5.4 Singular continuous spectral measures
5.4.2 Spectral measures of ∆ π
We are now ready to study the spectral measures of the adjacency operator∆πon the Schreier graphΓπ. We will find an explicit subspace of`2πfor which the spectral measures are singular continuous. More precisely, we will decompose`2π as the direct sum of the eigenspaces of
∆πplus a subspaceΦ, whose functions have purely singular continuous spectral measures.
We follow a strategy based on [60].
Let us start by relating the spectral measure of a functionf ∈`2π with that ofΠ∗f ∈`2π. To do so, we will use as main tool the transfer operator Lq,κ. An exposition of results concerning operators of this type can be found in [59].
Definition 5.4.6. Let q ∈ R[x]be the quadratic polynomial q(x) = (x−u)2 +t, with u, t∈R. Letκ :R\ {u} → Rbe a measurable function. We define the transfer operator Lq,κas
Lq,κα(y) = X
x∈q−1(y)
κ(x)α(x), for every measurable functionα:R\ {u} →Randy∈(t,∞).
Letµbe a positive measure on(t,∞). If, forµ-almost everyy∈(t,∞),κis positive on both preimages ofybyq, then, after performing a change of variables, there is a measure νonR\ {u}such that
Z
R\{u}
α dν= Z
(t,∞)
Lq,κα dµ.
for every positive measurable functionα:R\ {u} →R. We shall denote this measureνby L∗q,κµ.
In order to find the relation between the spectral measures of∆π associated withf ∈`2π andΠ∗f ∈`2π, we need the following general result about these transfer operators, the proof of which can be found in [59].
Lemma 5.4.7. LetHbe a Hilbert space andT a self-adjoint bounded operator ofH. Let K⊂ Hbe a closed subspace such thatq(T)K ⊂KandKandT KgenerateH. Suppose that there existα, β ∈Rsuch that, for everyv, w∈ H,hT v, wi=αhv, wi+βhq(T)v, wi.
Letv ∈ K and let µandµ0 be the spectral measures ofT and q(T)associated with v,
We may now apply Lemma 5.4.7 to our situation.
Proposition 5.4.8. Letf ∈`2π. Letµf andµΠ∗f be the spectral measures of∆πassociated
We now need to use a version of the Ruelle-Perron-Frobenius Theorem, which is stated in terms of full shifts on binary alphabets. In our case, we shall prove that the dynamical system given by the action ofGon its Julia setΛis conjugate to the shift on{0,1}N. Lemma 5.4.9. The dynamical system(Λ, G)is conjugate to the full shift on{0,1}N. Proof. By Remark 4.1.11, we know thatΛ⊂[−2,2(d−1)]. In fact,Λ⊂G−1(Λ) =I0∪I1,
Let us show that it is a bijection.
Letx, y∈Λhave the same associated sequence(εn)n≥0. Notice that
|G(x)−G(y)|=|x2−2(d−2)x−y2+ 2(d−2)y|=
=|x2−y2−2(d−2)(x−y)|=|x−(d−2) +y−(d−2)||x−y|.
However,xandyboth lie on the same side ofd−2, so
|x−(d−2) +y−(d−2)|=|x−(d−2)|+|y−(d−2)| ≥2p
d(d−2)≥3.
Hence|G(x)−G(y)| ≥3|x−y|. Ifx6=y, then for somen≥0the distance|Gn(x)−Gn(y)|
would be greater than the length of the intervalsI0, I1and soGn(x)andGn(y)would not lie in the same one, which contradicts the fact thatxandyhave the same associated sequence.
Therefore the map is injective. To prove that it is surjective, let us define the intervals Iε0...εn =
n
\
k=0
G−k(Iεk) =Iε0 ∩G−1(Iε1...εn),
forε0. . . εn ∈ {0,1}n+1, n ≥ 0. The preimage of any interval in [−2,2(d−1)] byG is a union of two intervals: one included inI0 and another inI1. We can use this fact to inductively verify that none of these intervals is empty. In addition, we have the inclusions
Iε0 ⊃Iε0ε1 ⊃ · · · ⊃Iε0...εn ⊃. . .
