Following the similarities found between linear subshifts and Schreier graphs of spinal groups acting on the binary tree, it is our goal now to extend the notions studied for linear subshifts to more general dynamical systems arising from Schreier graphs, not necessarily linear.
LetGbe a group generated by a finite setS, and letGact on a topological spaceX by homeomorphisms. Recall from Section 3.7 thatSch : X → G∗,S is the map assigning to each pointx∈ X its marked Schreier graph(Γx, x), and letX0 ⊂ X be the set of continuity points ofSch. We defined the space of Schreier graphs by
GG,X = Sch(X0),
withGacting onGG,X by shifting the marked vertex. Recall also that in the space of marked graphsG∗,S we consider the topology of local convergence, a basis of which is given by the cylinder sets
C(Γ,x)(r) ={(Γ0, x0)∈ G∗,S | B(Γ,x)(r)∼=B(Γ0,x0)(r)}, for given(Γ, x)∈ G∗,S andr≥0.
Definition 6.2.1. We call the pair(GG,X, G)aSchreier dynamical system.
Remark 6.2.2. Schreier dynamical systems are similar to subshifts in the sense that GG,X ={(Γ, x)∈ G∗,S | ∀r≥0, ∃x0 ∈ X0, (Γ, x)∈C(Γx0,x0)(r)}.
Recall that(GG,X, G)isminimalif the action ofGonGG,X is minimal, i.e., if all orbits are dense. Equivalently, if there is an isomorphic copy of any ball in the orbit of any graph in Ω. Formally, if
∀r ≥0, ∀(Γ, x),(Γ0, x0)∈ GG,X, ∃g∈G s.t.g(Γ0, x0)∈C(Γ,x)(r).
Proposition 6.2.3. Any Schreier dynamical system(GG,X, G)is minimal.
Proof. By continuity, it suffices to show that, for everyx, x0 ∈ X0andr ≥0, there exists g ∈ Gsuch that (Γx0, gx0) ∈ C(Γx,x)(r). LetU = Sch−1(C(Γx,x)(r))∩ X0, which is open since cylinders are open andSch is continuous on X0. We have x ∈ U, because (Γx, x)∈C(Γx,x)(r). Since the action ofGonX is minimal, there existsg∈Gsuch that
gx0 ∈U. Hence,(Γx0, gx0)∈C(Γx,x)(r).
Let us now establish some notions of complexity for Schreier dynamical systems, which are inspired by the analogy with linear subshifts.
Definition 6.2.4. We say that(GG,X, G)islinearly repetitiveif there is an isomorphic copy of any ball at a linear distance from the marked vertex of any graph inGG,X. Formally, if
∃C ≥0 s.t.∀r ≥0, ∀(Γ, x),(Γ0, x0)∈ GG,X,
∃g∈G, |g| ≤Cr, s.t.g(Γ0, x0)∈C(Γ,x)(r).
Notice that this definition is a strengthening of minimality. If we compare both conditions, for linear repetitivity, we needgnot only to exist but also to have length bounded linearly on r.
Definition 6.2.5. We say that(GG,X, G)satisfies theBoshernitzan condition(B)if there is aG-invariant, ergodic probability measureν onGG,X such that
lim sup
r→∞
rε(r)>0 whereε(r) = min{ν(C(Γ,x)(r))|(Γ, x)∈ GG,X}.
Notice that the setD(r) ={C(Γ,x)(r)|(Γ, x)∈ GG,X}is finite, soε(r)is well defined as a minimum. Let dr = |D(r)|be the number of possible r-balls that occur in GG,X. Notice also that because of minimality, allr-balls are found in any orbit. Sinceν is an invariant measure,ν(C(Γ,x)(r)) = 1/drfor any(Γ, x) ∈ GG,X. We can hence rewrite the Boshernitzan condition as
lim sup
r→∞
r dr
>0.
