A strongly aperiodic shift of finite type for the discrete Heisenberg group
aa r X i v : . [ m a t h . D S ] S e p A STRONGLY APERIODIC SHIFT OF FINITE TYPE FOR THEDISCRETE HEISENBERG GROUP
AYS¸E A. S¸AH˙IN, MICHAEL SCHRAUDNER, AND ILIE UGARCOVICI
Abstract.
We explicitly construct a strongly aperiodic subshift of finite typefor the discrete Heisenberg group. Our example builds on the classical aperi-odic tilings of the plane due to Raphael Robinson. Extending those tilings tothe Heisenberg group by exploiting the group’s structure and posing additionallocal rules to prune out remaining periodic behavior we maintain a rich projec-tive subdynamics on Z cosets. In addition the obtained subshift is an almost1-to-1 extension of a strongly aperiodic, minimal sofic shift. As a consequenceof our construction we establish the undecidability of the emptiness as well asthe extension problem for shifts of finite type on the Heisenberg group. Introduction
One of the fundamental differences between Z d symbolic dynamics for d “ d ą ˆ H p Z q “ x z y ¸ : x, y, z P Z + equipped with the usual matrix multiplication operation. This non-abelian nilpo-tent linear group can also be seen as a semi-direct product Z ⋊ A Z , where A “ p q induces an automorphism of Z leading to the group operation given by ` x, p y, z q ˘ ¨ ` a, p b, c q ˘ “ ` x ` a, ` p y, z q ` A x p b, c q ˘˘ “ ` x ` a, p y ` b, z ` c ` xb q ˘ . We provide an explicit construction of a strongly aperiodic shift of finite typefor H p Z q that is almost minimal. The main result in the paper is the following. Theorem 1.1.
The discrete Heisenberg group H p Z q admits a strongly aperiodicshift of finite type which is an almost 1-to-1 extension of a strongly aperiodic andminimal sofic system. The first author was partially supported by an Association for Women in Mathematics TravelGrant. The second author was supported by FONDECYT Project 1140015 and CONICYT PIABasal Grant AFB170001. The third author was partially supported by a Simons FoundationCollaboration grant.
We prove the theorem by providing a concrete construction of the shift of finitetype. We first show what tiles and local rules are sufficient to guarantee strong ape-riodicity and then explain how to modify the construction to control the topologicaldynamics. As a consequence of our explicit construction, we can show:
Corollary 1.2.
Both the emptiness problem as well as the extension problem areundecidable for shifts of finite type on H p Z q . Z background on aperiodic shifts of finite type. The existence of astrongly aperiodic shift of finite type was first established in the context of studyingtilings of the plane. Z shifts of finite type (SFTs) can be modeled by tiling spacesobtained from tiling R by aligned unit squares with colored edges. The SFT’slocal rules can be translated into the condition to have colors match across tileborders. The existence of a strongly aperiodic set – i.e. a set which tiles R , butonly by producing strongly aperiodic configurations – of such edge-colored squaresfor R is thus equivalent to the existence of a strongly aperiodic Z -SFT. Wang’sfamous conjecture [25] dating from the 1960s claimed that whenever a collection ofedge-colored unit square tiles could tile the entire plane, then those tiles would alsoallow for a periodic tiling, which in turn would imply decidability of the emptinessproblem asking whether a given finite collection of tiles will indeed tile the plane.Berger [2] was the first to disprove this conjecture with a huge set of over 20,000distinct tiles whereas subsequent authors have constructed examples using fewerand fewer tiles. In this paper we use extensively the well-known Robinson tilingsconstructed by Raphael Robinson in 1971 [24]. The most recent strongly aperiodic Z examples are given by a set of 13 square tiles found by Kari and Culik [18,9] in 1996 and by a preprint of Jeandel and Rao [16] establishing the minimalcardinality for strongly aperiodic edge-colored square tile sets to be 11. We notethat Berger’s construction of a strongly aperiodic set of square tiles led to a proof ofthe algorithmic undecidability of the emptiness problem for R . As a consequencethere is also no general method to determine whether or not a given two dimensionalshift of finite type is non-empty and whether or not a locally admissible pattern ona finite part of Z extends to an entire configuration. (See for example [23] for amore in depth treatment of these ideas and corresponding results.)1.2. Aperiodic shifts of finite type for general groups.
Related work ontilings for general groups basically comes in two flavors, considering either contin-uous or (finitely generated) discrete groups. In the 1990s Block and Weinbergerstudied weakly aperiodic tiling spaces for certain families of Lie groups [4], whereasMozes [21] constructed strongly aperiodic tile sets on symmetric spaces obtainedfrom some particular classes of semi-simple Lie groups. Later Goodman-Strauss[13, 14] extensively studied strongly aperiodic tiling spaces on the hyperbolic planeand Margenstern [20] proved the undecidability of the emptiness problem in thiscontext. Fusing techniques by Goodman-Strauss with properties of the planarKari-Culik tilings, in 2013 Aubrun and Kari [1] were able to produce a weakly ape-riodic tile set (with local matching rules) for the class of solvable Baumslag-Solitargroups, showing at the same time the undecidability of the emptiness problem forSFTs on these two-generator, one-relator groups. There is a more recent body of A tiling space is weakly aperiodic if none of its tilings has a co-compact symmetry, which ingeneral is strictly less than allowing no symmetry whatsoever (i.e. the strongly aperiodic setting). The example by Aubrun and Kari from [1] is actually believed to be even strongly aperiodic.
TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 3 work addressing the same questions for shifts of finite type on other types of finitelygenerated groups using tools from a variety of different areas.The authors established, several years ago, the existence of a strongly aperiodicSFT of H p Z q . That example is presented in this paper as the first step towards theproof of Theorem 1.1. Since that initial announcement a number of related resultshave been established using geometric group theory techniques. Carol and Penlandproved that admitting a strongly aperiodic shift of finite type is an invariant ofcommensurability of groups [6]. Meanwhile Cohen showed that the existence ofsuch subshifts is also a quasi-isometry invariant and that strongly aperiodic shiftsof finite type can not exist in groups with two ends [8]. In later work with Goodman-Strauss he nonetheless established the existence of strongly aperiodic shifts of finitetype on hyperbolic surface groups [5].Most recently Barbieri and Sablik have shown the existence of strongly aperiodicshifts of finite type on semi-direct product groups of the form Z ⋊ H , where H needs to be a group with decidable word problem [3]. Their proof involves a non-trivial extension of Hochman’s intricate work in [15] and subsumes the case of theHeisenberg group, but without giving an explicit instance of such systems.Our construction is significantly different and the difference in our approach, inparticular, allows us to control the topological dynamical properties of our example.Our arguments rely on a subtle relationship between the combinatorial structureof the Z subshift given by the Robinson tilings together with the geometry of theCayley graph of H p Z q . We explicitly describe both the alphabet and local rulesof the SFT, thus allowing us to completely understand several features (locally vs.globally admissible patterns, projective subdynamics) of our construction. As weshow in Section 5 the Cayley graph of the discrete Heisenberg group can be viewedas a countable union of Z cosets (drawn horizontally) which are connected byslanted (vertical) edges. We prove that it is possible to fill every other Z cosetby a valid Robinson tiling while forcing additional local rules along the connectingvertical edges in a consistent fashion that eliminates any kind of periodicity. Theidea of our approach is partly resembling work of Culik and Kari in [10] who con-structed a strongly aperiodic Z -SFT by extending their own strongly aperiodic Z tilings to the direct product Z » Z ˆ Z in a related way.The paper is organized as follows. In Section 2 we define subshifts and shiftsof finite type on finitely generated groups, briefly discuss relevant properties likeaperiodicity and minimality and recall the notion of Cayley graphs. Section 3describes the Z -SFT given by the Robinson tilings and a few technical properties ofthis SFT essential to our construction. Section 4 recalls some basic facts about thediscrete Heisenberg group H p Z q . In Section 5 we present the principal constructionof a specific strongly aperiodic SFT on this group exploiting the geometry of itsCayley graph as well as the rigidity of the Robinson SFT. Furthermore we commenton its dynamical properties and conclude the undecidability of the emptiness andthe extension problem for H p Z q -SFTs. Section 6 slightly modifies our previousconstruction to obtain an almost 1-to-1 SFT-extension of a particular minimalsofic system on H p Z q . AYS¸E A. S¸AH˙IN, MICHAEL SCHRAUDNER, AND ILIE UGARCOVICI Subshifts on finitely generated groups
Assuming a basic familiarity with symbolic dynamics we use this section tobriefly recall a few key notions and set some notation to be utilized in what follows.For additional background and more details we refer the reader to [7] and [19].Let G be a finitely generated, (countably) infinite group and let A be a finite(discrete) set, called the alphabet , whose elements are referred to as symbols . Thecartesian product A G – equipped with the prodiscrete topology – is the compactmetric space consisting of all functions ω : G Ñ A , called configurations . We usethe notation ω “ p ω g q g P G P A G where ω g : “ ω p g q denotes the symbol seen at aparticular element g P G . Similarly, for any finite subset F ( G , an element p P A F which can be thought of as p : F Ñ A , i.e. the restriction of a configuration to afinite support, is called a pattern of shape F . Again the notation p “ p p g q g P F P A F with p g : “ p p g q is used, while supp p p q “ F denotes its support. We call p a subpattern of another pattern q P A F of (finite) shape F ( G , respectively ofa configuration ω P A G , denoted by p Ď q , respectively p Ď ω if there exists anelement g P G such that F ¨ g Ď F and p “ q | F ¨ g , respectively p “ ω | F ¨ g . Thecountable set of all patterns is defined as A ˚ : “ Ť F ( G finite A F .Since G acts on itself by translations we obtain a natural G -action G σ y A G givenby homeomorphisms σ g : “ σ p g q : A G Ñ A G ( g P G determined coordinate-wise as ` σ g p ω q ˘ h : “ ω hg for all g, h P G . This symbolic G -action is known as the (right)shift and the pair p A G , σ q as the full G -shift over the alphabet A .A subset of A G preserved under the G -action σ is said to be shift invariant . A symbolic dynamical system for the group G and the alphabet A is then given by anyclosed, thus compact, and shift invariant set Ω Ď A G together with the restrictionof the shift action of G to Ω. The pair p Ω , σ | Ω q is