Hence, for any sequence(εn)n≥0, we have a decreasing sequence of closed, non-empty intervals, which implies that the intersection
\
n≥0
Iε0...εn
is non-empty. By construction, all points in this intersection must have(εn)n≥0as associated sequence, but by injectivity there can only be one. Therefore, the mapΛ→ {0,1}Ndefined above is a bijection.
Lemma 5.4.9 enables us now to use the following version of the Ruelle-Perron-Frobenius Theorem, whose proof can be found in [59, §2.2]. ForI ⊂R, letC(I)denote the space of continuous functions onI, endowed with the topology of uniform convergence, and letR∗+ denote the set of strictly positive real numbers.
Theorem 5.4.10. Letκ: Λ→R∗+be a Hölder function. ConsiderLq,κas an operator on C(Λ), and letλκbe its spectral radius. Then there exists a unique probability measureνκ onΛand a unique strictly positive functionlκ ∈C(Λ)such that
Lq,κlκ =λκlκ, L∗q,κνκ=λκνκ and Z
Λ
lκdνκ = 1.
Moreover, the spectral radius ofLq,κrestricted to the space of functions with zero integral with respect toνκis strictly smaller thanλκ. Also, for everyg∈C(Λ), we have
n→∞lim 1
λnκLq,κg= Z
Λ
gdνκ.
Let us now define the subspaceΦ, whose orthogonal complement in`2π will be the direct product of all eigenspaces of∆π. Letpidenote the vertex((d−1)N, i) ∈Γπ, fori∈X, and letϕi ∈`2πbe the function which takes value1atpi,−1atpi−1and0everywhere else (see Figure 5.8), fori∈X. We defineΦas the following subspace:
Φ =h∆nπϕi|n≥0, i∈Xi.
Γ(d−1)N, i
Γ(d−1)N, i−1
1
0
0 0
−1
Figure 5.8: The functionsϕi ∈`2π.
Lemma 5.4.11. For everyi∈X,
Π∗ϕi= (∆π+ 2)ϕi.
Proof. Notice that∆πϕiis supported in theA-pieces containingpiandpi−1. It takes value 1at theA-neighbors ofpiandpi−1, and value−1at theA-neighbors ofpi−1andpi. On the other hand,Π∗ϕitakes value1on the fullA-piece containingpiand−1on that containing pi−1. This proves the claim.
Recall that we set θ(x) = x+22 , and let k(x) = x+ 2, h(x) = 2(d−1)−x and ρ(x) = 12. Notice thatθ =kρand thathk = h◦G. From now on, for any measurable functionκ: Λ→R, we shall writeLκ =LG,κ.
Proposition 5.4.12. Letνρbe the probability measure onΛfrom Theorem 5.4.10, so that L∗ρνρ = νρ. Then, for every i ∈ X, the spectral measure of ∆π associated withϕi is
As a consequence of Lemma 4.2.7, we know thatµ({−2}) = 0. Therefore, for any measurable functionα: Λ→R, we have, using thatθ=kρ, 0as well. Consequently, using the relationhk=h◦G,
Z
By Proposition 5.4.4, any eigenfunction of∆π must be constant on theB-piece formed bypi, fori∈X, and therefore it must be orthogonal toϕi, fori∈X. We then conclude that
This implies that, for everyg∈C(Λ),
n→∞lim Lnρg= Z
Λ
g dνρ.
We have proven thatL∗ρ(h1µ) = 1hµ. Our goal now is to show that h1µis a finite measure, and then by unicity ofνρ, they must be multiples of one another.
Letg∈C(Λ)such thatgvanishes on a neigborhood of2(d−1)and0<R
Λ g
hdµ <∞.