Definition 6.2.6. We say that(GG,X, G)satisfies the condition(B0)if it satisfies the same condition as for linear repetitivity but, instead of for everyr≥0, for an increasing sequence going to infinity. Formally, if
∃C≥0, ∃(rn)n% ∞ s.t. ∀n≥0, ∀(Γ, x),(Γ0, x0)∈ GG,X,
∃g∈G, |g| ≤Crn, s.t.g(Γ0, x0)∈C(Γ,x)(rn).
Notice that linear repetitivity implies(B0).
Remark 6.2.7. We may now consider any minimal linear subshift(Ω, T)over an alphabet Aas a Schreier dynamical system. LetG=hta|a∈ Aibe the free product of|A|copies ofZ. The groupGacts onΩas follows
taω=
( T ω a=ω0
ω a6=ω0
.
Under this action, eachω ∈ Ωis stabilized by alltabuttω0. Assuming the subshift (Ω, T) is aperiodic, the associated Schreier graphs are as linear as in Figure 6.1. We can
tω−2 tω−1 tω0 tω1 tω2
T−2ω T−1ω ω T ω T2ω T3ω
Figure 6.1: Schreier graph of the action ofGonΩ, for the orbit of a pointω ∈Ω. Loops are omitted.
verify that the dynamical system given by(Ω, T)is equivalent to the Schreier dynamical system(GG,Ω, G).
In this setting, the conditions for linear repetitivity and(B)for(GG,Ω, G)coincide with the classical ones for the linear subshift(Ω, T).
Extending the notion of simple Toeplitz is more involved. Its definition involves the concatenation of words, which should be interpreted and redefined for graphs. Additionally, the group acting on a linear subshift is alwaysZ, which is no longer the case for Schreier dynamical systems. We now extend the notion of concatenation to graphs. LetGd∗,Σdenote the space of graphs with edge labels in an alphabetΣanddmarked vertices, ford≥1.
Definition 6.2.8. LetΣbe a finite alphabet andd1, d2≥1. LetΓ1 ∈ Gd1∗,ΣandΓ2 ∈ Gd2∗,Σ, with marked vertices(v0, . . . , vd1−1)and(w0, . . . , wd2−1), respectively. Theconcatenation ofΓ1 andΓ2at the verticesviandwj, for0≤i≤d1−1and0≤j≤d2−1, is the graph
Γ1 vi|wj Γ2= (Γ1tΓ2)/{vi=wj} ∈ G(d1+d2−2)∗,Σ, whose marked vertices are(v0, . . . ,vˆi, . . . , vd1−1, w0, . . . ,wˆj, . . . , wd2−1).
Γ1
v0 v1
v2
Γ2
w0 w1 w2
vi =wj Γ1 i|j Γ2 =
Figure 6.2: Concatenation ofΓ1andΓ2atviandwj.
This definition of graph concatenation will allow us to extend the notion of simple Toeplitz subshifts to Schreier dynamical systems.
Definition 6.2.9. Letd≥2and letSbe a finite, symmetric, generating set of a groupG.
LetΞ∈ G1∗,Sbe a finite graph, marked at the vertexξ, and letA ⊂ Gd∗,Sbe a finite alphabet of finite graphs, whose elements will be called letters. Let(Λk)k ∈ ANbe a sequence of letters. We recursively defineΓ(k)∈ G2∗,Sas follows:
• Γ(0)is the graph with one vertex and no edges. We consider it marked twice at that vertex.
• Fork≥0, letΓ0, . . . ,Γd−1beddisjoint copies ofΓ(k)∈ G2∗,S, with eachΓimarked at the vertices(γi, γi0). Let alsoΞ1, . . . ,Ξd−2 bed−2disjoint copies of the graph Ξ∈ G1∗,S, with eachΞi marked at the vertexξi. We set
Γ˜i =
( Γi γ0i|ξi Ξi 1≤i≤d−2 Γi i= 0, d−1 .
