Characterizations of periods of multidimensional shifts

Transcription

1 Chrcteriztions of periods of multidimensionl shifts Emmnuel Jendel, Pscl Vnier To cite this version: Emmnuel Jendel, Pscl Vnier. Chrcteriztions of periods of multidimensionl shifts <hl > HAL Id: hl Sumitted on 8 Mr 2013 HAL is multi-disciplinry open ccess rchive for the deposit nd dissemintion of scientific reserch documents, whether they re pulished or not. The documents my come from teching nd reserch institutions in Frnce or rod, or from pulic or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de documents scientifiques de niveu recherche, puliés ou non, émnnt des étlissements d enseignement et de recherche frnçis ou étrngers, des lortoires pulics ou privés.

2 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS EMMANUEL JEANDEL AND PASCAL VANIER Astrct. We show tht the sets of periods of multidimensionl shifts of finite type (SFTs) re exctly the sets of integers of the complexity clss NE. We lso show tht the functions counting their numer re the functions of #E. We lso give chrcteriztions of some other notions of periodicity. We finish the pper y giving some chrcteriztions for sofic nd effective sushifts. 1. Introduction A multidimensionl shift of finite type (SFT) is set of colorings of Z 2 given y locl rules. SFTs re one of the most fundmentl ojects in symolic dynmics [LM95]. One importnt question is to determine whether two SFTs re conjugte. One pproch to solve the prolem is y the study of invrints, i.e. quntities relted to sushifts tht sty invrint under conjugcy. The most significnt invrints for SFTs re the entropy, tht mesures the growth of the numer of vlid ptterns, nd the set of periodic points. These two invrints re reltively well known in the one-dimensionl cse: The entropy of n SFT is the logrithm of the spectrl rdius of some mtrix relted to the SFT, nd the set of periodic points reltes to the cycles of the multi-grph represented y the mtrix. As consequence, the set of integers n so tht there exists periodic point of period n is semi-liner set. The sitution ecomes more complex for multi-dimensionl SFTs. This is linked to the fct tht the emptiness prolem for SFTs ecomes undecidle [Ber66; Ro71], in prt due to the existence of nonempty SFTs with no periodic points. While the theory of one-dimensionl SFTs indeed reltes to the theory of finite utomt of computer science, mny properties of multidimensionl SFTs re to e understood using computility theory. The cse of the entropy ws recently solved y Hochmn nd Meyerovitch[HM10]: A rel λ 0 is the entropy of multidimensionl SFT if nd only if it is right pproximle, tht is if we cn compute sequence of rtionl numers converging to λ from ove. This ws one of the first rticles in series linking dynmicl properties of multidimensionl SFTs nd computility theory [Sim09; AS09]. In this rticle, we give chrcteriztion of the sets of periods of multidimensionl SFTs using complexity theory. For given point x in n SFT, let Γ x = { v Z 2 z Z 2, x(z + v) = x(z) } e the lttice of periods of x. We will study different notions of periodic points: c is strongly periodic of period n > 0 if Γ c = nz 2 c is 1-periodic of period v Z N \ {(0, 0)} if Γ c = vz c is horizontlly periodic of period n > 0 if n is the lest positive integer so tht nz {0} Γ c 1

3 2 EMMANUEL JEANDEL AND PASCAL VANIER All these notions cn redily e generlized for ny dimension d > 2. For given sushift X, let P X (resp. P 1 X, Ph X ) denote the set of strong periods (resp. 1- periods, horizontl periods) of X. The third notion seems it more peculir. It is introduced in this pper s first, somewht esier, result on which ll other results will e uilt. To give chrcteriztion of these sets in terms of complexity clsses, we will hve to see these sets s lnguges. If L N, denote y un(l) = { n n L}. If L Z N, denote un(l) = { p q (p, q) L} { p c q ( p, q) L}. We will prove: Theorem 1.1. For ny L N, there exists n SFT X such tht L = P X if nd only if un(l) NP. Theorem 1.2. For ny L Z N, there exists two dimensionl SFT X such tht L = P 1 X if nd only if un(l) NSPACE(n). Theorem 1.3. For ny L N, there exists two dimensionl SFT X such tht L = P h X if nd only if un(l) NSPACE(n). Here NP denotes s usul the clss of lnguges computle y nondeterministic Turing Mchine in polynomil time. NSPACE(n) denotes the clss of lnguges computle y nondeterministic Turing Mchine in liner spce. Plese note the slight difference etween theorem 1.1 nd the others: the two other theorems re vlid for fixed dimension d = 2. Theorem 1.1 needs however to e formulted for ll dimensions t once: given lnguge L NP the dimension of the SFT for which L is set of strong periods depends on L. It is in fct hrd to provide sttement vlid exctly in dimension d = 2. Intuitively, the reson is tht SFTs cn e seen s model of computtion. For most models of computtion, the spce complexity of prolem L is generlly the sme. However the time complexity of prolem depends on the exct definition of the model: the prolem to decide if word is plindrome is provly qudrtic in model of Turing Mchine with one tpe, ut ecomes liner if the Turing Mchine hs two tpes. The two-dimensionl SFTs eing ssimilted s new model of computtion, there is no reson for it to ehve like specific, lredy known, model of Turing Mchine. Tht s why we hve to use more roust clss, NP, which coincides for ll resonle models of computtion (See e.g. the Invrince Thesis [EB]). The prolem does not pper for spce clsses, s they re lredy roust. Rther thn the set of periodic points, nother interesting quntity is the numer of periodic points. This quntity mkes only sense for strongly periodic points. If X is n SFT, denote y N X the mp from {} to N tht mps n to the numer of points of strong period n. Then we will prove: Theorem 1.4. Let f : {} N, there exists n SFT X such tht = N X if nd only if f #P Here #P denotes s usul the clss of functions corresponding to the numer of ccepting pths of nondeterministic Turing Mchine working in polynomil time. This theorem gives first insight on the ehvior of the Zet function of multidimensionl SFT [Lin96]. These chrcteriztions in terms of complexity clsses led to some closure properties on the set of periods. NP nd NSPACE(n) re closed under intersection nd union, so the sets of periods re lso closed under intersection nd union. The