We know thatLnρgconverges uniformly to the constant functionR
Λg dνρ. Using this and
Therefore, the measure 1hµmust be finite, so it must be a multiple ofνρ. Let us show that
Finally, let us show thatµis singular continuous. We already know that it is supported onΛ, a Cantor set of zero Lebesgue measure, so its absolutely continuous part is trivial. In addition, the measureνρis atomless. Indeed, Theorem 5.4.10 implies that the action ofG onΛpreserves the measureνρ. Ifx∈Λwas such thatνρ({x})>0, thenνρ({Gn(x)}) = νρ({x})for anyn≥0, which is absurd asνρis a probability measure. In conclusion,µhas trivial absolutely continuous and discrete parts, so it is indeed singular continuous.
In order to prove the decomposition of the space`2πinto the direct sum of the eigenspaces of∆πandΦ, we will need to introduce another function besides theϕi. Before that, recall thatΦ =h∆nπϕi|n≥0, i∈Xior, equivalently,Φ =hp(∆π)ϕi |p∈C[x], i∈Xi.
Lemma 5.4.13. The subspaceΦis invariant under the operators∆π,Π∗andΠ.
Proof. First, notice that Φ is invariant under∆π by definition. Letp ∈ C[x], then, by Lemmas 4.2.1 and 5.4.11,
Π∗p(∆π)ϕi =p(G(∆π))Π∗ϕi=p(G(∆π))(∆π+ 2)ϕi ∈Φ, soΦis also invariant underΠ∗.
We observe thatΠϕi =ϕi. In addition, by Lemma 5.4.11, Π∆πϕi= Π(Π∗−2)ϕi = (d−2)ϕi.
Once again by Lemma 4.2.1, we have
ΠG(∆π)nϕi = ∆nπΠϕi = ∆nπϕi and
ΠG(∆π)n∆πϕi = ∆nπΠ∆πϕi = (d−2)∆nπϕi. Letp∈C[x], and let us decompose it asp(x) =PN
n=0anG(x)n+PN
n=0bnG(x)nx, for some coefficientsan, bn∈C. We then conclude
Πp(∆π)ϕi =
N
X
n=0
anΠG(∆π)nϕi+
N
X
n=0
bnΠG(∆π)n∆πϕi =
=
N
X
n=0
an∆nπϕi+ (d−2)
N
X
n=0
bn∆nπϕi ∈Φ, soΦis invariant underΠ.
Γ(d−1)N, i
1
1
1 1
1
Figure 5.9: The functionψ∈`2π.
We now introduce the functionψ∈`2π, defined byψ(pi) = 1for everyi∈X, and zero everywhere else. Observe thatψis orthogonal toϕifor alli∈X.
Lemma 5.4.14. We have
Π∗ψ= (∆π−(d−2))ψ.
Proof. On the one hand,Π∗ψis the function which takes value1on theA-pieces to whichpi belong, for everyi∈X, and zero everywhere else. On the other hand,∆πψtakes valued−1 onpiand value1on itsA-neighbors, for everyi∈X. Hence(∆π−Π∗)ψ= (d−2)ψ.
We briefly recall the functions θ(x) = x+22 , k(x) = x + 2, h(x) = 2(d−1)−x, ρ(x) = 12, and the relationsθ =kρ andhk = h◦G. Let us define as well the functions l(x) =x−(d−2),τ = lθ2 andσ= τk = lρ2.
Proposition 5.4.15. The spectral measure of ∆π associated with ψ is discrete. More precisely,ψis contained in the direct sum of the eigenspaces of∆π.
Proof. Letµ=µψ|Λbe the restriction toΛof the spectral measure of∆π associated with ψ. By Theorem 4.2.8, we have to show thatµ= 0. Letµ0 =µΠ∗ψ|Λbe the restriction toΛ of the spectral measure of∆πassociated withΠ∗ψ.
By Lemma 5.4.14, µ0 = l2µ, and by Proposition 5.4.8, µ0 = L∗θµ, so we obtain
+1 which is strictly smaller than1, provided thatd >2.
Let nowλσbe the spectral radius ofLσ, and letνσbe the probability measure onΛfrom
Recall that forf, g∈`2π, the spectral measureµf,g is the measure whose moments are h∆nπf, gi, for everyn≥0, so thatµf =µf,f.