Let(λ0, . . . , λd−1)be the marked vertices ofΛk. We defineΓ(k+1)∈ G2∗,S as Γ(k+1)= Λk λ0|γ0 Γ˜0 λ1|γ1 Γ˜1 λ2|γ2 . . . λd−1|γd−1 Γ˜d−1,
with marked vertices(γ00, γd−10 ).
Λk Γ0 λ0=γ0
γ00
Γ1 Ξ1
λ1=γ1 γ10 =ξ1
Γ2 Ξ2
λ2=γ2
γ02=ξ2
Γd−1
λd−1 =
γd−1 γd−10
Γ(k+1)
Figure 6.3: Construction of the graphΓ(k+1).
Notice that this construction is well defined, sinceΓ(k)∈ G2∗,S for everyk≥0. Indeed, this is true by definition fork= 0. Fork≥1, the graphsΓ˜ibelong toG1∗,Sfor1≤i≤d−2, and toG2∗,S ifi = 0, d−1. We concatenate a graph withdmarked vertices withd−2 graphs with one marked vertex and2graphs with two marked vertices, so the resulting graph has two marked vertices.
Now, once the sequence(Γ(k))kof finite graphs is defined, let(vk, vk0)be the marked vertices ofΓ(k) ∈ G2∗,S. For anyr ≥0, the ballBΓ(k),v0k(r)will be isomorphic forkbig enough. Hence, we have a sequence of marked graphs(Γ(k), v0k)∈ G1∗,S which converges to a marked graph(˜Γ(∞), v∞)∈ G1∗,S in the local convergence topology. LetΓˆ0, . . . ,Γˆd−1
beddisjoint copies ofΓ˜(∞), marked at the verticesv0, . . . , vd−1, and letΛ∈ Aoccurring infinitely often in(Λk)k, marked at the verticesλ0, . . . , λd−1. We finally define
Γ(∞)= Λλ0|v0 Γˆ0 λ1|v1 Γˆ1 λ2|v2 . . . λd−1|vd−1 Γˆd−1.
Notice that not every graph constructed in such a way is necessarily a Schreier graph.
Whenever it is, we will call it asimple Toeplitz graph. The coding information of a simple Toeplitz graph is the tuple(d,Ξ,(Λk)k). We set
GΓ(∞) ={(Γ, x)∈ G∗,S | ∀r≥0, ∃v∈Γ(∞), (Γ, x)∈C(Γ(∞),v)(r)}.
Asimple Toeplitz Schreier dynamical systemis a Schreier dynamical system(GG,X, G) such thatGG,X =GΓ(∞) for some simple Toeplitz graphΓ(∞).
Remark 6.2.10. Let (Ω, T) be a simple Toeplitz subshift defined by an alphabetA and the sequences(ak)k ∈ ANand(nk)k ∈ NN, withnk = 2for everyk ≥0. Suppose that ak 6= a0 for everyk ≥ 1. We may recover this subshift as a simple Toeplitz Schreier dynamical system by settingd = 2as follows. Fora0 ∈ A, we setΛa0 to be the graph with two verticesuandvand an undirecteda0-edge between them, marked at(u, v). For a∈ A \ {a0}, we setΛato be the graph with two verticesuandvand an undirecteda-edge between them, plusa0-loops onuandvfor everya0 ∈ A \ {a0, a}, marked at(u, v). Let A0={Λa|a∈ A} ⊂ G2∗,A. The sequence(Λak)k ∈ A0Ndefines a simple Toeplitz graph Γ(∞). If we letG=hta|t2a= 1,∀a∈ Aibe the free product of|A|copies ofZ/2Z, acting onΩas
taω =
T ω a=ω0
T−1ω a=ω−1
ω otherwise
,
thenGΓ(∞) = GG,Ω, and the orbits are the same as for(Ω, T). Notice that ford = 2the graphΞdoes not play any role in the construction ofΓ(∞).
