PROJECTIONAL ENTROPY AND THE ELECTRICAL WIRE SHIFT – PRELIMINARY VERSION 9 0 MICHAELSCHRAUDNER 0 2 Abstract. In this paper we present an extendible, block gluing Z3 shift of n finite type Wel in which the topological entropy equals the L-projectional a entropy for a two-dimensional sublattice L(Z3, even so Wel is not a full Z J extension of WLel. In particular this example shows that Theorem 4.1 of [3] 6 does notgeneralizetor-dimensionalsublatticesLforr>1. 1 Nevertheless we are able to reprove and extend the result about one- dimensional sublattices for general Zd shifts under the same mixing assump- ] tion as in [3] and by posing a stronger mixing condition we also obtain the S correspondingstatement forhigher-dimensionalsublattices. D . h t a m 1. Preliminaries [ The purposeofthis paperis toconstructanexampleshiftoffinite type showing 1 that Theorem 4.1 in [3] is no longer valid – contrary to an assertion made by the v authors – in the case of higher-dimensional sublattices. Nonetheless we get the 4 claimed result even for general Zd shifts instead of extendible shifts of finite type 9 4 by imposing a stronger mixing condition. The statement about one-dimensional 2 sublatticesalsoholdsforgeneralZd shiftsevenintheoriginalsetting,i.e.assuming 1. only the weaker mixing property of [3]. 0 We assume a basic familiarity with (multidimensional) symbolic dynamics, thus 9 here we just fix some notation. 0 Every finite alphabet A gives rise to a d-dimensional full shift AZd (d ∈ N), : v a space equipped with the product topology of the discrete topology on A which Xi supports a natural Zd (shift) action σ : Zd × AZd → AZd given by translation r (σ~ı(x))~ =(σ(~ı,x))~ :=x~ı+~ for all~ı,~∈Zd, x∈AZd. a Any closed σ-invariantsubset X ⊆AZd together with the restricted shift action σ|Zd X constitutes a Zd shift. If X can be defined using a finite set P ⊆ AF of allow×ed patterns on some finite non-empty shape F (Zd, so that X ={x∈AZd | ∀~ı∈Zd : x| ∈P}, it is called a Zd shift of finite type (SFT). ~ı+F In the following we will use L (X) to denote the set of patterns {x| |x∈ X} F F appearing in X on some fixed finite subset F ( Zd of coordinates. The language L(X) of a Zd shift X consisting of all finite patterns that occur as subwords of elements of X then is the union of L (X) over all F (Zd finite. F 2000 Mathematics Subject Classification. Primary: 37B50;Secondary: 37B10, 37B40. Keywordsandphrases. Zd;multidimensionalshiftoffinitetype;projectionalentropy;entropy minimal;non-degenerate. Theauthor wassupportedbyFONDECYTproject3080008. 1 2 MICHAELSCHRAUDNER The topological entropy of a Zd shift X measures the exponential growthrate of patterns and is defined in complete analogy to the one-dimensional setting as log|L (X)| h (X):= lim Cn top n→∞ |Cn| where C :={~ı∈Zd |k~ık ≤n}. Often we will write h(X) instead of h (X). n top Following [3] we now de∞fine sublattices, projectional entropy and degeneracy of Zd shifts: Ford∈Nand1≤r 2 and any sublattice L(Zd. Theorem 2.1 ([3, Theorem 4.1]). Let (X,σ) be an extendible, block strongly irreducible Z2 SFT and L(Z2 a 1-dimensional sublattice. Then h (X)=h (X) if top L and only if X =(X )Z. L However,toensureacertainpropertyoftheprojectionalshiftX ,theprimordial L proof uses Corollary 4.4.9 from [4], which is valid for one-dimensional shifts only. Hencethegeneralizationtohigher-dimensionalsublatticesisnotobviousandinfact is not true in general: In Section 3 we construct a Z3 SFT named the “electrical wire shift” that provides an example for whichthe abovetheorem does not hold in the case of a two-dimensional sublattice. Proposition 2.2. The electrical wire shift Wel (see Section 3) is an extendible, block strongly