called a G -shift space or simplya G -subshift . For notational convenience we refer to a shift space merely as Ω, orwhen we want to emphasize the acting group, as p Ω , G q .It is easy to establish an equivalent – but more combinatorial – definition of G -shift spaces. In fact any subshift p Ω , G q over A is characterized by selecting an (atmost countable) family of finite patterns F Ď A ˚ such that ω P A G is an elementof p Ω , G q if and only if σ g p ω q| supp p p q ‰ p for all g P G and p P F . To emphasizethe role of the chosen family of forbidden patterns F we introduce the notationΩ F : “ ω P A G : @ p P F : p Ę ω ( . Moreover we call p Ω , G q a G -shift of finite type ( G -SFT), if it is possible to obtain Ω “ Ω F for some finite family F . The image ofa G -SFT under a (topological) factor map is called a sofic G -shift .Adopting a complementary viewpoint, a pattern q P A ˚ is locally admissible in Ω F if none of its subpatterns is forbidden, i.e. @ p P F : p Ę q . Usually thisproperty is weaker than requiring the pattern q to be an actual subpattern of someconfiguration ω P Ω F , which would make q a globally admissible pattern. Givena set F , checking the local admissibility of a specific pattern is an easy (finite)task. On the other hand its global admissibility, known as the extension problem , isin general undecidable, even in the case of G -SFTs, for many groups G . Similarlythere is no universal decision procedure/algorithm for the emptiness problem askingabout the existence of any global configuration respecting all local constraints, i.e.to determine, given a (finite) set F , whether or not Ω F “ H .Both of these undecidability results are intimately related to the possibility ofconstructing non-empty aperiodic G -SFTs. Let Ω be a (non-empty) G -subshift with TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 5 ω P Ω. The G -orbit of ω is defined as the set of configurations t σ g p ω q : g P G u Ď Ω,whereas the stabilizer of ω is given as the subgroupStab G p ω q : “ t g P G : σ g p ω q “ ω u ď G of all group elements fixing ω . We say ω is weakly periodic if | Stab G p ω q | “ 8 ,whereas ω is called strongly periodic if its G -orbit is finite or equivalently if itsstabilizer is of finite index in G , i.e. | G : Stab G p ω q | ă 8 . Accordingly the (non-empty) subshift p Ω , G q is called weakly aperiodic if none of its configurations arestrongly periodic, that is | G : Stab G p ω q | “ 8 for all ω P Ω, while it is called stronglyaperiodic if all its configurations ω P Ω have trivial stabilizer Stab G p ω q “ t G u , thusexcluding even any weakly periodic behavior. A finitely generated group G is saidto admit a strongly aperiodic SFT if there exists a strongly aperiodic G -SFT.A G -subshift not containing any non-empty, proper, closed and shift-invariantsubset is referred to as minimal . Equivalently, every single G -orbit in a minimalshift p Ω , G q is dense. In the case of | Ω | “ 8 minimality excludes the existence ofstrongly periodic (but not necessarily weakly periodic) configurations.To finish this section we recall a tool from combinatorial group theory. Given afinitely generated group G , its (left) Cayley graph with respect to a finite generatingset S Ď G zt G u is the connected, locally finite, S -labeled digraph Γ G, S “ p V, E q whose set of vertices is V “ G and whose set of (directed) edges E Ď V ˆ V consistsof all pairs of the form p g, sg q for g P G and s P S . An edge e “ p g, sg q starts atvertex i p e q “ g , terminates at vertex t p e q “ sg and carries the generator s as itslabel. Note that there are exactly |S| edges starting respectively ending at eachgiven vertex g P G , making Γ G, S an |S| -regular digraph. We point out that theprodiscrete topology on A G mentioned above is induced by the usual product-spacemetric obtained from using the word metric on the S -Cayley graph (for an arbitrary(finite) set of generators S ), and that two configurations ω, ω P A G are consideredclose if they agree on all group elements inside a large ball – measured by the wordmetric – centered at the identity element 1 G .3. Robinson tilings of the plane
In this section we provide a brief review of the Robinson tilings introduced in[24], while referring the reader to [17, 22, 23] and the references therein for anydetails omitted in our exposition.Following [24, § § A Rob of 56 square tiles obtained from rotating the 14 decorated tiles shown inFigure 1 by multiples of 90 degrees. Following common nomenclature, we call(rotations of) the leftmost tile in both rows of Figure 1 a cross , while we collectivelyrefer to (rotations of) the 12 non-cross tiles as arms . In a valid tiling copies of thosetiles are placed edge to edge filling the plane subject to the following
Robinson rules :(R1) Across every edge, shared by a pair of adjacent tiles, arrows of each type(red and black) have to match head to tail.(R2) The parity check digits (0, 1 or 2) on opposite sides of an edge segment,shared by a pair of adjacent tiles, have to sum to 2.Note that both rules are local, in fact nearest neighbor, and completely symmet-ric with respect to the 4 cardinal directions (rows and columns) of Z . The secondrule in particular is forcing parity check digits in any row or column to be eitherconstant (digit 1) or periodically alternating (between digits 0 and 2). Moreover, AYS¸E A. S¸AH˙IN, MICHAEL SCHRAUDNER, AND ILIE UGARCOVICI ✲ ✲ ✲ ✲ ✲ ✲✲ ✲ ✲ ✲ ✲ ✲✛ ✛ ✛ ✛ ✛ ✛✛ ✛ ✛ ✛ ✛ ✛❄ ❄ ❄ ❄ ❄ ❄❄ ❄ ❄ ❄ ❄ ❄✲✲✛✛ ✻✻❄❄ ✻✻ ✲✲ ❄❄ ❄❄ ❄❄ ❄❄✲✲✛✛ ✲✲✛✛ ✲✲✛✛
Figure 1.