Proposition 6.2.11. LetGω be a spinal group withd ≥ 2, m ≥ 1 andω ∈ Ωd,m. The Schreier dynamical system(GG
ω,XN, Gω)is simple Toeplitz.
Proof. LetΞ,ΘandΛπ, forπ∈Epi(B, A), be as in Proposition 3.1.3. We can verify that the construction from Definition 6.2.9 with parameters(d,Ξ,(Λk)k)given byΛ0 = Θand Λk= Λωk−1, fork≥1, yields the same space of graphs asGG
ω,XN.
In Proposition 6.1.5, a characterization for linear repetitivity and(B)for linear simple Toeplitz subshifts was provided. We now want to give similar statements for the simple Toeplitz Schreier dynamical systems defined by spinal groups.
Let(GG,X, G)be a simple Toeplitz Schreier dynamical system, defined by(d,Ξ,(Λk)k).
We define a sequence of integers(mk)kas follows. For everyk ≥0,mk is the smallest integer such that {Λk, . . . ,Λk+mk−1} = {Λn}n≥k. For the Schreier dynamical system (GG
ω,XN,Gω)associated with a spinal groupGω, for everyk≥0,mkis the smallest integer such that{ωk, . . . , ωk+mk−1}={ωn}n≥k.
Theorem 6.2.12. Let Gω be a spinal group withd ≥ 2, m ≥ 1 andω ∈ Ωd,m, and let (GG
ω,XN, Gω)be its associated Schreier dynamical system. Then(GG
ω,XN, Gω)is linearly repetitive if and only if(mk)kis bounded.
Proof. Suppose that mk ≤ M for every k ≥ 0. Let r ≥ 0. As linear repetitivity is a local condition and the action is minimal, it suffices to consider two pointsξ, ξ0 ∈Γξ, for ξ ∈XN\Cof((d−1)N). We have to findg∈Gω, of length not greater thanCr, such that the ballsB=B(Γξ,ξ)(r)andB(Γξ,gξ0)(r)are isomorphic, for a constantCindependent ofξ, ξ0andr.
Let us consider the foldingsϕk: Γξ→Γk, for everyk≥0, given byη7→η0. . . ηd−1, and letR≥0be the smallest such thatϕRis injective onB. We will assumeR≥2, as the possibilities for smallerRare not many and can be checked directly or absorbed intoC.
Recall thatΓRξ =XRσR(ξ)is the copy ofΓRwhich containsξ. By definition ofR, it containsB. Moreover, we setBi =B ∩ϕ−1R (XR−1i), so thatB=∪d−1i=0Bi. Letsi be the smallest such thatϕsiis injective onBi, and sets= maxi∈Xsi. Notice thatsi ≤s≤R−1.
By minimality ofR,ϕR(B)contains the central pieceΛωR−2 ofΓR, and by definition of swe havediam(ϕR(B))≤2 diam(Γs) + 1. In addition, by minimality ofsandsi, we find thatdiam(ϕR(B))≥diam(Γs−1) + 2. We then have
diam(Γs−1) + 2≤2r ≤2 diam(Γs) + 1, 2s−1 ≤2r≤2s+1−1.
Let us assume first thats ≤R−2. In this case,ϕR(B)is contained in the subgraph Xs(d−1)R−2−s0X ofΓR, and soB andϕR(B)are isomorphic. By assumption, there exists somet, withs≤t≤s+M−1, such thatωt=ωR−2. By choosingtminimal, we may assumes≤t≤R−2. We define the following injective map
ψ: Xs(d−1)R−2−s0X → Γt+2
w(d−1)R−2−s0i 7→ w(d−1)t−s0i .
Notice that since ωt = ωR−2, ψ preserves the edges of the graphs, and so it is in fact an isomorphism with its image. In particular, sinceBis isomorphic toϕR(B), it is also isomorphic toψ(ϕR(B)).