We are now ready to prove the decomposition of the space`2π into the direct sum of eigenspaces of∆π andΦ.
Proposition 5.4.17. For everyf ∈Φ⊥, the spectral measureµf of∆π associated withf is discrete. The eigenvalues of∆π|Φ⊥ areS
n≥0G−n(d−2).
Proof. First, we know by Corollary 5.4.5 that the eigenspacesEx are nontrivial for every x ∈ S
n≥0G−n(d−2). LetP be the orthogonal projector toΦ⊥ in`2π. Lemma 5.4.13 implies thatP commutes with∆π,Π∗ andΠ. Let us show that, for everyf, g ∈ `2π, the
measureµP f,g is discrete and concentrated onS
n≥0G−n(d−2). It suffices to considerf finitely supported.
LetΓnπ = (Γn(d−1)N, X)be the subgraph ofΓπconsisting of vertices of the form(Xn(d−
1)N, X), for everyn≥ 0. Notice that onlydvertices inΓnπ have edges toΓπ\Γnπ, more precisely those having the form((d−1)n−10(d−1)N, i),i∈X. Let us call these vertices qin, and recall that pi were the vertices in the central B-piece, of the form((d−1)N, i), i∈X.
For everyi∈X,qinhas exactlyd−1B-neighbors, which lie outsideΓnπ. Let us call themri,jn , forj= 1, . . . , d−1. Intuitively, the subgraphΓnπis the same asS
i∈XBpi(2n−1), and S
i∈XBpi(2n) is the subgraph spanned by the union of Γnπ and {rni,j | i ∈ X, j = 1, . . . , d−1}. Let us call this other subgraphΓ¯nπ. See Figure 5.10 for a picture of these graphs.
p0
p1
p2 p3
pd−1
q0n
q1n
qn2 q3n
qd−1n r0,1n
rn0,2 rn0,3
rn0,d−1
Figure 5.10: The subgraphΓnπ containsqin, fori∈X. The subgraphΓ¯nπ contains also the verticesrni,j, fori∈Xandj= 1, . . . , d−1.
Consider now the subspaceLnof functions vanishing outsideΓ¯nπwhich are constant on ri,jn , forj= 1, . . . , d−1. Namely,
Ln= f ∈`2π
supp(f)⊂Γ¯nπ, ∀i∈X,∀j, j0= 1, . . . , d−1, f(ri,jn ) =f(rni,j0) . For every n ≥ 0, notice that both ΠLn+1 and Π∆πLn+1 are contained inLn. The former is immediate, and so is, for the latter, the inclusion of the support inΓ¯nπ. Iff ∈Ln+1
and we setαto be its value atqn+1i andβits value atri,jn+1, for allj= 1, . . . , d−1, then the value ofΠ∆πf atri,jn is equal toα+ (2d−3)βfor allj= 1, . . . , d−1, so it does not depend onj. See Figure 5.11 for an illustration of these facts.
Γn+1π measureµP f,gis discrete and concentrated onS
n≥0G−n(d−2). measureµP f,gis discrete and concentrated onS
n≥0G−n(d−2)for everyg∈H. However, by Lemma 4.2.5 we know thatsp(∆π|H⊥) = {d−2,−2}, so the measure µP f,g must
be discrete and concentrated onS
n≥0G−n(d−2)∪ {−2}for everyg ∈`2π. Finally, by Lemma 4.2.7, we know that the eigenspaceE−2is trivial, soµP f,g({−2}) = 0and therefore µP f,gis discrete and concentrated onS
n≥0G−n(d−2).
Combining the results of Propositions 5.4.12, 5.4.15 and 5.4.17, we obtain the following Theorem:
Theorem 5.4.18. LetGω be a spinal group withd≥3,m = 1andω∈Ωd,1, and let∆π be the adjacency operator on the Schreier graphΓπfrom Theorem 3.7.3, forπ ∈Epi(B, A) occurring infinitely often inω. The space`2πcan be decomposed asΦ⊕Φ⊥, withΦbeing the subspace spanned by the functions∆nπϕi, forn≥0andi∈X.Φ⊥is the direct sum of the eigenspaces of∆π. The spectrum of∆π|Φis singular continuous and the spectrum of
∆π|Φ⊥is pure point.