The alphabet used in the Robinson tilings (displayedtiles can still be rotated giving a total of 4 ¨ “
56 symbols).Observe that in arm tiles the convergence point of red side arrowsegments is always located towards the tip of the main arrow.since crosses – with their outward pointing arrows along all four edges – are notallowed to be adjacent to each other (due to the first rule), those two types ofrows/columns again have to strictly alternate in each tiling. As a consequence therules guarantee the appearance of rotations of the cross (i.e. leftmost tile) in thelower row of Figure 1 in an entire coset of 2 Z ˆ Z , sometimes called the alternatingcrosses constraint . Remaining locations in Z are filled by rotations of the other 13tiles with additional crosses (rotated copies of the leftmost symbol on the top rowin Figure 1) appearing in a highly regular fashion.Every element in the 2 Z ˆ Z coset filled by the alternating crosses constraintforms a corner of some 3 ˆ ˆ ˆ ˆ i P N it can be shown that inside every valid Z -Robinsontiling there exist correctly tiled p i ` ´ q ˆ p i ` ´ q square patches – usuallycalled level- i supertiles – containing cross tiles at precisely the sites in the set C i : “ Ť ij “ ` t j p k ` q ´ i : 0 ď k ă i ´ j u ˆ t j p k ` q ´ i : 0 ď k ă i ´ j u ˘ Ď r´ i ` , i ´ s . Note that again every level- i supertile forms part of somelevel- p i ` q supertile, forcing a hierarchy of bigger and bigger interlinked squares.We point out that even though the Robinson tilings are defined by rules governingpairs of adjacent tiles, these local constraints impose a rigid and global hierarchicalstructure. In particular, the nearest neighbor rules generate supertiles of arbitrarilyhigh levels and at the same time force an extremely regular appearance of underlyingcross tiles. The location and orientation of crosses, in turn, completely determinethe mutual arrangement of supertiles. As can be seen in Figure 3, showing part ofa typical Robinson tiling, this forces an immense rigidity which is indeed the keyingredient in proving the strong aperiodicity of all such tilings. This structure willalso play an important role in our construction in Sections 5 and 6.Hereinafter we denote by Ω Rob Ď A Rob Z the set of all valid Z -Robinson tilings. TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 7 ❄ ❄✛✛ ✻ ✻✲✲✻ ✻✲✲❄ ❄✛✛✲✲✛✛✻ ✻❄ ❄✛ ✲❄✻✛ ✲❄✻✻ ✻❄ ❄✲✲ ✛✛✲✛✻❄ ✲✛ ✻❄✻✲✻ ✲ ✛✛✛✛ ✲✲✲✲ ✛✛✛✛ ✲✲✲✲ ✛✛✛✛ ✲✲✲✲ ✛✛✛✛ ✲✲✲✲❄ ❄ ❄ ❄✻ ✻ ✻ ✻❄ ❄ ❄ ❄✻ ✻ ✻ ✻❄ ❄ ❄ ❄✻ ✻ ✻ ✻❄ ❄ ❄ ❄✻ ✻ ✻ ✻✛✛✛ ✲✲✲❄ ❄ ❄✻ ✻ ✻✲✲ ✛✛✻ ✻❄ ❄✛✛ ✲ ✲❄❄✻✻✻ ✻ ✻ ✻✻ ✻ ✻ ✻❄ ❄ ❄ ❄❄ ❄ ❄ ❄✲✲✲✲ ✲✲✲✲✛✛✛✛ ✛✛✛✛✲✲✲✛✛✛✲✲✲✛✛✛✻ ✻ ✻❄ ❄ ❄ ✻ ✻ ✻❄ ❄ ❄✛ ✛✲ ✲❄ ❄✻ ✻✛ ✛✲ ✲❄ ❄✻ ✻✛ ✲❄✻✻ ✻❄ ❄✲✲ ✛✛ ✻ ✻❄ ❄✲✲ ✛✛✻ ✻❄ ❄✲✲ ✛✛ ✻ ✻❄ ❄✲✲ ✛✛✲✲✛✛ ✲✲✛✛✻ ✻❄ ❄✻ ✻❄ ❄✲✲✛✛ ✲✲✛✛✻ ✻❄ ❄✻ ✻❄ ❄✲✲ ✛✛✲✲ ✛✛✲✛✲✛✻ ✻❄ ❄✻ ✻❄ ❄✻❄✻❄ ✻✛ ✻✻✻✛✛✛ ´ ´ ´ ´ ´ ´ ´ ´ Figure 2.
Supertiles of level 1 and 2 (parity check digits sup-pressed). Note that (black) arrows are pushing outward along theentire border of a supertile, thus forcing all neighboring tiles to bearms. Additionally, along a supertile’s periphery exactly two redarrows appear in the middle of two adjacent sides reverberatingthe orientation of its central cross.
Figure 3.
The hierarchical structure in the Robinson tilings(black arrows and parity check digits suppressed for visibility).4.
Basic properties of the Heisenberg group
In what follows we denote by H p Z q » Z ⋊ A Z the discrete Heisenberg group,given as a semi-direct product induced by the matrix A “ p q P GL p Z q . H p Z q is a 2-generator finitely presented, single-ended, non-abelian, nilpotent linear group AYS¸E A. S¸AH˙IN, MICHAEL SCHRAUDNER, AND ILIE UGARCOVICI with polynomial growth of order 4 (for details see [11]). Its binary operation takesthe form ` x, p y, z q ˘ ¨ ` a, p b, c q ˘ “ ` x ` a, ` p y, z q ` A x p b, c q ˘˘ “ ` x ` a, p y ` b, z ` c ` xb q ˘ corresponding to usual matrix multiplication in H p Z q ’s standard representationvia upper triangular matrices, i.e. where an arbitrary element p x, p y, z qq is seen asthe unipotent integer matrix ´ x z y ¯ for x, y, z P Z .We define three particular elements: x “ p , p , qq , y “ p , p , qq , z “ p , p , qq whose inverses are then given as x ´ “ p´ , p , qq , y ´ “ p , p´ , qq and z ´ “p , p , ´ qq . A general element p x, p y, z qq P H p Z q can be obtained as the product z z y y x x “ p x, p y, z qq . Hence we may fix S “ t x , y , z u as our set of generators for H p Z q . Since z “ xyx ´ y ´ , the elements x and y are sufficient to generate H p Z q and in fact yield the presentation H p Z q “ x x , y | y ´ xyx ´ “ xyx ´ y ´ “ yx ´ y ´ x y . For convenience we include the element z as an additional (third) generator, result-ing in the more commonly used presentation H p Z q “ x x , y , z | xyx ´ y ´ “ z , xz “ zx , yz “ zy y . Note in addition that z generates the center of H p Z q , which is isomorphic to aninfinite cyclic group. The subgroups x x , z y , x y , z y E H p Z q are normal and isomor-phic to Z , whereas the two infinite cyclic subgroups x x y and x y y are not normal in H p Z q .Figure 4 shows the S -labeled left Cayley graph of H p Z q . Here – and in allits visualizations in the remainder of this article – we view x y , z y -cosets of H p Z q as horizontal lattices isomorphic to Z , which are then connected through vertical(upward-pointing) edges representing the generator x . Due to being a semi-directproduct induced by the matrix A “ p q , the slope of those vertical edges dependsonly on the y -value of the vertices connected by them.5. Constructing a strongly aperiodic H p Z q -SFT To construct our strongly aperiodic example p Ω , H p Z qq , we first specify its al-phabet A . Next we define several sets of local rules governing how symbols canbe placed on the vertices of the S -Cayley graph forming valid configurations. Af-ter this step we show that the resulting H p Z q -SFT Ω ( A H p Z q has quite a rigidstructure. We describe its Z -projective subdynamics, and show that it is stronglyaperiodic.5.1. The first step of the construction: establishing weak aperiodicity.