4 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS 3 closure y union could of course e proven directly, y tking the disjoint union of the two SFTs. Note tht the closure y intersection is not trivil, s the clssicl construction y Crtesin product does not work s usefully s it would seem: It is not true tht the set of strong periods of X Y is the intersection of the sets of strong periods of X nd Y : If {2} (resp. {3}) is the set of strong periods of X (resp. Y ), then the set of strong periods of X Y will e {6} nd not s intended. Note in prticulr tht due to the peculir form of theorem 1.1, the SFT tht relizes the intersection of the strong periods of X nd Y cn e of higher dimension thn X nd Y. Finlly, since nondeterministic spce is closed under complementtion [Imm88; BDG88], 1-periods re closed under complementtion. The question whether the sets of strong periods re closed under complementtion is of course relted to the P vs NP prolem (More ccurtely, it is relted to the question NE = cone, nd to Asser s Prolem [JS74; DJMM]). This pper is orgnized s follows: The first section gives the necessry ckground oth in multidimensionl symolic dynmics nd complexity theory. We then proceed to prove ll four theorems. We first prove Theorem 1.3 on horizontl periods. The techniques used for this proof re the core of this rticle. The other three theorems then uild on this first proof, dding more nd more complex structures in the vrious constructions. We end this pper with discussion on similr results for multidimensionl sofic nd effective shifts rther thn SFTs. Some of the results of this pper were nnounced t the DLT conference in the extended strct [JV10]. 2. Preliminries 2.1. Symolic Dynmics. We give here primer on Multidimensionl Symolic Dynmics. See [LM95; Lin04] for more informtion Sushifts. Let Σ e finite lphet nd d > 0 n integer. A configurtion over Σ is coloring of Z d y Σ, tht is mp : Z d Σ. We denote y Σ Zd the set of ll configurtions over Σ, Σ Zd is lso clled the full shift on Σ. A pttern P is coloring of suset D Z d. D is the support of the pttern. A pttern is finite if D is finite. A pttern P of support D ppers in pttern P of support D if there exists position v D so tht v + D D nd P (v + z) = P (z) for ll z D. We will write P P to sy tht P ppers in P. Let F e set of finite ptterns. A pttern is dmissile for F if it contins no ptterns of F. The sushift X F defined y F is the set of ll configurtions where no pttern of F ppers: X F = {c Σ Zd P F, P c} A set X is sushift if there exists set F so tht X = X F. Sushifts cn lso e chrcterized y topologicl property, ut we will not need it here. A sushift of finite type (or shortly SFT) is sushift X F where F is finite. In this cse, we cn ssume tht ll ptterns of F re over the sme finite support D. In this setting, configurtion c is vlid if ll ptterns of support D ppering in c re not in F, such configurtion is clled point of X. The rdius of D is the smllest r so tht D [ r, r] d. The rdius of X F is the rdius of D. An effective sushift is sushift X F where F is recursively enumerle.

5 4 EMMANUEL JEANDEL AND PASCAL VANIER Let X nd Y e two d-dimensionl sushifts, lock code is mp F : X Y such tht there exists mp f : Σ V X Σ Y, with V = {v 1,..., v k } finite suset of Z 2 such tht for ny z Z: F (x) z = f(x z+v1,..., x z+vk ) A mp F : X Y is fctor mp if it is surjective lock code, Y is then clled fctor of X. A sushift is clled sofic if it is fctor of some SFT Periodic points. Let c Σ Zd e configurtion. A vector v Z d is vector of periodicity for c if c(z) = c(z + v) for ll z Z d. We write Γ c = {v Z d z Z d, c(z) = c(z+v)} for the set of vectors of periodicity of c. Γ c is of course lttice (i.e. (discrete) sugroup of Z d ). There re three cses for Γ c : Γ c = {0}: c hs no vector of periodicity. Γ c hs rnk d, then c is periodic. This corresponds to the notion of periodicity in dimension 1: finite orit. In prticulr, in this cse, one cn prove tht there exists n such tht nz d Γ c. If nz d = Γ c we will sy tht c is strongly periodic of strong period n. Γ c hs n intermedite rnk 1 k < d. We will sy tht c is k-periodic. If c is 1-periodic, then γ c = vz for unique vector v (upto opposite direction) which is clled the 1-period of c. S m,n + (0, 4r) S m,n S m,n (0, 4r) Figure 1. Representtion of S m,n for m = 5, n = 2 nd r = 1. Note tht for every point x in S m,n, the squre of size 2r nd of ottom right corner x is entirely contined in S m,n (S m,n + (0, 4r)) = D m,n. A configurtion which is 1-periodic in sushift of dimension d cn e seen s configurtion in sushift of dimension d 1. To mke this sttement exct, we need some definitions. Let X e n SFT of rdius r. Let (m, n) e integers with m n 0. Consider S m,n = {(x, y) Z 2 0 nx + my < 4mr}. nd D m,n = {(x, y) Z 2 0 nx + my < 8mr} = S m,n (S m,n + (0, 4r)) represented in figure 1. Now the numer of ptterns of support S m,n tht re (m, n)-periodic is finite, s ny such pttern is entirely determined y S ((0, m 1) Z) which contins t most 4rm points. So there re t most Σ 4rm such ptterns. Now consider the following directed grph G m,n (X):

6 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS 5 The vertices of G m,n (X) re ll ptterns of support S m,n tht re (m, n) periodic. There is edge from P to P if the pttern P P of support D m,n defined y P P (z) = P (z) if z S m,n nd P P (z) = P (z (0, 4r)) otherwise is vlid for X. Now it is cler tht there exists ijection etween configurtions of X with (m, n) s period nd i-infinite pths in G m,n (X). It is due to the fct tht Z 2 = i (S m,n + (0, 4r)i) nd tht ny squre of size 2r whose ottom right corner is in S m,n is in D m,n (this is where the hypothesis m n is used) so tht ny squre of size 2r in Z 2 is contined in D m,n + (0, 4r)i for some i. Thus the entire informtion is contined in this grph. It is then esy to otin the following consequences: Lemm 2.1. Let X e two-dimensionl SFT of rdius r. If X contins configurtion which is periodic of period (m, n), then it contins configurtion which is fully periodic. (If the finite grph G m,n (X) contins n infinite pth, it contins cycle) More precisely, if X contins configurtion with (m, n) s period, m n > 0, then it contins fully periodic configurtion with (m, n) nd (0, p) s periods, for some 0 < p Σ 4rm. We could lso otin chrcteriztion of 1-periods with the grph: Lemm 2.2. X dmits (m, n) s 1-period if nd only if the grph G m,n (X) contins pth u 0... u k so tht u i = u 0 for some i < k. u i+1 u 1 u j = u k for some i j < k For ech (m, n ) so tht d N, (m, n) = d (m, n ), there exists some pttern u l which is not (m, n ) periodic. We cn of course choose k 3m Σ 4rm. The first three conditions ensure tht there exists configurtion c which is 1-periodic nd dmits (m, n) s period. The lst condition ensures tht (m, n) is indeed the lest period of c Aperiodic SFTs nd determinism. Let X e Z d SFT, X is periodic when no point of X dmits periodicity vector. There re severl well known such SFTs for Z 2, most of them come from the study of tilings, see e.g. Berger [Ber66; Ber64], Roinson [Ro71], Kri [Kr96; Kr92]. We will need two dimensionl periodic SFTs with prticulr property in this pper : determinism. A two dimensionl SFT is north-west deterministic if for ny two symols, t positions (i, j) nd (i+1, j +1) there is t most one symol c llowed t position (i+1, j). Such n SFT ws constructed y Kri [Kr92], his prticulr SFT will e used in section 5. Est-determinism cn e determined in the sme wy: for ny two symols, t positions (i, j) nd (i, j + 1) there is t most one symol c llowed t position (i + 1, j) Computtionl Complexity. In this section we provide some ckground on computtionl complexity nd its links with sushifts of finite type. More informtion out computtionl complexity nd computility cn e found in [BDG88; AB09; Rog87].

