1
dΠ∗ is an isometry from`2σξ toK, asΠΠ∗ =d, which implies thatsp(G(∆ξ)|K) = sp(∆σξ)by Lemma 4.2.1. Now setL={f ∈`2σξ |∆ξΠ∗f ∈ K}and letf ∈L. Letp∈Γσξ, and letqj,rj be itsAandB-neighbors, respectively, for j= 1, . . . , d−1. Letα,βj,γjdenote the values off atp,qjandrj, respectively, and set β = Pd−1
j=1βj andγ = Pd−1
j=1γj. Recall from the proof of Lemma 4.2.1 that, for every i∈X, we have
∆ξΠ∗f(ip) =
β+ (d−1)α, i= 0 2(d−1)α, i6= 0, d−1 γ+ (d−1)α, i=d−1
.
Since∆ξΠ∗f ∈ K, it must be constant over the vertices of Γξ of the form ip, i ∈ X.
This means thatβ = γ = (d−1)α. Similarly, for everyj = 1, . . . , d−1, we deduce (d−1)βj =β−βj +αand(d−1)γj =γ−γj+α. Equivalently, thatβj =γj =αfor
everyj = 1, . . . , d−1. Thusf is constant, and hencef = 0, soL= 0.
Now letg∈∆−1ξ K∩K, so bothg,∆ξg∈K. There existsh∈`2σξ such thatg= Π∗h, and so∆ξΠ∗h= ∆ξg∈K. Consequently,h∈L, but thenh= 0and thereforeg= 0, so
∆−1ξ K∩K= 0. We may now use Lemma 4.2.3 to obtain
sp(∆ξ|H) =G−1(sp(G(∆ξ)|K)) =G−1(sp(∆σξ)).
Proposition 4.2.4 establishes a relation between the part of the spectrum of ∆ξ (∆π) which corresponds to the subspace H in `2ξ (`2π) to the whole spectrum on `2σξ (`2π), via the quadratic mapG. We now want to find the part of the spectrum corresponding to the orthogonal complement ofH, denotedH⊥. Once we know both parts, we will be able to find an explicit description of the spectrum of∆ξ(∆π).
Lemma 4.2.5. sp(∆ξ|H⊥) = sp(∆π|H⊥) ={d−2,−2}.
αd−1 α0
α3 α1
α2
(d−1)p 0p
3p 1p
2p βj0
βj2 βj1 βj3
βd−1j γj0
γj2 γj1 γj3 γd−1j
Figure 4.4: Values of the functionf ∈H⊥at the verticesip,iqj andirj ofΓξ, fori∈X andj= 1, . . . , d−1.
Proof. Once again the proof for∆π is analogous so we do not include it. Letf ∈H⊥. Let p ∈Γσξ, and letqj,rj be itsAandB-neighbors, respectively, forj = 1, . . . , d−1. For eachi∈X, setαi =f(ip),βji =f(iqj),γji =f(irj), as in Figure 4.4. Becausef ∈H⊥, we have
0 =hf,Π∗δpi=hΠf, δpi= Πf(p) =X
i∈X
f(ip) =X
i∈X
αi. Likewise, we obtain, for everyj= 1, . . . , d−1,
X
i∈X
βji =X
i∈X
γji = 0.
As in the proof of Lemma 4.2.1, we will find∆ξf(ip)and∆2ξf(ip). In order to simplify notation, we will writeβ =Pd−1
j=1βj0andγ =Pd−1
j=1γjd−1. Using the equalities above, we
have
Furthermore, again using the equalities we have established,
∆2ξf(ip) = end, we need the following result.
Lemma 4.2.6. 2(d−1)and−(d−1)2−1are not eigenvalues of∆ξ(∆π).
Proof. The analogous proof for∆π is omitted. Letf ∈ `2ξ be an eigenfunction of ∆of eigenvalue2(d−1). Sincef ∈`2ξ,fmust be bounded and reach its maximum value on some vertices. LetΣbe the set of vertices ofΓξattaining that maximum. Ifp∈Σandqjare its neighbors,j= 1, . . . ,2(d−1)becauseΓξis2(d−1)-regular, then2(d−1)f(p) = ∆f(p) = P2(d−1)
j=1 f(qj). Butf(qj)≤f(p), and therefore,f(qj) =f(p)for allj= 1, . . . ,2(d−1), soqj ∈Σfor everyj = 1, . . . ,2(d−1). SinceΓξis connected,f must be constant, and sincef ∈`2ξ,f must be zero.
