Projectional entropy and the electrical wire shift

0.2 MB·English

by  Michael H. Schraudner

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1 that Theorem 4.1 in [3] is no longer valid – contrary to an assertion made by the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 <dletU ={~u(1),...,~u(r)},V ={~v(1),...,~v(d r)}(
−
Zd be two disjoint sets of integer vectors such that U∪˙ V is a linearly independent
set whose integer span span (U∪˙ V) = ~u(1),...,~u(r),~v(1),...,~v(d r) equals Zd.
Z (cid:10) − (cid:11)Z
Then L := span (U) = ~u(1),...,~u(r) ( Zd is called an r-dimensional sublattice
Z (cid:10) (cid:11)Z
of Zd. (Using ~u(1),...,~u(r) as generators, L is isomorphic to Zr.) The set L :=
′
span (V) = ~v(1),...,~v(d r) constitutes a complementary (d −r)-dimensional
Z (cid:10) − (cid:11)Z
sublattice.
Let X be some Zd shift and L be any r-dimensional sublattice (1 ≤ r < d).
By projecting points of X onto L we obtain a Zr shift X := {x| | x ∈ X} on
L L
which the Zr shift action is given as σ| . Now the L-projectional entropy of
L XL
X is the topological entropy of the Zr sh×ift X and we denote this quantity by
L
h (X):=h (X ).
L top L
Given a Zd shift X and some r-dimensional sublattice L (1 ≤ r < d) we form
a new Zd shift by taking the cartesian product of X with itself along some com-
L
plementary sublattice L′: (XL)Zd−r := QL′XL = {(x(w~) ∈ XL)w~ L′}. For every
~u∈L,~v ∈L′ the symbol at coordinate ~u+~v ∈Zd in the point (x(w∈~) ∈XL)w~ L′ is
given by (x(~v)) . Note that by shift-invariance this construction is independe∈nt of
u~
the complementary sublattice L we chose. Obviously X ⊆ (X )Zd−r. In the case
′ L
that X = (X )Zd−r, i.e. X is a full Zd r extension of X , we call X degenerate
L − L
(with respect to L).
Fixing ~u,~v ∈ Zd the finite set B := {~ı ∈ Zd | ∀1 ≤ k ≤ d : ~u ≤~ı ≤ ~v }
k k k
is called a (rectangular/cuboid) block and we will use the notation B = [~u,~v] to
denote this set. Moreover we set ~ ∈ Zd to be the vector with all its components
equal to 1, thus for n∈N we hav1e [−n~,n~]={~ı∈Zd |k~ık ≤n}.
0
We saya Zd SFT X is extendible, if fo1r an1yblockB =[~v,w~∞](Zd everyallowed
configuration P ∈AB actually is in L (X), i.e. every locally valid pattern P on a
B
block B can be extended to a point in X.
Wefinishthissectionbyrecallingthreeuniformmixingproperties. Thefirstone
consideringonlypairsofcubes [−n~,n~]comesfrom[3],whereasthe second,more
1 1
homogeneous looking one that takes into account arbitrary blocks was introduced
in [1].
A Zd shift X is called block strongly irreducible [3] if there exists a constant
s∈N such that whenever~ı,~∈Zd and m,n∈N satisfy that the distance between
the blocks~ı+[−m~,m~] and~+[−n~,n~] (with respect to the maximum-metric
on Zd) is larger tha1n s1any two patte1rns1P ∈ L (X), P ∈ L (X)
1 [ m~,m~] 2 [ n~,n~]
can be put together, i.e. there exists a point x ∈−X1wit1h x| −=1P1 and
~ı+[ m~,m~] 1
x| =P . − 1 1
~+[ n~,n~] 2
A−Zd1sh1ift X is calledblock gluing [1] if there exists a constant g ∈N such that
0
whenever the two blocks B = [~u(1),~v(1)], B = [~u(2),~v(2)] ( Zd have a distance
1 2
PROJECTIONAL ENTROPY AND THE ELECTRICAL WIRE SHIFT – PRELIMINARY VERSION3
largerthang anypairofpatternsP ∈L (X), P ∈L (X)canbeputtogether,
1 B1 2 B2
i.e. there exists a point x∈X with x| =P and x| =P .
B1 1 B2 2
In factit is nothardto show thateventhoughdefined seemingly different, both
notions actually coincide and we will use them interchangeably.
Lemma 1.1. A Zd shift is block strongly irreducible if and only if it is block gluing.
The third mixing property – putting a strictly stronger condition on X – that
will be used in our results is the uniform filling property (UFP) introduced in [7].
AZdshiftX hastheUFP[7]ifthereexistsafillinglengthl ∈N suchthatwhen-
0
everwetakeapointy ∈X andapatternP ∈L (X)onsomeblockB =[~u,~v](Zd
B
there exists a point x∈X with x| =P and x| =y| .
B Zd [u~ l~,~v+l~] Zd [u~ l~,~v+l~]
\ −1 1 \ −1 1
2. Main results on projectional entropy
In [3] the following theorem about 2-dimensional SFTs was proved and it was
claimed [3, page 250] that the same result holds for general dimensions d > 2 and
any sublattice L(Zd.
Theorem 2.1 ([3, Theorem 4.1]). Let (X,σ) be an extendible, block strongly irre-