irreducible Z3 SFT such that h (Wel)=h (Wel) for the sublattice top L L:=h~e ,~e i (Z3. Nevertheless Wel ((Wel)Z. 1 2 Z L We deferthe proofofProposition2.2to Section3whereallnecessaryproperties of Wel are shown during its construction. Existence of the electrical wire shift shows that the argument of Johnson, Kass and Madden used in the proof of their Theorem 4.1 does NOT extend to sublattices of dimension 2 or higher without additional conditions. One possible such assumption is a stronger mixing property like the UFP. Imposing this we obtain the following general theorem. (Note that we do not assume our Zd shift to be extendible or of finite type.) Theorem 2.3. Let X bea Zd shift with theuniform fillingproperty and let L(Zd be any r-dimensional sublattice (1 ≤ r < d), then h (X) = h (X) if and only if top L X =(X )Zd−r. L Proof. The implication “X =(X )Zd−r, thus h (X)=h (X)” is trivial. L top L FortheconverseweusethatifX hastheUFP,thesameistrueforthecartesian product (X )Zd−r. (Observe that we do not claim X seen as a Zr shift has to L L have the UFP though.) To show this, suppose X has filling length l ∈ N and L ( Zd is some (d−r)- ′ dimensionalsublattice complementaryto L. For any point y = y(w~) ∈X ∈ (cid:0) L(cid:1)w~ L′ ∈ 4 MICHAELSCHRAUDNER (X )Zd−r we can take a family y(w~) ∈ X of preimages under the projection L (cid:0) (cid:1)w~ L′ onto L such that for all w~ ∈ L wee have y(w~∈)| = y(w~) = y| . Similarily for ′ w~+L w~+L any finite pattern P ∈ L (X )Zd−r onesome block B = [~u,~v] (~u,~v ∈ Zd) we B(cid:0) L (cid:1) have a point z = z(w~) ∈ X ∈ (X )Zd−r realizing P, i.e. z| = P. Again (cid:0) L(cid:1)w~ L′ L B we have a family of preimages ∈z(w~) ∈ X with z(w~)| = z| . (cid:0) (cid:1)w~ L′ (w~+L) B (w~+L) B Applying the uniform filling propeerty in X f∈or every peair y(w~), ∩z(w~)| with w~∩∈ B L we get a family of points x(w~) ∈ X such thatex(w~)|e = z(w~)| and ′ (cid:0) (cid:1)w~ L′ B B x(w~)| = y(w~)| e . Now∈we can projecet each x(w~e) onto the Zd [u~ l~,~v+l~] Zd [u~ l~,~v+l~] ceoordin\at−es1in w~1+Leto obt\ain−a1poin1t x(w~) ∈ X . (More preciselyewe first shift L x(w~) by w~ ∈ L ( Zd, then project σ (x(w~)) ∈ X onto L and define x(w~) as the ′ w~ iemage σ (x(w~))| ∈ X .) Putting all tehe x(w~) together we finally form a point w~ L L x = x(w~)e∈ X ∈ (X )Zd−r and we can easily check that x| = P and (cid:0) L(cid:1)w~ L′ L B x| =y|∈ . Thus l is also a filling length for (X )Zd−r. Zd [u~ l~,~v+l~] Zd [u~ l~,~v+l~] L It\is−k1nown1 that th\e u−n1iform1filling property implies entropy minimality. For a detailed proof of this technical result see Lemma 2.7. Hence (X )Zd−r having the L UFP forces a strict entropy inequality h (X) < h (X )Zd−r = h (X ) = top top(cid:0) L (cid:1) top L h (X)wheneverX ((X )Zd−r isapropersubsystem. Thisfinishes ourproof. (cid:3) L L Remark 2.4. In [1, Appendix C] we defined the meandering streams shift and we proved it to be a non entropy minimal Z2 SFT which is corner gluing in each of the 4 corners (in NE-, NW-, SE-, SW-direction). Using a similar constructionas for the electricalwire shift, this example shows thatwe cannotweakenthe mixing assumption in Theorem 2.3 to 4-corner-gluing. In the case of 1-dimensional sublattices we get the corresponding generalization of Theorem 2.1 for Zd shifts even assuming only the weaker (original) mixing as- sumptionusedin[3]. Theproofofthefollowingtheoremwasfoundincollaboration with R. Pavlov [5]. Theorem2.5. LetX beablockgluingZd shiftandletL(Zd beany1-dimensional sublattice, then h (X)=h (X) if and only if X =(X )Zd−1. top L L Proof. Again one of the implications is trivial. For the converse let X be block gluing with gluing constant g ∈N and let L=hw~i be generated