For the second claim, we know thatk∆k=|S|= 2(d−1), and since(d−1)2+ 1>
2(d−1)for everyd >2,−(d−1)2−1cannot be insp(∆).
Recall from Definition 3.2.3 thatΓ1η is the subgraphXσ(η)ofΓξ, forη∈Cof(ξ). We will call any such subgraph anA-piece. Similarly,Λnη is the subgraph(d−1)n0Xσn+2(η)of Γξ, forn≥0andη∈Cof(ξ). We will call such subgraph ann-piece, and, more generally, we will call anyn-piece aB-piece. Finally, for convenience, we will sometimes callA-pieces (−1)-pieces. In Figures 4.3 and 4.4,B-pieces are drawn in blue, whileA-pieces are drawn
in black.
Notice that every vertex belongs to exactly oneA-piece and every vertex not fixed by Bbelongs to exactly oneB-piece. Moreover, everyA-piece hasdvertices, joined together only byA-edges. More precisely, for everyk= 1, . . . , d−1, any vertexiηfrom anA-piece inΓξ has anak-edge to(i+k)η in the sameA-piece. Similarly, anyB-piece has alsod vertices, joined together by onlyB-edges. For everyb∈B\ {1}, any vertex(d−1)n0iη from aB-piece inΓξhas ab-edge to(d−1)n0(i+k)ηin the sameB-piece, ifωn(b) =ak.
For the graphsΓπ, we may extend these notions in a natural way.A-pieces (equivalently (−1)-pieces) are subgraphs of the form(Γ1η, i) = (Xση, i), for η ∈ Cof((d−1)N)and i∈X. n-pieces are subgraphs of the form(Λnη, i) = ((d−1)n0Xσn+2(η), i), forn≥0, η ∈ Cof(ξ) andi ∈ X. We define the subgraph ((d−1)N, X)to be the only ∞-piece.
Again, we call anyn-piece aB-piece, forn≥0orn=∞.
Intuitively,AandB-pieces inΓπare exactly those within thedcopies ofΓ(d−1)N, with the exception of the new∞-piece which joins thedcopies together.
Lemma 4.2.7. The eigenspaceE−2is trivial, whileEd−2is the subspace ofH⊥given by
Ed−2={f ∈`2ξ|Πf = 0, f constant onB-pieces}.
Proof. Once more, we provide only the proof for∆ξ, as that for∆πis analogous. Let us first show that bothEd−2andE−2 are contained inH⊥. Letf ∈Ed−2. Then, by Lemma 4.2.1,
∆σξΠf = ΠG(∆ξ)f =G(d−2)Πf = (−(d−1)2−1)Πf.
But Lemma 4.2.6 then impliesΠf = 0. In addition,Π∆ξf = (d−2)Πf = 0. Now, for as before,f ∈H⊥. This shows thatH does not contain any eigenfunction of eigenvalue d−2nor−2. corresponds to the following system of equations:
all of whose solutions are constant vectors. Hence,f is constant onB-pieces.
Finally, let us showE−2 = 0. Letf ∈E−2. Again, we know thatΠf = 0. For every
But∆ξf =−2f, soP
j∈Xf(ipj) = 0, or equivalently thatf has zero sum on allB-pieces.
For the remaining part of the proof, we will consider the subgraphsΓnη =Xnσn(η)of Γξ, forn≥1,η ∈Cof(ξ)(see Definition 3.2.3). Notice that such subgraphs can only be connected toΓξ\Γnη by two vertices:(d−1)n−10σn(η)and(d−1)nσn(η). We shall call them the extremes ofΓnη. We claim that for every subgraphΓnη ofΓξ,f is antisymmetric at its extremes, i.e., thatf((d−1)n−10σn(η)) =−f((d−1)nσn(η)). Indeed, we proceed by induction onn.
Ifn= 1, the subgraphsΓ1η forη∈Cof(ξ)are preciselyA-pieces. SinceΠf = 0, the sum of the values off at itsdvertices is0. However,f vanishes on vertices whose first digit is different from0, d−1, as they are fixed byB, and is hence antisymmetric on the remaining two vertices, which are the extremes ofΓ1η.