Let A Rob be the set of 56 Robinson tiles as depicted in Figure 1. Symbols fromthis alphabet will be used to fill every other x y , z y -coset of H p Z q , to set up thesame rigid hierarchical structure of crosses appearing in level- i supertiles ( i P N ) asseen in the original Z -Robinson tilings. Remaining x y , z y -cosets will then be filledwith symbols from a disjoint alphabet A Count actually consisting of pairs fromtwo disjoint sets. Denote by A Bin : “ t , , , u a binary alphabet containing twodistinct versions of the digits 0 and 1 and let A Seg be an alphabet consisting of 4unit square tiles as shown in Figure 5. The square on the left is called a blank tile.
TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 9 (-1,(-2,-1))(-1,(-2,0)) (-1,(-2,1)) (-1,(-2,2))(0,(-2,-1)) (0,(-2,0)) (0,(-2,1)) (0,(-2,2))(1,(-2,-1)) (1,(-2,0)) (1,(-2,1)) (1,(-2,2))(2,(-2,-1)) (2,(-2,0)) (2,(-2,1)) (2,(-2,2))(-1,(-1,-1))(-1,(-1,0)) (-1,(-1,1)) (-1,(-1,2))(0,(-1,-1)) (0,(-1,0)) (0,(-1,1)) (0,(-1,2))(1,(-1,-1)) (1,(-1,0)) (1,(-1,1)) (1,(-1,2))(2,(-1,-1)) (2,(-1,0)) (2,(-1,1)) (2,(-1,2))(-1,(0,-1)) (-1,(0,0)) (-1,(0,1)) (-1,(0,2))(0,(0,-1)) (0,(0,0)) (0,(0,1)) (0,(0,2))(1,(0,-1)) (1,(0,0)) (1,(0,1)) (1,(0,2))(2,(0,-1)) (2,(0,0)) (2,(0,1)) (2,(0,2))(-1,(1,-1)) (-1,(1,0)) (-1,(1,1)) (-1,(1,2))(0,(1,-1)) (0,(1,0)) (0,(1,1)) (0,(1,2))(1,(1,-1)) (1,(1,0)) (1,(1,1)) (1,(1,2))(2,(1,-1)) (2,(1,0)) (2,(1,1)) (2,(1,2))(-1,(2,-1)) (-1,(2,0)) (-1,(2,1)) (-1,(2,2))(0,(2,-1)) (0,(2,0)) (0,(2,1)) (0,(2,2))(1,(2,-1)) (1,(2,0)) (1,(2,1)) (1,(2,2))(2,(2,-1)) (2,(2,0)) (2,(2,1)) (2,(2,2))
Figure 4.
The S -labeled left Cayley graph of the discrete Heisen-berg group where dotted, dashed, and solid edges correspond togenerators x , y and z respectively.Moreover we refer to the northwest-southeast pointing black line in the second andfourth tile as the diagonal, and the vertical line in the third and fourth tiles as theforward line segment. Forming the Cartesian product of those two alphabet sets weobtain the 16 element set A Count : “ A Bin ˆ A Seg , called the counter-alphabet , whosesymbols are thus ordered pairs p b, s q with b P A Bin and s P A Seg . One may think ofthe second component s as being a decoration superimposed on top of the binarydigit b giving it the option to send additional signals throughout the configuration(propagating black lines). In the final step of our construction local constraints onthese symbols will be used to implement a family of synchronized binary countersthat destroy all periodicity left in the previous steps. The alphabet of our H p Z q -SFT Ω is thus the disjoint union A : “ A Rob \ A Count and every configuration in Ωcan be seen as a function from H p Z q to the finite alphabet set A avoiding certainpatterns. Figure 5.