Define now g ∈ Gω to be the labeling of a shortest path in Γt+2 from ϕt+2(ξ0) to ψ(ϕR(ξ)), and setB0=B(Γξ,gξ0)(r). We have
ϕt+2(B0) =B(Γ
ξ,ϕt+2(gξ0))(r) =B(Γ
ξ,gϕt+2(ξ0))(r) =
=B(Γ
ξ,ψ(ϕR(ξ)))(r)∼=ψ(ϕR(B))∼=B.
This implies thatϕt+2(B0)is contained in the subgraphXt0XofΓt+2, and so thatB0and ϕt+2(B0)are isomorphic. Hence,B ∼=B0. Moreover,
|g| ≤diam(Γt+2) = 2t+2−1≤2s+M+1−1≤2M+3r.
Assume nows=R−1. For everyg∈Gωsuch thatgϕR(ξ0) =ϕR(ξ)inΓR, we set B0 =B(Γξ,gξ0)(r), and we have
ϕR(B0) =B(Γξ,ϕR(gξ0))(r) =B(Γξ,gϕR(ξ0))(r) =
=B(Γξ,ϕR(ξ))(r) =ϕR(B).
Notice also that ϕR must also be injective on B0, as otherwise we would have 2r = diam(B0) > diam(ϕR(B0)) = diam(ϕR(B)) = 2r. Let u be the vertex (d−1)R in ΓR. We have eitheru∈ϕR(B)oru6∈ϕR(B).
Ifu6∈ϕR(B), then letg∈Gωbe the labeling of a shortest path fromϕR(ξ0)toϕR(ξ) inΓR. In this case, we trivially haveB ∼=B0, and furthermore
|g| ≤diam(ΓR) = 2R−1 = 2s+1−1≤8r.
Ifu∈ϕR(B), letvbe its unique preimage inB. We proceed differently depending on whetherBfixesvor not.
Assume first thatv is fixed byB, which may only happen if d ≥ 3. Let g be the labeling of a shortest path fromϕR+1(ξ0)toϕR(ξ)1inΓR+1. SinceϕRis injective on both BandB0, so isϕR+1. Moreover,ϕR+1(B) ⊂ XRifor somei ∈ X while by definition ϕR+1(B0)⊂XR1. The unique preimagev0ofuinB0is also fixed byB, which implies that B ∼=B0. In addition,
|g| ≤diam(ΓR+1) = 2R+1−1 = 2s+2−1≤16r.
Suppose now thatvis not fixed byB, and belongs to a copy ofΛωN, for someN ≥R.
By assumption, there exists somet, withR ≤t ≤ R+M −1, such thatωt = ωN. We may choosetminimal so thatt ≤ N. Defineg ∈ Gω as the labeling of a shortest path fromϕt+1(ξ0)toϕR(ξ)(d−1)t−R0inΓt+1. Again asϕRis injective on bothBandB0, so isϕt+1. Moreover,ϕt+1(B) ⊂ XR(d−1)t−R+1 andϕt+1(B0) ⊂ XR(d−1)t−R0by
definition. Notice thatBandB0are isomorphic if and only if the loops atvandv0are labeled by the same generators. The former has loops labeled by generators inKer(ωN), while the latter has loops labeled by generators inKer(ωt). Since we chosetsuch thatωt = ωN, B ∼=B0. Furthermore,
|g| ≤diam(Γt+1) = 2t+1−1≤2R+M−1 = 2s+M+1−1≤2M+3r.
This concludes the proof of the implication⇐, as in all cases we found an elementg∈Gω
for which the ball of radiusraroundgξ0is isomorphic to the ball of radiusraroundξ, and if we takeC= 2M+4, we have|g| ≤Cr.
Conversely, assume that the Schreier dynamical system is linearly repetitive and(mk)k is not bounded. LetCbe the linear repetitivity constant, and setM such thatC≤2M−3− 1. Since (mk)k is not bounded, there exist n ≥ 0 and N ≥ n+M such that ωN 6∈
{ωn, . . . , ωn+M−1}. We assume moreover thatN is minimal and setr = 2n+2−1.