7 6 EMMANUEL JEANDEL AND PASCAL VANIER Usully to model computtion, Turing mchines re used. Despite its power, this model is quite simple to descrie : shortly, Turing mchine is device with finite memory ut tht cn red/write on n infinite tpe t the position of its hed. Formlly, it is tuple (Q, Γ, B, Σ, δ, q 0, H) where: Q is finite set of sttes, Γ is the tpe lphet, the finite set of symols tht cn pper on the tpe, B Γ is the Blnk symol, Σ Γ \ {B} is the input lphet, q 0 is the initil stte, H is the set of hlting sttes, δ : Q \ F Γ Q Γ {,, } is the trnsition function. The symols, nd stnd for moving the hed to the right, left nd to let it where it is respectively. If prolem cn e nswered y Turing mchine, then it is clled decidle nd otherwise undecidle. The most fmous undecidle prolem is the Hlting Prolem: deciding whether given Turing mchine hlts with itself s n input. Another well known undecidle prolem is the Domino Prolem: given set of Wng tiles, does it tile the plne? A prolem is clled recursively enumerle (r.e.) if there exists Turing mchine enumerting its elements nd co-recursively enumerle (co-r.e.) when its complement is r.e. A complexity clss is clss of prolems decidle y Turing mchine such tht some resource is ounded. The usul restrictions of the resources re on time or spce : the clsses TIME(f(n)) re the clsses of prolems decidle in time f(n), where n is the size of the input nd f fonction from N to N. the clsses SPACE(f(n)) re the clsses of prolems decidle in spce f(n). Turing mchines my e nondeterministic, which mens tht the trnsition function is multivlued. In this cse, the input is ccepted if there exists sequence of trnsitions leding to n ccepting stte. Time nd spce complexity clsses re lso defined in the cse of nondeterministic Turing mchines : the clsses NTIME(f(n)) re the clsses of prolems nondeterministiclly decidle in time f(n), where n is the size of the input nd f fonction from N to N. the clsses NSPACE(f(n)) re the clsses of prolems nondeterministiclly decidle in spce f(n) As sid erlier, tilings nd recursivity re intimtely linked. In fct, it is quite esy to encode Turing mchines in tilings. Such encodings cn e found e.g. in [Kr94; Ch08]. Given Turing mchine M, we cn uild tiling system τ M in figure 2. The tiling system is given y Wng tiles, i.e., we cn only glue two tiles together if they coincide on their common edge. This tiling system τ M hs the following property: there is n ccepting pth for the word u in time (less thn) t using spce (less thn) w if nd only if we cn tile rectngle of size (w+2) t with white orders, the first row contining the input. Note tht this method works for oth deterministic nd nondeterministic mchines.

8 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS 7 s s s q 0 q 0 s s s s s s s s s s s s q 0 s q 0 q 0 s h h Figure 2. A tiling system, given y Wng tiles, simulting Turing mchine. The mening of the lels re the following: lel s 0 represents the initil stte of the Turing mchine. The top-left tile corresponds to the cse where the Turing mchine, given the stte s nd the letter on the tpe, writes, moves the hed to the left nd to chnge from stte s to s. The two other tiles re similr. h represents hlting stte. Note tht the only sttes tht cn pper in the lst step of computtion (efore order ppers) re hlting sttes.

9 8 EMMANUEL JEANDEL AND PASCAL VANIER 3. Horizontl periodicity in SFTs of dimension 2 nd spce complexity In this section, we give proof for theorem 1.3. We first prove tht the unry lnguge corresponding to horizontl periods cn e recognized in liner spce y nondeterministic Turing mchine (lemm 3.1) nd then the reciprocl (lemm 3.2). Lemm 3.1. Let L N e the set of horizontl periods of two-dimensionl SFT X, then un(l) NSPACE(n). Proof. Let X = X F e 2-dimensionl SFT on the lphet Σ. We will construct nondeterministic Turing mchine ccepting 1 n if nd only if n + 1 is horizontl period of X. The mchine hs to work in spce O(n), the input eing given in unry. Let r e the rdius of X, point is in X if nd only if ll its r r locks hve no su-pttern contined in F. Furthermore, we cn prove tht if there exists point of horizontl period n, then there lso exists such point, with verticl period t most Σ rn. Here is n lgorithm, strting from n s n input tht checks whether n is horizontl period of some point of X: Initilize n rry P of size n so tht P [i] = 1 for ll i. First choose nondeterministiclly p Σ rn Choose r i-infinite rows (c i ) 0 i r 1 of period n (tht is, choose r n symols). For ech r + 1 i p, choose i-infinite row c i of period n (tht is, choose n symols), nd verify tht ll r r locks in the rows c i... c i r+1 do not contin foridden pttern. At ech time, keep only the lst r rows in memory (we never forget the r first rows though). (Verifiction of the lest period) If t ny of the previous steps, the row c i is not periodic of period k < n, then P [k] 0 For i r, verify tht ll r r locks in the rows c p i... c p c 0... c i r 1 do not contin foridden pttern. If there is some k such tht P [k] = 1, reject. Otherwise ccept. This lgorithm needs to keep in memory only 2r rows nd the rry P t ech time, hence is in spce O(n). Lemm 3.2. Let L N e lnguge such tht un(l) NSPACE(n), then there exists two-dimensionl SFT X such tht n L if nd only if there exists point c X with horizontl period n. Proof. Tke nondeterministic Turing mchine M ccepting un(l) in liner spce. Using trditionl tricks from complexity theory, we cn suppose tht on input 1 n the Turing mchine uses exctly n + 1 cells of the tpe (i.e. the input, with one dditionl cell on the right) nd works in time exctly c n for some constnt c (depending only on the Turing mchine M). We re going to construct n SFT X such tht 1 n L if nd only if n + 4 is horizontl period of some point of X. The modifiction to otin n + 1 rther thn n + 4, nd thus prove the lemm, is left to the reder (siclly ftten the verticl lines of presented in lemm 3.3 elow so tht they sor 3 djcent tiles), nd serves no interest other thn technicl. The proof my siclly e split into two prts:

10 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS 9 First produce n SFT Y c such tht every point of horizontl period n looks like grid of rectngles of size n y c n delimited y horizontl nd verticl mrkers (see fig. 3) nd whose horizontl periods re N\{0, 1}. Lemm 3.3 shows how to construct such n SFT. The Turing mchine M is then encoded inside these rectngles on lyer C: the nondeterministic trnsitions re synchronized on every line to ensure the computtions inside the rectngles re the sme. The min difficulty lies in the first prt, while the second prt is strightforwrd nd does not need further explntion. Now we prove tht 1 n L if nd only if n + 4 is horizontl period of the the SFT X. For the component C to e vlid, the input 1 p 4 (4 = 1 ( symol of Y c ) + 1 (left order for the TM) + 1 (right order for the TM) + 1 (lnk mrker on the end of the tpe)) must e ccepted y the Turing mchine, hence 1 p 4 L Finlly, due to the synchroniztion of the nondeterministic trnsitions, the C component is lso p-periodic. As consequence, our tiling is p-periodic, hence n = p 4. Therefore 1 n+4 L Conversely, suppose 1 n L. Consider the coloring of period n + 4 otined s follows (only period is descried): The component A consists of n + 3 correctly tiled columns of our periodic Est-deterministic SFT, with n dditionl column of. Note tht periodic points exist. The component C corresponds to successful computtion pth of the Turing mchine on the input 1 n, tht exists y hypothesis. As the computtion lsts less thn c n steps, the computtion fits exctly inside the n c n rectngle. We then dd ll other lyers ccording to the rules to otin vlid configurtion, thus otining point of X of period exctly n + 4. Lemm 3.3. There exists n SFT Y k such tht for ny point y Y k of horizontl period p, y is composed of rectngles of size p k p 1 with mrked oundries. Furthermore, ll integers n 2 re horizontl periods. We will construct the SFT y superimposing severl components (or lyers) ech of them ddressing specific issue Y k = A C k T : A will llow us to force periodic tilings to hve columns seprted y verticl lines, C k will mke the horizontl lines nd nd t the sme time force the regulrity of the verticl ones, T will force tht within horizontl period, only one verticl line cn pper. The components nd their rules re s follows:

11 10 EMMANUEL JEANDEL AND PASCAL VANIER Proof. Figure 3. The shpe of the se SFT Y k : whenever point of Y k is periodic, it must hve the ove shpe where the width of the rectngles is exctly the period p nd their height k p 1. w0,9 w1,9 w2,9 w3,9 w4,9 w5,9 w6,9 w7,9 w8,9 w9,9 w10,9w11,9 w0,9 w1,9 w2,9 w3,9 w4,9 w5,9 w6,9 w7,9 w8,9 w9,9 w10,9w11,9 w0,8 w1,8 w2,8 w3,8 w4,8 w5,8 w6,8 w7,8 w8,8 w9,8 w10,8w11,8 w0,8 w1,8 w2,8 w3,8 w4,8 w5,8 w6,8 w7,8 w8,8 w9,8 w10,8w11,8 w0,7 w1,7 w2,7 w3,7 w4,7 w5,7 w6,7 w7,7 w8,7 w9,7 w10,7w11,7 w0,7 w1,7 w2,7 w3,7 w4,7 w5,7 w6,7 w7,7 w8,7 w9,7 w10,7w11,7 w0,6 w1,6 w2,6 w3,6 w4,6 w5,6 w6,6 w7,6 w8,6 w9,6 w10,6w11,6 w0,6 w1,6 w2,6 w3,6 w4,6 w5,6 w6,6 w7,6 w8,6 w9,6 w10,6w11,6 w0,5 w1,5 w2,5 w3,5 w4,5 w5,5 w6,5 w7,5 w8,5 w9,5 w10,5w11,5 w0,5 w1,5 w2,5 w3,5 w4,5 w5,5 w6,5 w7,5 w8,5 w9,5 w10,5w11,5 w0,4 w1,4 w2,4 w3,4 w4,4 w5,4 w6,4 w7,4 w8,4 w9,4 w10,4w11,4 w0,4 w1,4 w2,4 w3,4 w4,4 w5,4 w6,4 w7,4 w8,4 w9,4 w10,4w11,4 w0,3 w1,3 w2,3 w3,3 w4,3 w5,3 w6,3 w7,3 w8,3 w9,3 w10,3w11,3 w0,3 w1,3 w2,3 w3,3 w4,3 w5,3 w6,3 w7,3 w8,3 w9,3 w10,3w11,3 w0,2 w1,2 w2,2 w3,2 w4,2 w5,2 w6,2 w7,2 w8,2 w9,2 w10,2w11,2 w0,2 w1,2 w2,2 w3,2 w4,2 w5,2 w6,2 w7,2 w8,2 w9,2 w10,2w11,2 w0,1 w1,1 w2,1 w3,1 w4,1 w5,1 w6,1 w7,1 w8,1 w9,1 w10,1w11,1 w0,1 w1,1 w2,1 w3,1 w4,1 w5,1 w6,1 w7,1 w8,1 w9,1 w10,1w11,1 w0,0 w1,0 w2,0 w3,0 w4,0 w5,0 w6,0 w7,0 w8,0 w9,0 w10,0w11,0 w0,0 w1,0 w2,0 w3,0 w4,0 w5,0 w6,0 w7,0 w8,0 w9,0 w10,0w11,0 Figure 4. A periodic point of A. Here w i,j re symols of the lphet of W. The first component A is prtly composed of n periodic Est-deterministic SFT W, whose symols will e clled white symols. We cn tke the one from J. Kri [Kr92] 1. To otin A from W, we dd to the lphet new symol. With the dditionl foridden ptterns: no white symol shll e ove or elow, two cnnot pper next to ech other horizontlly. With this construction, periodic point of A of period p must hve columns of white symols seprted y verticl lines of. This is due to the fct tht W forms n periodic SFT. For the moment nothing forids more thn one gry column to pper inside period. Figure 4 shows possile form of periodic point t this stge. 1 Note tht in the pper the SFT is descried y Wng tilings nd tht it is NW-deterministic. However, it is strightforwrd to modify the rules in order to get n Est-deterministic periodic SFT. This exct SFT will e studied lter on in section 5