ducible 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 sublat-
tices 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 min-
imal, 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 en-
tropy 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)<htop(X).
| <{z0 }
Since Y (X was arbitrary, X indeed is entropy minimal. (cid:3)
3. The construction of the electrical wire shift Wel
We build Wel in three steps: First we construct an extendible, block gluing Z2
SFTW whichmodelsasystemofstraightwiresrunninginaZ2 plane. Awiremay
branchintomultiplesubwiresandthosecanunifyagain. Howeverthereareneither
electricalsourcesnor consumers(sinks), thus once present,a wire has to goonfor-
ever without starting or ending at a certain coordinate in Z2. In a second step we
slightlymodifyW keepingtheextendabilityandtheblockgluingpropertybyinde-
pendently replacingthe occurencesofa particularsymbolwith elements ofa set of
k ≥2 distinct but completely interchangeable copies of this symbol. Doing this we
getafamilyofnewZ2 SFTsW whichhavelargerentropyandinparticularareno
k
longer entropy minimal. Finaflly we put together Z2 configurations of W building
2
ourelectricalwire shiftWel (WZ. AgainWel will be extendible andbflock gluing.
2
Points of Wel projected onto tfhe two-dimensional sublattice L := h~e ,~e i ( Z3
1 2 Z
will look like arbitrary Z2 configurations of W , but by imposing an “electrical”
2
condition along the ~e -direction we excludefthe possibility of having certain Z2
3
configurations of W sitting immediately next to each other. These restrictions
2
force Wel to be afproper subsystem of the full Z extension WZ. Nevertheless we
2
f
can show that h (Wel)=h (W )=h (Wel).
top top 2 L
f
Step 1 (The Z2 wire shift). The formal construction of the wire shift W involves
an alphabet A of 7 symbols which we think of as square tiles of unit length as
W
displayedinFigure1. Wewillrefertosymbol1astheblanksymbolandtosymbols
2 up to 7 containing finite segments of wires (thicklines) as the wire symbols.
These symbols can be placed next to each other only in a way that conserves
wires: Precisely the symbols 2,3,4,7 with a wire present at their left edge are
allowed to sit to the right of a symbol 2,3,4,6 having a wire at its right edge and
exactly the symbols 4,5,6,7 with a wire present on their lower edge can appear
PROJECTIONAL ENTROPY AND THE ELECTRICAL WIRE SHIFT – PRELIMINARY VERSION7
s s s s
1 2 3 4 5 6 7
Figure 1. The alphabet A of the wire shift W
W
aboveasymbol3,5,6,7havingawireonitsupperedge. Analogouslyonlysymbols
1,5,6areallowedtotherightof1,5,7andoneof1,2,3ispossibleabove1,2,4. Note
that by posing these nearest-neighbor restrictions as adjacency rules we define a
non-trivial,evenstronglyessential(i.e.allsymbolsoccurandallallowedtransitions
are realized in some point) Z2 SFT.
Lemma 3.1. The wire shift W is extendible and block gluing.
Proof. Since W is a nearest-neighbor SFT and the horizontal resp. vertical tran-
sitions only depend on the wires that are (are not) present at the vertical resp.
horizontal borders of each symbol we may consider only the configurations along
the boundaries of arbitrary blocks.
Let B = [~u,~v] ( Z2 be a rectangular block in Z2 and P ∈ L (W) any allowed
B
configuration on B. We define a point w ∈ W as follows: w| := P, and w| := 1
B ~ı
for all~ı ∈ Z2 \ [~u−~,~v+~]∪(~u−~ +Z~e )∪(~v +~ +Z~e ) . The remaining
(cid:0) 1 1 (cid:1)
1 1 1 1
coordinatesarefilledwithwiresymbolsaccordingtoFigure2,wherewiresegments
drawn in grey may or may not be necessary, depending on the symbols along the
border of the configuration P. It is easily checked that this configuration w truely
is a point in W. Therefore we have extended an arbitraryvalid patternP on some
finite rectangle B to the whole of Z2, which proves W to be extendible.
r r r r r r r r r
r r
P on B
r r
r r
r r r r r r r r r
Figure 2. Extending finite configurationsP on rectangular blocks B
SimilarilywemaytaketwopatternsP ∈L (W),P ∈L (W)onrectangular
1 B1 2 B2
blocks B ,B ( Z2 where we suppose B ,B are separated by a distance larger
1 2 1 2
8 MICHAELSCHRAUDNER
than 2. There are two cases: Either B , B are separated by a distance >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