Let nowξ ∈ XN\Cof((d−1)N) andξ0 ∈ Cof(ξ)be two vertices ofΓξ, such that ϕN+2(ξ) = (d−1)N00 =: vandϕN+2(ξ0) = (d−1)N−1000 =: v0. Because of linear repetitivity, there existsg∈Gωwith|g| ≤Crsuch thatB:=B(Γξ,ξ)(r)∼=B(Γξ,gξ0)(r) =:
B0.
Notice thatr = diam(Γn+2) ≤ diam(ΓN+1). AsϕN+2(ξ) belongs to the subgraph XN00ofΓN+2, this implies thatϕN+2 is injective onB, and hence that it must be also injective onB0. We had thatB ∼=B0, so in particular
B(Γ
N+2,v)(r) =ϕN+2(B)∼=ϕN+2(B0) =B(Γ
N+2,gv0)(r).
At the same time,
|g|+r ≤(C+ 1)r≤2M−3(2n+2−1)≤2n+M−1−1<2N −1 =d(v, v0).
ThereforeϕN+2(B0) must be confined to the subgraphXN+10 ofΓN+2. However, the central piece of ΓN+2, a copy of ΛωN, is contained in ϕN+2(B), so its image by the isomorphism must be mapped to a copy ofΛωs withinXN+10, for somes ≤ N −1. If n≤ s≤N −1, we haveωs =ωN, which is a contradiction by minimality ofN, and if s < nthenϕN+2cannot be injective onB0, which is also a contradiction.
Theorem 6.2.13. Let Gω be a spinal group withd ≥ 2, m ≥ 1 andω ∈ Ωd,m, and let (GGω,XN, Gω) be its associated Schreier dynamical system. Then(GGω,XN, Gω)satisfies (B0)if and only if(mk)khas a bounded subsequence.
Proof. The proof is analogous to the proof of Theorem 6.2.12. For the implication⇐, let (mkn)nbe the bounded subsequence of(mk)k. We call a radiusr ≥0admissible if there exists somen≥0such that2kn−1 ≤2r ≤2kn−1, and define the sequence of radii from
the definition of condition(B0) as the increasing sequence of admissible radii, which is clearly unbounded.
Letrbe an admissible radius, with2k−1 ≤2r ≤2k−1withk=knfor somen≥0, and define the sequence of radii from the definition of condition(B0)as the increasing sequence of admissible radii, which is clearly unbounded. We proceed as for linear repetitivity and defineRands. The inequality2s−1≤2r≤2s+1+ 1then impliesk−1≤s≤k.
For the cases≤R−2, we precise that thetis chosen such thatk≤t≤k+M−1, instead, but we obtain the inequalitys ≤t ≤ s+M, which later allows us to conclude that|g| ≤2M+4r. Ifs=R−1, we proceed again as for linear repetitivity. The inequality k−1≤R−1≤kallows us to find the same bounds for the length ofgin each case.
The proof of the converse is analogous to the proof of Theorem 6.2.12 too. Assume that the Schreier dynamical system satisfies(B0), which sets the constantCand defines the sequence of admissible radii, and letM be as above. If(mk)k does not have a bounded subsequence, then there exists somen≥0such that, for everyk≥n,{ωk, . . . , ωk+M−1} 6=
{ωl}l≥k. Now possibly the radius2n+2−1is not admissible, but in that case we letrbe an admissible radius such thatr ≥2n+2−1and setn0 such thatr ≤2n0+2−1. Asn0 ≥n, there exists some minimalN ≥n0+M−1such thatωN 6∈ {ωn0, . . . , ωn0+M−1}. From this point on, we proceed as in the proof of Theorem 6.2.12, but writingn0in the place of n.
Corollary 6.2.14. Letd≥2andm≥2, and letΩLRandΩB0 be the subsets ofΩd,mfor which(GG
ω,XN, Gω)is linearly repetitive and satisfies(B0), respectively. Ifµis the uniform Bernoulli measure onEpi(B, A)N, we have
µ(ΩLR) = 0, µ(ΩB0) = 1.