12 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS ) ) P 2 = { 1 0, 0 1, 0 1, 1 1} c) Figure 5. ) The trnsducer corresponding to k = 2 nd the symols of the corresponding SFT. A vlid pttern for C 2 is given in ) nd c): the reker symol in ) is from the A lyer nd corresponds lwys to the 1 symol of C 2. The second lyer of symols C k = P k {, } will produce horizontl lines so tht points of period np will consist of rectngles of size p k p, delimited y the symols nd. The ide is s follows: suppose ech horizontl segment etween two verticl lines is word over the lphet P k = {{0,... k 1} {0, 1}}, tht is, represents numer etween 0 nd k p 1 1 written in se k. It is then esy with locl constrints to ensure tht the word on the next line is + 1 mod k p 1. The {0, 1} component represents the crry. The lphet P k is composed of P k nd of crry 1 tht will e superimposed to the symol only. See figure 5 for trnsducer in the cse k = 2 nd its reliztion s n SFT. With the {, } sucomponent, we mrk the lines corresponding to the numer 0, so tht one line out of k p 1 is mrked. This line is the only one where 0 1 is trnsformed into 1 0 on the right of verticl line of. A is foridden to pper on the right or left of : this forces ech column to hve counters tht re resetted t the exct sme moment, nd thus to hve the exct sme size. Figure 6 shows some typicl tiling t this stge: the period of tiling is not necessrily the sme s the distnce etween the rectngles, it my e lrger. Indeed, the white symols in two consecutive rectngles my e different. Component T is formed of the sme lphet s W of component A, recll tht W is n Est-deterministic SFT. The foridden ptterns re tht two different symols cnnot e horizontl neighors. In ddition to tht, we forid of elements of component T on the right of to e different to the ones of component W. Tht mens tht the symols of the first column to the right of verticl line of re exctly the sme for ech verticl line of. The SFT W eing Est-deterministic, this mens tht the symols of the columns etween verticl lines of re exctly the sme for ech column, these re the oundries of the rectngles. At this stge, periodic point necessrily hs regulr rectngles on ll the plne, whose width correspond to the period, s shown in figure 6.

13 12 EMMANUEL JEANDEL AND PASCAL VANIER w0,3 w1,3 w2,3 w3,3 w0,3 w1,3 w2,3 w3,3 w0,3 w1,3 w0,3 w1,3 w0,3 w1,3 w0,3 w1,3 w0,2 w1,2 w2,2 w3,2 w0,2 w1,2 w2,2 w3,2 w0,2 w1,2 w0,2 w1,2 w0,2 w1,2 w0,2 w1,2 w0,1 w1,1 w2,1 w3,1 w0,1 w1,1 w2,1 w3,1 w0,1 w1,1 w0,1 w1,1 w0,1 w1,1 w0,1 w1,1 w0,0 w1,0 w2,0 w3,0 w0,0 w1,0 w2,0 w3,0 w0,0 w1,0 w0,0 w1,0 w0,0 w1,0 w0,0 w1,0 w0, 1w1, 1 w2, 1w3, 1 w0, 1w1, 1 w2, 1w3, 1 w0, 1w1, 1 w0, 1w1, 1 w0, 1w1, 1 w0, 1w1, 1 w0, 2w1, 2 w2, 2w3, 2 w0, 2w1, 2 w2, 2w3, 2 w0, 2w1, 2 w0, 2w1, 2 w0, 2w1, 2 w0, 2w1, 2 w0, 3w1, 3 w2, 3w3, 3 w0, 3w1, 3 w2, 3w3, 3 w0, 3w1, 3 w0, 3w1, 3 w0, 3w1, 3 w0, 3w1, 3 w0, 4w1, 4 w2, 4w3, 4 w0, 4w1, 4 w2, 4w3, 4 w0, 4w1, 4 w0, 4w1, 4 w0, 4w1, 4 w0, 4w1, 4 ) ) Figure 6. ) An exmple of vlid periodic point with C 2, the distnce etween two consecutive verticl lines is now constnt. However the period is not necessrily the width of the rectngles. ) Once we dd component T, the width of the rectngles is now exctly the period. Note tht we only show here the component A nd the sucomponent {, } of C 2. Now we prove tht if y Y k is periodic of period n, then it necessrily is formed of verticl lines of t distnce n of ech other nd of horizontl lines of t distnce k n 1 of ech other. Consider point of Y k of period n: Due to component A, verticl line of must pper. The period is succession of verticl lines of nd white columns. Due to component C k, the verticl lines of re spced y distnce of p, where p n. Furthermore, there re horizontl lines of t distnce k p 1 of ech other. Due to component T, the tiling we otin is horizontlly periodic of period p, thus p = n.

14 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS Strong periodicity in SFTs nd time complexity In this section we prove theorem 1.1 on strong periods. In squre n n one cn only emed computtions ending in time inferior to n. However, given n n squre of symols, one cnnot check tht it hs no foridden ptterns in less thn n 2 time steps with Turing mchine. The nlogue for higher dimensions holds: ny 2d-dimensionl cue of n 2d symols cn emed computtions in time n d, nd checking such cue needs n 2d time steps. Thus the clss NTIME 1 (n d ) for unry inputs 2 cn e cptured y 2d-dimensionl cues nd only d-dimensionl cues re checkle in time n d The gp here is not surprising: while spce complexity clsses re usully model independent, this is not the cse for time complexity, where the exct definition of the computtionl model mtters. An exct chrcteriztion of strongly periodic SFTs for d = 2 would in fct e possile, ut messy: it would involve Turing mchines working in spce O(n) with O(n) reversls, see e.g [CM06]. Here the solution comes from the fct tht periodic SFTs of ll dimensions would e cptured y the time complexity clss NP 1 = d N NTIME 1(n d ), the gp eing filled y the infinite union. This is theorem 1.1, whose proof elow will e, s efore, divided in three prts : We first show tht one cn check whether d dimensionl SFT is strongly periodic of period p in time p d nd thus tht the prolem is in NTIME 1 (n d ) (lemm 4.1). For the converse, we first construct se SFT with mrked cues (lemm 4.2) in similr wy s for horizontl periods, All tht is left to prove then is how the Turing mchines re encoded inside these cues (lemm 4.3). It is interesting to note tht the complexity clss NP 1 lso chrcterizes spectr of first order formul, see [JS74]. Lemm 4.1. For ny d dimensionl SFT X, un(p X ) NP: there exists Turing mchine M NP 1 tht given p s n input, determines whether p is strong period of X. Proof. It suffices to tke Turing mchine tht nondeterministiclly guesses p d cue nd then checks whether it contins ny foridden ptterns nd whether p is the strong period of X : for this lst prt, it hs to check tht for ll k < n, k is not period. Lemm 4.2. There exists d-dimensionl SFT Y d such tht : Any periodic point is strongly periodic. Any strongly periodic point y Y d of period p is constituted of djcent d-cues p d with mrked orders. Every integer p 2 is strong period of Y d. Proof. As efore, the construction will e sed on some periodic SFT, with some dded symols to rek the periodicity nd force regulr structure. The SFT Y d is mde of three lyers A S T : 2 Note tht the clss NTIME1 (n d ) for unry inputs corresponds to the clss NTIME 2 (2 dn ) for inry inputs nd tht complexity clsses re usully defined on inry inputs. So NP 1 is not the fmous NP clss, lthough NP = P would imply NP = P.