Remark 5.4.19. This Theorem together with the results of Sections 3.7 and 5.3 allows to observe that the spectral type of the adjacency operator is not preserved under the Gromov-Hausdorff convergence in the space of marked graphs. Indeed, let us consider the example of Schreier graphs of the Fabrykowski-Gupta group. Using the notation from Section 3.7, we have
(Γπ, pi) = lim
n→∞(Γ1N,(d−1)n01N),
withπ ∈Epi(B, A)mappingbtoaandpi = ((d−1)N, i)∈Γπ. The graphΓ1N has pure point spectrum, by Theorem 5.3.9, and the marking of the graph does not change the spectral measures. Nonetheless, Theorem 5.4.18 implies thatΓπ has a Kesten spectral measure with nontrivial singular continuous part. A possible explanation for the appearance of the singular continuous component might be the fact thatΓπexhibits a natural nontrivial symmetry, given by the maps(ξ, i)7→(ξ, j), for anyi, j∈X. On the other hand, no marking ofΓ1N yields a similar type of symmetry.
Complexity
Linear subshifts constitute a remarkable subject of study in the area of symbolic dynamics, and spinal groups have previously been related to their Schreier graphs [38, 63, 39], although only for the binary case. In this chapter we review some notions of low complexity related to linear subshifts and extend them to the context of Schreier dynamical systems. We moreover characterize when they are satisfied by the dynamical systems arising from spinal groups.
6.1 Linear subshifts
LetAbe a finite alphabet,A∗ = ∪n∈NAn the free monoid of finite words onA, and let AZ be the set of all two-ended sequences of symbols in A, equipped with the product topology. Forω ∈ AZandn∈N, we callWn(ω) ={ωi. . . ωi+n−1|i∈Z} ⊂ Anthe set of all possible subwords ofω of lengthn, and we writeW(ω) = ∪n∈NWn(ω) ⊂ A∗. If u∈ A∗oru∈ AN, we also denote byWn(u)the set of all subwords ofuof lengthnand W(u) =∪n∈NWn(u)⊂ A∗.
Consider the shift operator T : AZ → AZ, given by (T ω)n = ωn+1. A subshift is a pair (Ω, T), where Ω ⊂ AZ is closed and invariant under T. For n ∈ N, we set Wn=Wn(Ω) =∪ω∈ΩWn(ω)⊂ An, the set of all subwords inΩof lengthn. The shiftT defines an action ofZonΩ. We call a subshiftminimalif this action is minimal, i.e., if all orbits are dense. There exist various measures of complexity of minimal subshifts, let us introduce two of them.
Definition 6.1.1. We say that a subshift(Ω, T)islinearly repetitiveif
∃C ≥1, ∀n≥0, ∀u∈Wn, ∀w∈WCn, u∈Wn(w).
In other words, if there is a constantC≥1for which every subword of lengthnoccurs in every subword of lengthCn.
Linear repetitivity is a strong form of minimality, and it has been widely studied (see [17], [25] and [50]).
Definition 6.1.2. We say that a subshift(Ω, T)satisfies theBoshernitzan condition(B)if there exists a T-invariant ergodic probability measureν onΩsuch that
lim sup
n→∞
nε(n)>0,
whereε(n) = min{ν(C(u)) |u ∈ Wn}andC(u) = {ω ∈ Ω |ω0. . . ωn−1 = u}. Here C(u)is the cylinder ofu, i.e., the set of sequences havinguat the positions0ton−1, and ε(n)is the minimum measure of these sets over all subwordsu. This value represents the minimal probability among all the possible subwords of lengthn.
This condition was introduced by Boshernitzan in [13], and it was shown to imply unique ergodicity for minimal subshifts in [14].