The alphabet A Seg containing four square tiles withdifferent combinations of up to two black line segments.We continue by specifying the first set of local rules. These will be nearestneighbor rules determining which symbols can be adjacent within a x y , z y -coset of H p Z q . For every configuration ω P Ω, if an element h P H p Z q is filled with a symbol ω p h q P A Rob , then both elements y h and z h have to be filled with symbols from A Rob as well. Similarly, if ω p h q is a symbol from A Count again both its neighboring symbols ω p y h q and ω p z h q have to be in A Count . Those two rules force each single x y , z y -coset to be entirely filled with symbols of either A Rob or A Count .Since the two infinite order elements y and z commute, the subgroup x y , z y ď H p Z q generated by them is isomorphic to Z and hence we may identify every x y , z y -coset x y , z y x x “ p x, p y, z qq P H p Z q : y, z P Z ( (for x P Z fixed) with a copy of Z where the direction of y (assumed to point to the right) generates horizontal rowsand the direction of z (pointing forward) generates horizontal columns. The nextset of nearest neighbor rules affects symbols from A Rob as follows: Suppose element h P H p Z q is filled with a Robinson tile ω p h q P A Rob . The symbol ω p y h q P A Rob seen at coordinate y h – still in the same x y , z y -coset – then has to be a Robinsontile which is allowed to be horizontally adjacent to ω p h q sitting on its left. Similarlythe symbol ω p z h q P A Rob seen at coordinate z h – again in the same x y , z y -coset – isforced to be a Robinson tile which is allowed to sit directly above ω p h q in a valid Z -Robinson tiling. Overall those rules imply that the configuration seen on any x y , z y -coset filled with symbols from A Rob is in fact a valid Z -Robinson tiling.The subsequent set of local rules induces further constraints on x y , z y -cosetsfilled with symbols from A Count . Suppose element h P H p Z q is filled with a symbol ω p h q “ p b h , s h q P A Count , whose second component s h P A Seg has a diagonalsegment. In this case, both symbols ω p zy ´ h q “ p b zy ´ h , s zy ´ h q P A Count and ω p z ´ y h q “ p b z ´ y h , s z ´ y h q P A Count have to continue this diagonal line, so thattheir second component’s tile – s zy ´ h and s z ´ y h respectively – again has to have ablack diagonal segment connecting upper-left and lower-right corners. Analogouslyfor the forward segment, i.e. s h P A Seg with a black forward segment connectingits top to its bottom edge forces the existence of forward line segments in the tilesseen in the second component of both symbols ω p z h q P A Count and ω p z ´ h q P A Count . Hence the presence of a single symbol from A Seg with a diagonal/forwardsegment forces the continuation of the corresponding diagonal/forward line acrossall diagonally/forward adjacent symbols within the x y , z y -coset. The purpose ofthose lines is to synchronize information in the final step of our construction andwe will come back to the consequences of this segments have to continue rule later.For now, let us continue by explaining some of the nearest neighbor constraintswe need to impose along the x -direction. Since we want configurations in adjacent x y , z y -cosets to alternate between Robinson tilings and certain configurations fromthe counter-alphabet A Count we simply enforce the following. If for any h P H p Z q , ω p h q P A Rob is a Robinson tile, then ω p x h q P A Count has to be a counter-symbol,while conversely ω p h q P A Count implies ω p x h q P A Rob .The final constraints in this first step of our construction are local rules forcingthe propagation of crosses along direction x : Assume that the symbol ω p h q seen at h P H p Z q is a Robinson tile, i.e. ω p h q P A Rob . If it is a cross tile, the first componentof the counter-symbols placed at x h as well as at x ´ h has to be one of the bold-face digits or , whereas if the symbol ω p h q is a non-cross Robinson tile, firstcomponents in both those counter-symbols have to be the non-bold-face version ofa digit 0 or 1. Symmetrically, seeing a counter-symbol with its first component or at position h P H p Z q forces arbitrarily oriented crosses at locations x h and x ´ h ,while a counter-symbol ω p h q “ p b h , s h q P A Count with b h P t , u is only allowed incombination with ω p x h q and ω p x ´ h q both being non-cross Robinson tiles. Hencebold-face digits in the first component of counter-alphabet symbols are basicallyused to connect locations with crosses, while non-bold-face digits transport across TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 11 the information about sites containing non-cross Robinson symbols sitting adjacentin the x -direction in neighboring x y , z y -cosets.While understanding the effects of all previous local rules is rather straight-forward, here it is definitely time to pause for a moment and check that this lastset of constraints does not render our construction void by making it infeasibleto fill the entire Heisenberg group with symbols without violating those rules. Toproduce valid configurations in our H p Z q -SFT Ω we have to show the a priori notobvious fact that a sequence of valid Z -Robinson tilings can be chosen for everyother x y , z y -coset, so that crosses do align exactly in the described fashion. Thisargument is the content of the following lemma. Lemma 5.1.
There are families of regular Z -Robinson tilings indexed by Z , whichcan be used to tile every other x y , z y -coset, allowing us to fill in the remaining sitesof H p Z q with symbols from the counter-alphabet A Count and respecting all localrules specified so far.Proof.
Simply note that there is a Z -Robinson tiling ρ P p Ω Rob , Z q which has itslevel- i supertiles ( i P N arbitrary) centered at coordinates p i ` i ` Z q ˆ p i ` i ` Z q ( Z and thus sees crosses at exactly the coordinates in C “ lim i Ñ8 C i “ tp , qu Y ď i P N ` p i ´ ` i Z q ˆ p i ´ ` i Z q ˘ ( Z . For each x P Z , place a copy of the tiling ρ into the x y , z y -coset x y , z y x x , i.e. let ω ` p x, p y, z qq ˘ : “ ρ p y, z q . Next define a set of locations for bold-face digits B : “ tp , qu Y ď i P N ` p i ´ ` i Z q ˆ i Z ˘ ( Z and for every x P Z ` ω to the remaining x y , z y -cosets x y , z y x x by fillingin symbols from A Count as follows: ω ` p x, p y, z qq ˘ : “ , ˘ iff p y, z q P B ` , ˘ iff p y, z q R B .
It is now straightforward to check that this gives a H p Z q -configuration ω P A H p Z q respecting all rules specified so far. Note that for each x P Z multiplication by thegenerator x on the right transforms the set of crosses into the set of bold-face digits,i.e. x ¨ p x, p y, z qq : p y, z q P C ( “ p x ` , p y, z qq : p y, z q P B ( while for x P Z ` x ¨ p x, p y, z qq : p y, z q P B ( “ p x ` , p y, z qq : p y, z q P C ( . (cid:3) Lemma 5.1 shows that the constraints defined so far do not preclude the existenceof weakly periodic behavior. In fact the particular configuration ω constructed inthe proof has the property that x x y ď Stab H p Z q p ω q .Before pursuing the construction of our H p Z q -SFT Ω we point out that the rudi-mentary system defined above already allows us to prove Corollary 1.2 establishingthe undecidability of the extension and emptiness problem for H p Z q -SFTs: Proof of Corollary 1.2.