by w~ ∈Zd. We 0 Z claim that X is block gluing as well. L For this let P ∈ L (X ) and P ∈ L (X ) be any two finite words on L- 1 B1 L 2 B2 L intervals B = [u(1),v(1)] := {jw~ | u(1) ≤ j ≤ v(1)},B = [u(2),v(2)]( L. Thus we 1 2 havepointsy,z ∈X withy| =P ,z| =P andtakingpreimagesofthosegives L B1 1 B2 2 y,z ∈ X such that y| = y, z| = z. Define Zd-blocks B := [u(i)w~,v(i)w~] ( Zd L L i (eie=1,2). If we assueme v(1)+eg < u(2) those blocks B ,Be are at least a distance 1 2 g+1apartfromeachother. SinceX isblockgluingtheereeexistsapointx∈X with x|Be1 = y|Be1 and x|Be2 = z|Be2. Projecting x onto L we get a point x :=ex|L ∈ XL wehichreealizestheetwopateternsP ,P exactelyatB ,B andhenceg isalsoeagluing 1 2 1 2 constant for X . L NowrecallthatforZshifts tobe blockgluingisequivalenttohavingthe specifi- cationproperty–foradefinitionsee[2,Section21]. Nextweexploitthatexpansive dynamical systems with specification are intrinsically ergodic [2, Theorem 22.15]. Hence X carries a unique measure of maximal entropy usually called the Bowen L PROJECTIONAL ENTROPY AND THE ELECTRICAL WIRE SHIFT – PRELIMINARY VERSION5 measure µ ∈ M (X ) which has full support [2, Theorem 22.10, Proposition B Max L 22.17]. Then the product measure µ := µZd−1 ∈ M (X )Zd−1 is the unique B Max(cid:0) L (cid:1) maximal measure (ref ?) on the caertesian product (X )Zd−1. Since we assume L h (X) = h (X) = h (X )Zd−1 , every measure ν ∈ M (X) of maximal top L top(cid:0) L (cid:1) Max entropy for X is at the same time a measure of maximal entropy for (X )Zd−1. L Therefore ν = µ and (X )Zd−1 = supp(µ) = supp(ν) ⊆ X implies X = (X )Zd−1 L L as claimed. e e (cid:3) Remark 2.6. Observe that for sublattices L ( Zd of dimension r ≥ 2 the Zr shift X in general does not inherit the block gluing property from X. For one L such example use the electrical wire shift Wel and the 2-dimensional sublattice L generatedby~e and~e +~e . ChoosingN ∈N,apatternofblanksalongthediagonal 1 1 2 B := {n(~e +~e ) | 0 ≤ n ≤ N} ( L (which is a rectangular block in the Z2 shift 1 1 2 Wel) forces blanks on all of {m~e +n~e | 0 ≤ m,n ≤ N} ( L. Therefore it never L 1 2 can be put together with a non blank at B = {N~e }, even though the distance 2 1 between the blocks B and B equals N and thus can be made larger than any 1 2 given gluing constant. So in the proof of Theorem 2.5 we implicitly had to use the special geometry of a one-dimensional lattice and this is the reason why the result is necessarily different from the one obtained for a higher dimensional sublattice. The following technical fact about the topological entropy of subsystems of Zd shifts having the UFP seems to be known at least for SFTs [6], though we are not aware of an explicit demonstration in the literature. For completeness we include a proof which also gives the result in the case of general Zd shifts. Lemma 2.7. Every Zd shift X having the uniform filling property is entropy minimal, i.e. every non-empty proper subsystem of X has strictly smaller (topological) entropy. Proof. Let Y ( X be a proper subsystem of X. Hence there exists a pattern P ∈ L(X)\L(Y), say of shape B = [~,n~] for some n ∈ N. Assume X has the 1 1 UFP with a filling length l∈N and put B :=[(1−l)~,(n+l)~]. We prove the following bound on the neumber of va1lid Y-pat1terns in comparison to the number of valid X-patterns on a large hypercube C(N) := [~,N(n+l)~] 1 1 (N ∈N). 