15 14 EMMANUEL JEANDEL AND PASCAL VANIER lyer A will force strongly periodic points to hve mrked lines, lyer S will force strongly periodic points to hve mrked d-cues, nd finlly, lyer T will force the strongly periodic points to e composed of d-cues of side p, the strong period. We will now detil ech component nd the strongly periodic points thus otined. Let W e 2-dimensionl NW-deterministic periodic SFT 3, we define 2-dimensionl SFT A : the lphet of A is composed of the lphet of W with the dditionl symols,,, the foridden ptterns of W re kept nd the following re dded: ove nd ellow my only pper or, on the left nd right of my only pper or, on the left nd right of there my only e, ove nd elow, there my only e. The (strongly) periodic points of A hve necessrily one of the new symols,,. Tht is to sy they re necessrily formed of either n infinity of lines of, either n infinity of lines of, or y n infinity of squres with sides mrked y nd nd corners y. The d-dimensionl SFT A is otined from the 2-dimensionl SFT A y keeping the lphet, keeping the rules for the first two dimensions, nd then force the symols next to ech other long ll other directions to e equl. In the sequel, we will cll A the plne with the rules of A. The second lyer S will force the A plne of periodic points to e formed of squres nd will mrk the frontiers of the d-cues. For 2 i d, we define d 1 SFTs S i with the sme lphet formed y the symols,,,,, : The djcency rules for S i on the plne defined y e 1, e i re tht two symols cn e next to ech other iff their orders mtch: left/right mtchings correspond to ±e i nd ove/elow to ±e 1. The rules pplying to the other dimensions re tht if there is symol t x Z d, then there must lso e symol t x ± e k, for k 1, i. We superimpose the S i s in order to otin S: t ech position, the symols on ll S i components must ll e tken from only one of the sets {, } nd {,,, }. See figure 7 for n exmple of how S i nd S j cn e superimposed. Then to otin A S we dd superimposition rules only with the S 2 sucomponent of S, which hs its rules on the A plne nd forms squres on it. The rules re tht cn only e superimposed to, (resp. ) cn only e superimposed to (resp. ) nd the other symols cn only e superimposed to white symols. As consequence, the symols,, on the A plne must form squres on the strongly periodic points. Figure 8 shows how the A plnes of component S 2 nd component A re superimposed. The strongly periodic points of the resulting SFT re points tht hve d-cues whose corners re mrked y (,..., ). The oundries of the d-cues re mrked y the nd symols: if the side of the d-cues 3 For this lemm we cn tke ny such SFT, however we will use Kri s [Kr92] tileset lter to further the construction.

16 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS 15 e 1 e j e i Figure 7. How S i nd S j re superimposed. e 1 e i Figure 8. The A plnes of components S 2 (left) nd A (right) nd how they re superimposed: lyer S 2 forces the symols, nd of component A to form squres. is n, nd there is corner t p = (p 1,..., p d ) Z d, then for ny point q = (q 1,..., q d ) Z d, or on component S k is equivlent to p k q k mod n nd or is equivlent to p 1 q 1 mod n.

17 16 EMMANUEL JEANDEL AND PASCAL VANIER w0,4 w0,4 w0,4 w1,4 w2,4 w3,4 w4,4 w0,4 w1,4 w2,4 w3,4 w4,4 w0,3 w0,3 w0,2 w0,2 w0,1 w0,1 w0,0 w0,0 w0,4 w0,4 w0,4 w1,4 w2,4 w3,4 w4,4 w0,4 w1,4 w2,4 w3,4 w4,4 w0,3 w0,3 w0,2 w0,2 w0,1 w0,1 ) w0,0 w0,0 ) Figure 9. In ), the wy L right nd L dig synchronise the first column of the periodic ckground for ll squres. In ), the wy U up nd U dig synchronise the top line of the periodic ckground for ll squres. However, the squres formed on the A plne of A my not hve the sme periodic ckground, nd thus there could e more thn one squre in period. Thus the period is multiple of the size of the d-cues. We wnt now to prevent this from hppening nd force the strong period to e exctly the size of the d-cues. Component T = L right L dig U up U dig is here to ddress this lst prolem: y synchronizing the periodic ckground etween squres on A it will force the lest distnce etween two (,..., ) to e the strong period. To do this, since W is NW-deterministic, we only need to trnsmit to the neighoring squres the upper line of symols nd the leftmost one, see figure 9. Ech sucomponent s lphet is copy of the lphet of W nd the rules re s follows: On L right, the symols t z re the sme s the symols t z ± e 2. The only superimposition rule is tht symol on the right of on A must e the sme s on L right. This component synchronises the leftmost column of symols of ll horizontlly ligned squres. On L dig, the symols t z re the sme s the symols t z±(e 1 +e 2 ). The only superimposition rule is tht symol on the right of on A must e the sme s on L dig. This component synchronises the leftmost column of symols of ll digonlly ligned squres. On U up, the symols t z re the sme s the symols t z ± e 1. The only superimposition rule is tht symol ove on A must e the sme s on U up. On U dig, the symols t z re the sme s the symols t z±(e 1 +e 2 ). The only superimposition rule is tht symol on A must e the sme s on U dig. The construction is now finished. Now tke strongly periodic point of Y d with strong period p. By construction, it necessrily is constituted of identicl djcent hypercues of side p. The oundries of the hypercues eing mrked y symols