Proof. First notice thatω ∈ΩLRif and only if(mk)kis bounded. If we set ΩMLR ={ω ∈Ωd,m | ∀k≥0, {ωk, . . . , ωk+M−1}={ωn}n≥k}},
thenΩLR=∪M≥1ΩMLR. BecauseΩMLRcannot contain any sequence with constantM-prefix, we have
µ ΩMLR
≤µ
G
ω0...ωM−1
not constant
ω0. . . ωM−1ΩMLR
= EM −E
EM µ ΩMLR ,
whereEdenotes|Epi(B, A)|. Hence,µ ΩMLR
must be zero, and asΩLR is the countable union of sets of measure zero, it must have measure zero itself.
Similarly, notice thatωB0 if and only if(mk)khas a bounded subsequence. Setting ΩMB0 ={ω∈Ωd,m | ∀k≥0, ∃l≥k, {ωl, . . . , ωl+M−1}={ωn}n≥l}},
we haveΩB0 =∪M≥1ΩMB0. In this case, forM ≥E, we can bound the measure ofΩMB0from below by adding the restriction that theM-prefix contains all epimorphisms inEpi(B, A).
Hence, we reduce the problem to finding the probability that a givenM-tuple ofEpi(B, A)M contains all the epimorphisms. Using the union bound, we see that the probability that the tuple does not contain all of them is not greater than(|E| −1)|E|−1
|E|
M−1
, and therefore
µ ΩMB0
≥µ
G
{ω0,...,ωM−1}=
=Epi(B,A)
ω0. . . ωM−1ΩMB0
≥
≥ 1−(|E| −1)
|E| −1
|E|
M−1!
µ ΩMB0 . This coefficient tends to1asM increases, which shows thatµ(ΩB0) = 1.
Theorems 6.2.12 and 6.2.13 are intended as analogous statements of Proposition 6.1.5 for the Schreier dynamical systems arising from spinal groups. For linear repetitivity, we obtain indeed a parallel result. Nevertheless, while in Proposition 6.1.5 the weaker analog for linear simple Toeplitz subshifts is condition(B), in Theorem 6.2.13 it is condition(B0).
These conditions are in fact equivalent in the context of linear simple Toeplitz subshifts.
However, we do not know whether this is the case for simple Toeplitz Schreier dynamical systems in general. A reasonable starting point to solve this question would be finding out whether this is true for spinal groups.
Visualization tool for Schreier graphs of automata groups
In the study of Schreier graphs of groups acting on rooted trees, the first step is always to understand those associated with the action on the finite levels of the tree. These finite graphs can be computed algorithmically with little effort. However, understanding their underlying structure is key if one wants to extrapolate to other possibly infinite Schreier graphs.
During the realization of this thesis, and with the aim of making this task simpler, an interactive graphical tool to visualize Schreier graphs of automata groups was developed. It can be found athttps://unige.ch/~perezper.
In this appendix, we provide a brief description of the tool as well as some screenshots to illustrate it, but we actively encourage the reader to visit the URL and play around with the examples that are included.
The first dropdown menu contains a list of examples of automata groups. After choosing one, and selecting a level of the tree from the second dropdown, clicking onVisualize
will create a view where the Schreier graph of the chosen automata group with respect to the stabilizer of any vertex of the selected level will be displayed.
The vertices will rearrange themselves in a way that minimizes the distance between neighbors but forcing a separation so that the graph is readable. Notice that loops are not drawn.
Any vertex can be dragged and dropped from their position, and the rest of the graph will adapt consequently.
a new panel appears. After choosing the number of states and the size of the alphabet, two tables must be filled, in order to specify the output and successor of each state and input. The graph will be displayed after clicking onVisualize. We provide here an example where the Hanoi towers group on3pegs is specified as custom automaton.
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