Assume the claim to be true for n ≥ 1, and consider Γn+1η , for η ∈ Cof(ξ). We decompose Γn+1η as dsubgraphs Γnζ(i), i ∈ X, where ζk(i) = ηk for all k ∈ N\ {n}
and ζn(i) = i. Each of these subgraphs has extremes pi = (d−1)niσn+1(η) and qi = (d−1)n−10iσn+1(η),i∈X, and theqiform aB-piece. Our goal is to show thatf(p0) =
−f(pd−1).
Notice that the verticespi,i6= 0, d−1, are fixed byB, and sof(pi) = 0,i6= 0, d−1. By induction hypothesis we havef(pi) =−f(qi)for everyi∈X, so in particularf(qi) = 0, for i 6= 0, d−1. Finally, the qi form a B-piece, so P
i∈Xf(qi) = 0, which implies f(q0) =−f(qd−1). Now we conclude
f(pd−1) =−f(qd−1) =f(q0) =−f(p0).
0
0 0
0
0
0
α −α α −α
Figure 4.5: Any(−2)-eigenfunctionf is antisymmetric on the extremes of the subgraphΓnξ, for alln≥1andη ∈Cof(ξ). Therefore it must be zero.
Hence iff(p) =α6= 0for somep∈Γξ, we can find an infinite path onΓξon whichf has alternating valuesα,−α, which is a contradiction with the fact thatfis square summable.
Thenf = 0and soE−2 = 0.
Notice that ford= 2the eigenspaceEd−2 is also trivial. We may now conclude with the explicit computation of the spectrum of the adjacency operator∆ξ(∆π) on the Schreier graphsΓξ(Γπ).
Theorem 4.2.8. LetGω be a spinal group withd≥3,m = 1andω∈Ωd,1. Let∆be the adjacency operator on any Schreier graphΓin the space of graphsGGω,XN, i.e., eitherΓξ forξ∈XNorΓπforπ∈Epi(B, A)repeating infinitely often inω. Then,
sp(∆) = Λ∪ [
n≥0
G−n(d−2), (4.3)
whereG(x) =x2−2(d−2)x−2(d−1)andΛis its Julia set, which is a Cantor set of zero Lebesgue measure. In particular,sp(∆)does not depend onξnorπ.
Proof. We only provide the proof for∆ξ, since for∆πit is analogous. By induction onn≥ 0, and in parallel for all the graphsΓξwithξ∈XN, let us show thatG−n(d−2)⊂sp(∆ξ).
The case n = 0 is a consequence of Lemma 4.2.5. Suppose thatSn
k=0G−k(d−2) ⊂ sp(∆σξ)for somen≥1. Then, by Proposition 4.2.4,Sn+1
k=0G−k(d−2)⊂sp(∆ξ). Hence S
n≥0G−n(d−2)⊂sp(∆ξ). Sincesp(∆ξ)is closed, andΛis the adherence of this set, Λ⊂sp(∆ξ)too.
Now letx∈sp(∆ξ)such thatx 6∈S
n≥0G−n(d−2). In that case, Proposition 4.2.4 and Lemma 4.2.5 imply thatGn(x)∈sp(∆σn(ξ))for everyn≥0. Therefore the sequence Gn(x)is bounded, which means thatx∈Λ.
Remark 4.2.9. The mapGis the same map from Remark 4.1.11, so it satisfiesG(x) = ψ(x) +d−2 =F(x−(d−2)) +d−2, whereψandFare the maps from Theorem 4.1.10 withm= 1.
As with Corollary 4.1.12, the Cantor spectrum in Theorem 4.2.8 allows us to conclude an immediate consequence about spectral measures.
Corollary 4.2.10. LetGωbe a spinal group withd≥3,m= 1andω∈Ωd,1. Let∆be the adjacency operator on any Schreier graphΓin the space of graphsSch(XN), i.e., either Γξforξ ∈XNorΓπ forπ ∈Epi(B, A)repeating infinitely often inω. Then any spectral measure of∆has trivial absolutely continuous part.
Proof. The support of any spectral measure of∆is contained insp(∆). Since the Lebesgue measure ofsp(∆)is zero, its absolutely continuous part must be trivial.