18 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS 17 (,..., ) nd (,..., ) nd the corners y (,..., ). It is strightforwrd to see tht ny p 2 is strong period of the constructed sushift. Lemm 4.3. Let L N e lnguge such tht un(l) NTIME(n d ), there exists 2d-dimensionl SFT X such tht L = P X nd such tht ll periodic points re strongly periodic. Proof. Let M e Turing mchine recognizing L in nondeterministic time n d. We need to construct n SFT X M whose strong periods re exctly the ccepted inputs of M. We Using lemm 4.2, ll tht is left to prove is how to restrict the periods to the integers ccepted y M. In order to do this, we will encode computtions of M inside the 2d-cues of Y 2d : on unry input, M tkes t most n d time steps to ccept or reject, so 2d-cue of side n hs exctly the right mount of spce to encode such computtion. The ide is to fold the spce-time digrm of the Turing mchine so tht it fits into the cue while still preserving the locl constrints. Such folding hs lredy een descried y Borchert [Bor08] nd cn lso e deduced from Jones/Selmn [JS74]. We then hve to mke sure tht the nondeterministic trnsitions re identicl in ll 2d-cues of point. Let us now descrie this in more detils. In spce-time digrm of M with input n, tpe cells hve coordintes (t, s) with t n d, s n d, where t is the time step nd s the position in spce. We now hve to trnsform ech cell (t, s) into cell of the 2d-cue of size n, so tht two consecutive (in time or spce) cells of the spce-time digrm remin djcent cells of the cue. So we trnsform s nd t into elements of 0, n 1 d with reflected n-ry code (lso clled reflected Gry-codes), see [Flo56; Knu05], this corresponds exctly to folding the time/spce. The vector (t 0,..., t d 1 ) 0, n 1 d will represent the integer t = i n i where { t i = i when j>i t j is even (n 1) t i otherwise see [Knu05, Formul (51)]. The next positition is given y the prity of the sum of the stronger weighed digits. In order to trnform this in locl constrints, it will suffice to encode prities of positions in the cue with some lyers P t nd P s for time nd spce respectively. Lyer P t is mde of severl sulyers P i = {0, 1}, one for ech direction e i, 2 i d. We now give the rules, recll tht the oundries of the cue re mrked. Without loss of generlity, we my suppose tht there is corner in position 0. This corner hs 0 on ll lyers P i. The rules re the following : if there the symol p t position z 0, n 1 d on sulyer P i, then there must e p + 1 mod 2 t position z + e i nd p t positions z + e j, with j i. These rules do not pply when the next position is t the oundry of the 2d-cue. The lyer P s is similr, except it is on dimensions d + 1 to 2d. An exmple for three dimensionl folding cn e seen on figure 10. Now tht we hve encoded the Turing mchines inside the 2d-cues, their size cn only e one of its ccepted inputs. However, recll tht the Turing mchines encoded re nondeterministic, therefore we hve to synchronize the trnsitions etween the different hypercues, otherwise the periods my e multiples of ccepted inputs. In order to do tht, we dd new component N, which is constituted of the following sulyers, whose lphets re ech copy of the possile trnsitions of M:

19 18 EMMANUEL JEANDEL AND PASCAL VANIER 1 0 Figure 10. Folding of three dimensionl cue, the red on the prity lyer stnds for 0 nd the white for 1. The direction where to look for the next cell is given y the sum of the prities. The first sulyer, ℸ, will propgte the trnsition of time-step to ll cells of the sme time-step, tht is to sy the rest of the tpe. A cell where trnsition hppens imposes the symol on ℸ to e the trnsition hppening. The symol propgtes long spce: if there is symol l on ℸ t position z Z d, then there must e exctly the sme symol t position z ± e i, for d + 1 i 2d. We lso hve set of sulyers ℶ i, one for ech time dimension, 1 i d. Component ℶ i hs the following rules : the symol on ℶ i t position z is identicl to the one t position z + (e i + e d+1 ). When the cell is on order of the 2d-cue on dimension 1, the symols on ℶ i nd ℸ hve to e identicl. For the construction of lemm 4.2, this mens tht S i contins or. Figure 11 shows how this synchronistion is done. As in lemm 3.2, n + 4 is (strong) period if nd only if n is ccepted y M. Agin, to otin exctly n it suffices to ftten the symols on the orders. 5. Counting the numer of periodic points in SFTs In theorem 1.1 we hve seen tht the sets of strong periods of SFTs re exctly the sets of integers recognized nondeterministic polynomil time: we cn go one step further nd give chrcteriztion of the sequences p n (X) n N where p n (X) corresponds to the numer of points in X with period n. In the previous construction this numer ws relted to the numer of ccepting pths of the Turing mchine : the numer of possile periodic ckgrounds possile for ech squre of A mkes it hrd to chrcterize. The following theorem is consequence of forcing the periodic ckground of squres of the sme size to e unique: Theorem 5.1. For ny SFT X, let N X : N e the function defined y N X ( n ) = p n (X)/n d where d is the dimension of X. We hve then the following: {N X X n SFT} = #P Note tht the function p n (X) hs een normlized y n d, this is due to the fct tht there re exctly n d shifted versions of sme strongly periodic point of period n.

20 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS 19 ei ej ed+1 Figure 11. The Synchroniztion of nondeterministic trnsitions etween the 2d-cues : here is the projection in 3 dimensions e d+1, e i, e j, with 1 i, j d. ℸ is represented in lue nd ℶ j is represented in red. On the top, we represented component S j of the construction of lemm 4.2. Lyers ℸ et ℶ j synchronise together when we re on the side of squre on S j. Proof. To prove tht #P {N X X n SFT} we hve to fix the periodic ckground for the squres of our previous construction: the numer of strongly periodic points of period n with corner t 0 will then e exctly the numer of ccepting pths of the Turing mchine M rn on n. In order to do tht, insted of tking ny NW-deterministic periodic SFT W, we will tke Kri s SFT [Kr92] nd show tht it is esy to fix the top nd leftmost orders of the squres: this will determine the rest of the squre. Kri s set of tiles is exctly the sme s Roinson s [Ro71], see figure 12, except tht it hs one supplementry lyer with digonl rrows, see figure 13. A vlid tiling with this tileset cn e seen on figure 14, the top nd left orders determine the whole squre. These orders re lmost trivil nd cn e extended to ny length, still forcing n dmissile pttern. Conversely {N X X n SFT} #P: checking whether n P X is NP 1. The numer of ccepting pths of the Turing mchine of the proof of lemm 4.1 is exctly the numer of periodic points of period n. In order to normlize y n d, we lso force this mchine to only keep the guessed fillings of the n d cue tht re the smllest mong their trnsltes for the lexicogrphic order.

21 20 EMMANUEL JEANDEL AND PASCAL VANIER () () Figure 12. Roinson s periodic tileset : ) cross nd ) rms. The tileset lso includes the rottes of these tiles. If the min rrow of n rm is horizontl (resp. verticl) we will cll it horizontl (resp. verticl) rm. Two tiles cn e neighors if nd only if outgoing rrows mtch incoming ones. () Hor () Ver (c) Ver Hor Ver Hor Ver Hor Figure 13. Kri s ddition to mke it NW-deterministic : new lyer with digonl rrows tht hve to mtch t their extremities. On horizontl rms only, we superimpose the tile () nd on verticl ones only, the tile (). The tiles (c) re superimposed on crosses only.