1 Nd (PB) (cid:12)LC(N)(Y)(cid:12)≤(cid:0)1−(cid:12)LBe(X)(cid:12)− (cid:1) ·(cid:12)LC(N)(X)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) For this let J = {~ | 1 ≤ i ≤ Nd} := {~ ∈ ~ +(l+n)N d | k~k ≤ N(l+n)} i 0 and for every subset I ⊆J define 1 ∞ LI (X):={x| |x∈X ∧∀~∈J \I : x 6=P} . C(N) C(N) ~+B Note that L (X) = {x| | x ∈ X ∧∀~ ∈ J : x 6= P} and LJ (X) = ∅C(N) C(N) ~+B C(N) L (X). Now for every~ı∈I ⊆J we have C(N) UFP (cid:12)LIC(N)(X)(cid:12) ≤ (cid:12)L~ı+Be(X)(cid:12)·(cid:12)(cid:8)x|C(N) (cid:12)x∈X ∧x|C(N) ∈LIC(N)(X)∧x|~ı+B =P(cid:9)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) and using L e(X)=Le(X) we get a lower bound ~ı+B B (*) (cid:12)(cid:8)x|C(N) (cid:12)x∈X∧x|C(N) ∈LIC(N)(X)∧x|~ı+B =P(cid:9)(cid:12) (cid:12) (cid:12) (cid:12) ≥(cid:12)LBe(X)(cid:12)−1·(cid:12)LIC(N)(X)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 6 MICHAELSCHRAUDNER Now we can estimate the number of valid Y-patterns on C(N). (cid:12)LC(N)(Y)(cid:12)≤(cid:12)L∅C(N)(X)(cid:12)=(cid:12)L{C~(1N})(X)\(cid:8)x|C(N) (cid:12)x∈X ∧x|~1+B =P(cid:9)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) =(cid:12)LC{~(1N})(X)(cid:12)−(cid:12)(cid:8)x|C(N) (cid:12)x∈X ∧x|C(N) ∈L{C~(1N})(X)∧x|~1+B =P(cid:9)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (*) ≤ (cid:0)1−(cid:12)LBe(X)(cid:12)−1(cid:1)·(cid:12)L{C~(1N})(X)(cid:12)≤(cid:0)1−(cid:12)LBe(X)(cid:12)−1(cid:1)2·(cid:12)L{C~(1N,~)2}(X)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ≤...≤(cid:0)1−(cid:12)LBe(X)(cid:12)−1(cid:1)|J|·(cid:12)LJC(N)(X)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 1 Nd =(cid:0)1−(cid:12)LBe(X)(cid:12)− (cid:1) ·(cid:12)LC(N)(X)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) This proves (PB) and putting this bound into the definition of topological entropy yields htop(Y)≤Nlim Ndlog(cid:0)1−(cid:12)(cid:12)LBeN(Xd()l(cid:12)(cid:12)−+1(cid:1)n)+dlog(cid:12)(cid:12)LC(N)(X)(cid:12)(cid:12) →∞ 1 1 = (l+n)d log(cid:0)1−(cid:12)(cid:12)LBe(X)(cid:12)(cid:12)− (cid:1)+htop(X)2 along 1 2 direction ~e (suppose this is the vertical direction in Figure 2) or along direction 2 ~e . In the firstcase we surroundeachpattern P (i=1,2)with a wire as in Figure 1 i 2. Note that the ~e -separation of at least two coordinates is large enough to do 2 this without causing any conflict in placing symbols. Hence – filling all remaining coordinates with blanks – there is a valid point w ∈ W realizing both patterns, i.e. w| = P and w| = P . For the second case, B ,B being separated along B1 1 B2 2 1 2 direction~e by a distance >2, we just have to rotate the picture in Figure 2 by 90 1 degrees, which can be done as the alphabet and the transition rules are invariant under symbol rotation, and proceed as before. This proves W being block gluing (at gap g =2) as claimed. (cid:3) Remark 3.2. Since the wire shift W is block gluing andhasanalphabet withmore than one symbol its topological entropy is strictly positive. Calculations give the estimate log1.75 < h (W) < log1.964. Moreover every non-trivial continuous top factorofW is block gluing againandthus has to havestrictly positive entropy,i.e. W has topologically completely positive entropy. Step 2 (Splitting the blank symbol). Next we modify W by splitting the blank symbol into k ≥ 2 distinct, but completely interchangeable copies, thus the new alphabet A := {1 | 1 ≤ i ≤ k} ∪˙ {2,3,4,5,6,7} has (k + 6) symbols. The k i adjacencyrulesconcerningwire-conservationstayunchangedanditcanbechecked that the above argument showing extendability and the block gluing property for W (Lemma 3.1) does not at all depend on the number of distinct types of blank symbols and thus just carries over. Hence we immediately get Corollary 3.3. For k ≥ 2 the new Z2 SFTs W again are extendible and block k gluing (at gap g =2). f In addition we are able to calculate the topological entropy and show that W k f is no longer entropy-minimal. Lemma 3.4. For every k ≥ 2 the topological entropy of W equals logk. In k particular each W contains a proper Z2 SFT