By construction, every valid configuration of the H p Z q -SFT described above has to see a Robinson tiling inside every other x y , z y -coset. Exploiting the nested squares formed by red arrows in those tilings as an infra-structure we may then use supplementary symbol decorations and additional localrules to embed arbitrary Turing machine computations following the same tech-niques as in [24, § A Rob using two distinct shades of red arrows to alternately color intersecting contours ofsupertiles of contiguous level. Those contours of size 2 i ` i P N ) form a hierarchyof non-intersecting nested squares, smaller ones being completely contained insidebigger ones. The region enclosed by the square contour of a level-2 i supertile butlying outside all contours of smaller (even) level contained within is referred to asa board of level 2 i . Each such board contains 2 i ` p i ` q unobstructed siteslocated at the intersections of an unobstructed row with an unobstructed column.Now a second layer of tiles can be superimposed on this decorated Robinsontiling using boards of level 2 i to simulate the first 2 i ` i ` i ` i ` i ` i board.Since there is no bound on the board size, there is also no bound on the lengthof the (finite) TM run coded within. A straight-forward reduction from the haltingproblem then gives the desired undecidability result: There is a valid way of fillingarbitrarily large boards, i.e. extending a locally admissible pattern into a globallyadmissible configuration on the entire x y , z y -coset, if and only if the simulated TMdoes not halt. (cid:3) Alignment of supertiles in the z direction. In this subsection we showthat the propagation of crosses constraint imposes some additional rigidity on the Z -Robinson tilings which can be seen in our construction. Lemma 5.2.
In each x y , z y -coset filled with symbols from the Robinson alphabet A Rob , the local rules specified so far force level- i supertiles (for all i P N ) to be fullyaligned along the z -direction.Proof. Let us fix a x y , z y -coset, say x y , z y x x ( H p Z q for some x P Z , containing sym-bols from A Rob . The Robinson rules force the existence of some level-1 supertile,i.e. a 3 ˆ TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 13 at its center as well as its 4 corners and arms in the remaining 4 tiles along itsborder. Assume this level-1 supertile is centered on coordinate p x, p y, z qq P H p Z q ( y, z P Z ). The alternating crosses constraint then implies that the entire subcoset x y , z y z z ´ y y ´ x x is filled with cross tiles all of which are corners of level-1 super-tiles. Now the propagation of crosses along the x -direction guarantees the presenceof cross tiles in all of x x , y , z y z z ´ y y ´ x x . Moreover it also allows us to deducethe position of the level-1 supertiles adjacent (along the z -direction) to the onecentered on p x, p y, z qq as follows.The cross tile at p x, p y ´ , z ´ qq can only be the corner of a level-1 supertilewith center either at p x, p y ´ , z ´ qq or at p x, p y, z ´ qq . (Having its center at p x, p y ´ , z ´ qq or at p x, p y, z ´ qq is impossible due to the ordinary Z -Robinsonrules, because this would force the cross tile at p x, p y ´ , z ´ qq to be part of twodistinct level-1 supertiles.) Since the central position of a supertile is always filledwith a cross, the first case would see cross tiles at both coordinates p x, p y, z qq , thecenter of the level-1 supertile we started with, and p x, p y ´ , z ´ qq . Howeverthese crosses would need to propagate along the x -direction, enforcing cross tiles at p x ´ , p y, z ´ y qq and p x ´ , p y ´ , z ´ ´ p y ´ qqq “ p x ´ , p y ´ , z ´ y qq .As noted before, the entire set x x , y , z y z z ´ y y ´ x x contains cross tiles and thusthere would be crosses at p x ´ , p y ´ , z ´ ´ y qq and p x ´ , p y ´ , z ` ´ y qq aswell. Clearly this leaves no valid possibility to fill coordinate p x ´ , p y ´ , z ´ y qq with a Robinson symbol. Hence the level-1 supertile containing p x, p y ´ , z ´ qq asone of its corners has to be centered on p x, p y, z ´ qq and is therefore aligned alongthe z -direction with the one centered on p x, p y, z qq . A symmetric argument showsthat the level-1 supertile containing coordinate p x, p y ´ , z ` qq has to be centeredat p x, p y, z ` qq . This local alignment then propagates across the entire z -direction,enforcing one level-1 supertile centered at each coordinate in x z y z z y y x x .Proceeding by induction on the level of supertiles we establish the general resultusing a similar reasoning: A level- p i ` q supertile centered on p x, p y, z qq contains 4level- i supertiles centered on p x, p y ˘ i , z ˘ i qq . Due to the induction hypothesisthose give rise to two infinite families of aligned level- i supertiles respectively cen-tered on x z i ` y z z ´ i y y ˘ i x x , which in turn have to group together to form morelevel- p i ` q supertiles. Again there are only two possibilities for the center coordi-nate of the one containing the level- i supertile centered on p x, p y ´ i , z ´ ¨ i qq ,namely p x, p y ´ i ` , z ´ i ` qq or p x, p y, z ´ i ` qq . However having cross tilesat p x, p y ´ i ` , z ´ i ` qq and p x, p y, z qq implies by the propagation constraintadditional crosses at ` x ´ , p y ´ i ` , z ´ i ` ´ p y ´ i ` qq ˘ “ ` x ´ , p y ´ i ` , z ´ y q ˘ and at p x ´ , p y, z ´ y qq which still is not compatible with the presence of crosstiles already fixed by the alignment of level- i supertiles. Note that the family ofaligned level- i supertiles centered at x z i ` y z z ´ i y y ´ i x x mentioned above enforceslevel- i supertiles centered on x z i ` y z z ´ y ´ i y y ´ i x x ´ , making it impossible to fillthe y -segment p x ´ , p y ´ i ` ` k, z ´ y qq : 1 ď k ă i ` ( with symbols from A Rob without violating any Robinson rules. Therefore we areonce more left with the option of having aligned level- p i ` q supertiles centeredat p x, p y, z qq and p x, p y, z ´ i ` qq (and following a symmetric argument also at p x, p y, z qq and p x, p y, z ` i ` qq ). (cid:3) Let ω P A H p Z q be a configuration satisfying all previous constraints. As a directconsequence of Lemma 5.2, all z -columns x z y y y x x “ tp x, p y, z qq P H p Z q : z P Z u p x, y P Z q containing Robinson symbols have to see crosses spaced periodically at distance 2 i for a uniquely determined i P N Y t8u . (Here i P N guarantees the existence of z P Z such that t z P Z : ω p x, p y,z qq is a cross tile u “ z ` i Z , while i “ 8 meansthat ˇˇ t z P Z : ω p x, p y,z qq is a cross tile u ˇˇ ď i only depends on the value of y and is kept constantacross an entire x x , z y -coset in ω .5.3. Pruning out weakly-periodic behavior: introducing counters.