22 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS 21 Figure 14. A vlid tiling y Kri s NW-deterministic tileset. The top nd left orders determine the whole squre. The digonl rrows re not represented ut cn e esily deduced.

23 22 EMMANUEL JEANDEL AND PASCAL VANIER 6. 1-periodic points in SFTs of dimension 2 We now go ck to idimensionl SFTs, nd focus on 1-periodicity. Recll tht point is 1-periodic when it only hs coliner vectors of periodicity. We prove theorem 1.2: the sets of 1-periods of SFTs re exctly the sets of vectors of N Z \ {0} tht re in NSPACE 1 (n). Lemm 6.1. Let X e n SFT of dimension 2, then un(p 1 X ) NSPACE(n). Proof. We hve to construct Turing mchine M which on input v = (m, n) N Z \ {0} decides in spce v if v is 1-period. We hve seen in susection how to construct the grph G v (X) nd tht v is 1-period iff this grph contins two mutully ccessile cycles. This grph does not fit, however, in spce v. The lgorithm tht we will use is similr to the one introduced in lemm 3.1, it will just need to check the existence of two different completions tht cn e glued together. We suppose without loss of generlity tht m n nd tht r is the rdius of the SFT X. Initilize n rry P of size mx(m, n) such tht P [i] = 1 for ll i s nd oolen D to flse. Nondeterministiclly sizes t 1, t 2 3m Σ 4rm, the sizes of the two cycles. Prllely choose 2 2r horizontl lines (l i ) 0 i r 1 nd (l i ) 0 i r 1 of length m, ech sequence of lines forms rectngle of 2r m symols of Σ. We now do these steps in prllel: For ll 2r < i t 1, nondeterministiclly choose line l 1 of length m nd check tht there is no foridden pttern in the lines l i,..., l i 2r n. For this step, it suffices to keep the 2r lst lines l i,..., l i 2r nd 2r symols on ech of the preceeding n lines. We lso nondeterministiclly choose lines l j for 2r < j < t 2 t the sme time nd check if they form n invlid pttern. It is importnt to choose line l i t the sme time s line l j until min(t 1, t 2 ) is reched: whenever l i is different from l i, ssign true to D. At ech of these steps, check if there is periodicity vector (m, n ) such tht (m, n ) k = (m, n), if it is not the cse then P [k] 0. Once the lst lines l t1 nd l t 2 hve een guessed, chck tht there is no foridden pttern on the ptterns formed y the symols rememered on lines: l t1 2r n,..., l t1, l 0,..., l 2r l t 2 2r n,..., l t 2, l 0,..., l 2r l t1 2r n,..., l t1, l 0,..., l 2r l t 2 2r n,..., l t 2, l 0,..., l 2r This checks tht the two cycles found re mutully ccessile. If ny of the steps efore filed or if there exists k, m, n < m such tht k (m, n ) = (m, n) nd P [k] = 1, or if D is flse then reject. Accept otherwise. This lgorithm only needs to keep 4rm + 2rn symols of Σ, s well s P nd D in memory. Lemm 6.2. For ny constnt k N, there exists 2-dimensionl SFT Y k such tht ny one periodic point of period (m, n) is formed of m k m 1 rectngles with

24 CHARACTERIZATIONS OF PERIODS OF MULTIDIMENSIONAL SHIFTS 23 mrked orders, s in figure 15. Furthermore, Y k dmits s 1-periods ny (m, n) such tht 0 < n < m. Figure 15. The mrked rectngles of the 1-periodic configurtions. Proof. As in the preceeding proofs, the construction will e done in successive steps, y superimposition of severl lyers A, C c, R, S : Agin, the first component A is sed on n periodic Est-deterministic SFT W. The lphet of A is Σ A = (Σ W {, }) {,,, }, we cll the symols of Σ W the white symols. Agin, the other symols llow to rek periodicity, the rules re the following: The rules etween the symols of Σ W remin unchnged. White symols my hve or nother white symol ove, ut two my not e ove/elow ech other. The constrints on whites re trnsmitted over the symols: removing line of nd gluing the white symols ove nd elow must not produce ny foridden pttern. The rules etween the symols {,,,, } re Wng rules. Only the white sides of symols {,,,, } my touch white symol. At this stge, the configurtions with periodicity vector necessrily hve n infinity of verticl lines. Verticl lines my eventully e met y extremities of finite horizontl lines, see figure 16. An infinity of horizontl lines leds to n periodic configurtion. () () Figure 16. There re severl possiilities for 1-periodic points of A: n infinity of verticl lines (), or n infinity of verticl lines seprted y finite horizontl lines ().

25 24 EMMANUEL JEANDEL AND PASCAL VANIER Component C k is k-ry counter, exctly s in lemm 3.3. Now tht this counter hs een dded, the points hving periodicity vector necessrily contin n infinity of verticl lines, necessrily joined y horizontl lines t distnce k n 1 when they re distnt y n. In points with periodicity vector, component R forces columns formed y the nerest verticl lines to ll e of the sme width, nd the offset etween horizontl lines of two neighoring columns to lwys e the sme. The first is done y first projecting ech horizontl line to the left nd to the right until it reches the next verticl line. Between projections on the sme side, we gin put counter C k, this forces the sizes of the two columns to e identicl. To mke the offset etween horizontl lines constnt for ll columns, we project two signls of slope 1 on oth the left nd the right of the horizontl lines. These signls propgte normlly on white symols nd cnnot touch verticl line directly: they hve to first pss on horizontl line. As they do this, their direction chnges nd their slope ecomes 1. They must then touch the next verticl line t the exct sme time s the projection from two columns to the left/right, t which time the signl stops, see figure 17. Right lignment signl Left lignment signl Left projection Right projection Figure 17. How component R opertes. Ech horizontl line sends projection to the next verticl line on ech side. On ech side, etween projections, there is counter C k. Another signl is send digonlly, with slope 1, it cn only touch horizontl line, t which point it chnges its direction nd needs to rerech the sme verticl line t the sme time s the projection from two columns to the left/right. At this stge, the points with periodicity vector re necessrily constituted of rectngles of identicl size, trnslted y vector (m, n ) from one column to nother. (m, n ) is not necessrily periodicity vector, however there must exist k such tht k(m, n ) is. We now rrnge for the width m of the rectngles nd their offset n to form the periodicity vector: we set the periodic ckground in the columns to e