subsystem of fufll entropy. k f Proof. Astherearenorestrictionsonthetypesofblanksymbolsthatcanbeplaced nexttoeachotherhorizontallyorvertically,W containsthefullshiftonitskblanks k f as a subsystem and thus h (W )≥logk. top k f 1 1 r r r r 1 2 , , , , Figure 3. W has corner condition 2 2 f To show the reversed inequality we look at the case k = 2 and check that W 2 actually has corner condition 2, i.e. for any given configuration on the coordinaftes ~ı+{(−1,−1),(−1,0),(0,−1)}thereareexactlytwosymbolspossibleatcoordinate ~ı ∈ Z2 (see Figure 3). For k > 2, given a corner configuration of the first type PROJECTIONAL ENTROPY AND THE ELECTRICAL WIRE SHIFT – PRELIMINARY VERSION9 (left-most) in Figure 3 filling in one of the blanks yields precisely k possibilities, whereas for corner configurations of the other three types the number of possible symbols stays 2. Hence none of the cases give more than k choices for the next symbol. Following from this any configuration on the lower left border R := {~ı ∈ C |~ı = −n ∨~ı = −n} of a big square C := [−n~,n~] for n ∈ N allows for at n 1 2 n most k4n2 ways to fill in the remaining 4n2 coordina1tes1of C . Summing over all n configurations on R we get a coarse estimate (cid:12)LCn(Wk)(cid:12)≤ X k4n2 ≤(cid:12)Ak(cid:12)|R|·k4n2 =(k+6)4n+1·k4n2 . (cid:12) f (cid:12) (cid:12) (cid:12) f R(Wk) L Putting this bound into the definition of topological entropy gives htop(Wfk)=nl→im∞log(cid:12)(cid:12)L(cid:12)CCnn((cid:12)Wfk)(cid:12)(cid:12) ≤nl→im∞log(cid:0)(4kn+2+6)44nn++1·1k4n2(cid:1) =logk . (cid:12) (cid:12) Soh (W )=logkasclaimedandthefullshiftonthekblanksymbolsconstitutes top k a properfSFT subsystem of full entropy. (cid:3) Remark 3.5. WedonotknowwhetherthewireshiftW fromStep1itselfisentropy minimal or not. However the above shows that there is a conceptual change in entropy between the case of only one blank in W with h (W) > log1 and the top case of k ≥ 2 blanks in W with h (W ) = logk. So having (at least) the k top k two blanks is enough to cofmpensate thefentropy of the underlying shift W, thus generating a proper subsystem of equal entropy. It seems that this phenomenon does occur in many Zd shifts (see e.g. the meandering streams shift in [1]) where we can eventually get non entropy minimal systems by independently splitting an apropriate symbol (in the presence of a fixed point) or a couple of symbols (in the presence of some periodic point). To see that none of the SFTs W is actually topologically conjugate to a full k f shift it suffices to count the fixed points of W – there are k+2. k f Lemma3.6. NoneofthemodifiedwireshiftsW (k ≥2)isdegeneratewithrespect k to any sublattice. The same is true for W. f Proof. Let L=h~ui (Z2 be any 1-dimensionalsublattice; then at least one of the Z twostandardbasevectors~e ,~e ∈Z2doesnotbelongtoL. Fixingacomplementary 1 2 sublattice L =h~vi (Z2 we may assume that~e =m~u+n~v (resp.~e =m~u+n~v) ′ Z 1 2 with m,n ∈ Z and n 6= 0. As W contains points seeing an arbitrary symbol (cid:0) k(cid:1)L f of A at a particular coordinate in L we may pick w(1),w(2) ∈ W such that k (cid:0)fk(cid:1)L w(1) = 7 and w(2) = 2. Now if W were degenerate with respect to L, i.e. W = ~0 mu~ fk fk w(~ı) ∈ W , a family w(~ı) with w(~0) := w(1) and w(n~v) := w(2) (cid:8)(cid:0) (cid:0) k(cid:1)L(cid:1)~ı L′(cid:9) (cid:0) (cid:1)~ı L′ would givefa valid∈point w := w(~ı) ∈∈ W . However this point w would see a symbol 7 at the origin and a(cid:0)symb(cid:1)~oı∈lL2′ at fcokordinate ~e (resp. ~e ), which is not 1 2 possible due to the transition rules forcing wire conservation. (cid:3) Looking at the projectionalentropies of W we get that the infimum of h (W ) k L k taken over all sublattices L ( Z2 is attainfed exactly in the two principal difrec- tions. Moreoverthisinfimumisstrictlyboundedawayfromthetopologicalentropy h (W ). top k f 10 MICHAELSCHRAUDNER Lemma 3.7. The projectional entropy of W (k ≥ 2) with respect to the hori- k zontal respectively vertical sublattice L = hf~e i ( Z2 resp. L = h~e i ( Z2 is 1 1 Z 2 2 Z h (W ) = h (W ) = log 2+ k + √k2 4k+8 . For every other sublattice L ( Z2 L1 k L2 k (cid:0) 2 −2 (cid:1) f f the projectional entropy is h (W )=log(k+6). L k f Proof. For L the projection W is the 1-dimensional subshift given by the 1 (cid:0)fk(cid:1)L1 nearest neighbor transition conditions that ensure wire conservation in the hori- zontaldirection,i.e.asequence(w ) ∈A Z ofsymbolsisvalid,ifandonlyifthe i i Z k ∈ wire is conserved in each subword w w (i ∈ Z). Thus the projectional entropy i i+1 is given as the logarithm of the Perron-eigenvalue of the corresponding transition matrix. A simple calculation gives λ =2+ k + √k2 4k+8. P 2 −2 As the definition of the wire shifts W is completely symmetric with respect to k the~e - and~e -direction the same holdfs for the sublattice L . 1 2 2 For other sublattices L(Z2 different coordinates in L are never horizontallyor vertically adjacent. Thus we can freely place symbols on all coordinates in L. It is then easily checked that any configurationon L can be extended obeying the rules on wire conservationto get a configurationon all of Z2 (recall Figure 3 to see that every corner can be filled). This gives a valid point of W . Hence W = A Z k (cid:0) k(cid:1)L k f f and hL(Wk)=log(cid:12)Ak(cid:12). (cid:3) f (cid:12) (cid:12) In the following we are going to build a Z3 SFT and for this we will take the ~e -direction as vertical, whereas the ~e - and ~e -axes are situated in a horizontal 3 1 2 plane. Therefore from now on “above” resp. “below” refers to an increase resp. decrease of the ~e -component of Z3 coordinates. 3 Step 3 (The Z3 electrical wire shift). Let L := h~e ,~e i ( Z3. To get our Z3 1 2 Z SFT Wel we stockpile infinitely many Z2 configurations of W as horizontallayers 2 in a point in Wel, i.e. Wel ⊆ w ∈ A Z3 ∀n ∈ Z : w| f ∈ W = WZ. In (cid:8) 2 (cid:12) L+n~e3 2(cid:9) 2 (cid:12) f f order to get Wel being a proper subset of WZ we pose the following “electrical” 2 f restriction on the allowed vertical transitions: Whenever a wire symbol 2 resp. 5 appears at a coordinate~ı ∈ Z3 then at coordinates~ı±~e we are only allowed to 3 see an arbitrary blank or a symbol 5 resp. 2. If we have a symbol a ∈ {3,4,6,7} at some coordinate then there cannot be any wire symbol directly above or below; instead we have to see one of the blanks there. We can think of this condition as forbidding parallel wires next to each other in the ~e -direction, as they may cause 3 “interference” of signals. However note that wires may and will cross on vertically adjacent horizontallayers (placing a symbol 2 above a symbol 5 or vice versa). As this extra condition is still given by nearest-neighbor restrictions, Wel ( WZ is a 2 f SFT. The projection map π : Wel →W , w 7→w| restricting points in Wel to the L 2 L f horizontal sublattice L is surjective (taking any W -point as a configuration on L 2 f and filling Z3\L with blanks gives rise to a valid point in Wel), so Wel =W and L 2 then Wel ((Wel)Z. f L Lemma 3.8. Wel is extendible and block gluing. Proof. The argument is a slight elaboration of the proof we presented for Lemma 3.1. Taking any pattern P ∈ L (Wel) on some finite 3-dimensional cuboid B := B [~u,~v] ( Z3, to build a valid point w ∈ Wel with w| = P, we surround every B

Projectional entropy and the electrical wire shift

Similar Projectional entropy and the electrical wire shift