Toeplitz subshifts constitute a remarkable family of subshifts which provides examples of aperiodic subshifts but yet is simple enough to be studied in detail. A typical example of the so-calledsimple Toeplitz subshiftsis the substitutional subshift which encodes the Schreier graphs of Grigorchuk’s group. It was introduced in [55] and has been studied in a slightly different form in [38] and [37]. Some very explicit results about combinatorics and complexity of simple Toeplitz subshifts are proven in [63].
Definition 6.1.3. Let(ak)k ∈ ANbe a sequence of letters and(nk)k∈NNsuch thatnk≥2 for allk∈N. We definep(0) ∈ A∗as the empty word and, recursively, fork≥0,
p(k+1)=p(k)akp(k). . . akp(k),
where the wordp(k)appearsnktimes and the letterakappearsnk−1times. Becausep(k) is a prefix ofp(k+1)for everyk≥0, the sequence(p(k))kconverges to a one-ended infinite wordp(∞)∈ AN. Thesimple Toeplitz subshiftassociated with the coding sequences(ak)k and(nk)kis
Ω ={ω∈ AZ|W(ω)⊂W(p(∞))},
i.e., the set of all two-ended words whose subwords all occur as subwords ofp(∞).
There is an equivalent definition of simple Toeplitz sequences via a hole-filling procedure, which can be found in [63]. By construction, we can verify that for anyω ∈Ωin a simple Toeplitz subshift and for anyi∈Z, there existsn∈Zsuch thatωi+nkare all equal, for all k∈Z.
Proposition 6.1.4([52]). Simple Toeplitz subshifts are minimal and uniquely ergodic.
We bring our attention to the simple Toeplitz subshifts defined by (ak)k ∈ AN and (nk)k ∈NNsuch thatnk= 2for everyk∈N. In this case, fork≥0we have
p(k+1)=p(k)akp(k).
By a result in [63], we can characterize when simple Toeplitz subshifts are linearly repetitive or satisfy(B). Even though this characterization is given in general, here we will consider only the aforementioned subfamily of simple Toeplitz subshifts withnk= 2for everyk≥0.
LetA˜be the eventual alphabet of(ak)k, i.e., the set of letters which occur infinitely often in(ak)k. Let alsomk≥1be the smallestmfor which{ak, . . . , ak+m−1}= ˜A, for every k∈N.
Proposition 6.1.5([63]). Let (ak)k ∈ AN and(nk)k ∈ NNsuch thatnk = 2for every k∈Nand consider the simple Toeplitz subshift(Ω, T)they define. Then
• (Ω, T)is linearly repetitive if and only if(mk)kis bounded.
• (Ω, T)satisfies(B)if and only if(mk)khas a bounded subsequence.
As illustrated by this result, in the context of simple Toeplitz subshifts, the Boshernitzan condition(B)is a weaker analog of linear repetitivity.
Example 6.1.6. The Schreier graphs associated with Grigorchuk’s group (d= 2,m= 2) can be encoded as a simple Toeplitz subshift, which can also be obtained via a substitution.
Let A = {a, x, y, z} and(ak)k ∈ AN defined as a0 = a, ak = x, y, z whenever k is congruent with1,2,3modulo3, respectively, for everyk ≥1. Let(Ω, T)be the simple Toeplitz subshift associated with this sequence and(nk)k, withnk = 2for everyk ≥ 0.
In [38], this subshift was used in order to study spectral properties on the Schreier graphs of Grigorchuk’s group. We can alternatively construct this subshift with the following substitution onA:
τ(a) =axa, τ(x) =y, τ(y) =z, τ(z) =x.
The sequence(τk(a))k converges to a one-ended infinite wordη ∈ AN. The set of all two-ended wordsω∈ AZwhose subwords are subwords ofηis equal toΩ. The resulting simple Toeplitz subshift(Ω, T)is minimal, uniquely ergodic, linearly repetitive and satisfies (B).
It is moreover shown in [39] that Schreier graphs of all spinal groups withd= 2can be encoded by simple Toeplitz subshifts, however only few of them are substitutional.