Havingestablished the existence of valid configurations in Lemma 5.1 we now focus ourattention on pruning out periodic behavior. To this end we impose a few additionallocal rules on sites filled with symbols from A Count , forcing collaboration betweencorresponding x y , z y -cosets to form binary counters of arbitrarily large sizes whichwill run in a heavily synchronized way.Fixing a value y P Z observe the global structure of the configuration seen in the x x , z y -coset x x , z y y y “ tp x, p y, z qq P H p Z q : x, z P Z u . Since generators x and z commute and are of infinite order, we might again thinkof such a coset as an isomorphic copy of Z . Forgetting about the slant in the x -direction shown in the S -Cayley graph in Figure 4 for now we refer to z asbeing horizontal, pointing to the right and x as vertical, pointing upward. Thecross-propagating x -columns seen in the restriction ω | x x , z y y y of a configuration ω P A H p Z q obeying all previous rules necessarily partition x x , z y y y into infinite vertical(slanted) strips whose uniform horizontal width 2 i ( i P N Y t8u ) is given by theperiodic occurrences of crosses in aligned Robinson supertiles. Within each suchstrip we will run a binary counter whose current value, represented by symbolsfrom the counter-alphabet A Count , is stored in the z -row segment confined betweentwo successive cross-propagating x -columns and increases by 1 when going up twolayers (to the next A Count -filled z -row). Strips of finite width 2 i obviously limitthe range of their binary counter, which after reaching its maximal value 2 i ´ x x , z y y y with counters of width 2 “
4. Note that cross-propagating x -columns in distinct x x , z y -cosets come from the central crosses of different levelsupertiles and are thus spaced at different periods 2 i . Since supertiles of arbitrarilyhigh level appear in every Robinson tiling there is no upper bound on the width ofthe strips seen in a valid configuration ω P Ω and thus also no global upper boundon the maximal counter value, i.e. the number of steps it takes some counters toperiodically repeat.The actual construction of binary counters using only local rules is a bit technical,but well-known for Z -SFTs or even Z -cellular automata. Here we just providea short description of the necessary constraints, while encouraging the reader tocheck that those rules indeed implement the counters we need. First, the presenceof a cross-propagating x -column to the right should trigger addition of 1 on thecounter’s least significant bit, i.e. the first component of the counter-symbol sittingimmediately to the left of the x -column containing crosses has to change its value TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 15 Figure 6.
Binary counters of width 4 as seen in an x x , z y -coset.(A counter’s most significant bit, distinguished as a bold-face digit,is always stored in a cross-propagating x -column, whereas its leastsignificant bit is stored immediately to the left of the contiguouscross-propagating x -column to the right.) Width 4 counters over-flowing on level 6 from the bottom, trigger diagonal segments ontheir most significant bits as well as forward segments across theentire counter. Width 2 counters (on neighboring x x , z y -cosets)overflowing on levels 2 , ,
10 and 14 trigger additional diagonallines passing through the shown coset on those rows.in every counter step. Consequently, seeing symbols ω p h q “ p b h , s h q P A Count with b h P t , u and ω p z h q “ p b z h , s z h q P A Count with b z h P t , u around somesite h P H p Z q , symbol ω p x h q “ p b x h , s x h q P A Count is only allowed to haveits first component b x h ‰ b h . Note that since counter-symbol and Robinson-tile cosets alternate, the next value of the counter is stored two z -rows above,therefore compelling a vertical offset by x instead of just x . Addition of bitsthen has to be executed locally proceeding left, across the entire counter strip, moving a possible carry along until reaching the next cross-propagating x -column(containing the counter’s most significant bit). Looking at a particular site h P H p Z q with z -neighboring symbol ω p z h q “ p b z h , s z h q P A Count such that b z h P t , u ,this procedure is taken care of by allowing only certain combinations of symbolsfrom A Bin to appear in the first components b g of sites g P t h, z h, x h, x z h u (alsooccupied by symbols from A Count ). The list of 16 allowed locally admissible patternsis shown in Figure 7. b x h b x z h b h b z h P , , , , , , , , , , , , , , , + . Figure 7.
Whenever site z h P H p Z q contains a counter symbol p b z h , s z h q P A Count with b z h P t , u , we allow exactly 16 locallyadmissible ways to assign binary symbols to surrounding sites h , x h and x z h . Remark . For reasons that will become relevant in Section 6 we have to com-ment on the evolution of infinite width binary counters, which – according to thehierarchical structure of an arbitrary Robinson tiling – may exist in at most one x x , z y -coset. Such an exceptional coset either contains a single cross-propagating x -column, dividing it into two “halfplanes” each of which is filled by one of two sep-arate infinite width counters or – in the absence of any cross-propagating x -column– the entire x x , z y -coset is occupied by a single infinite width binary counter.In the former case the counter running in the left halfplane still has a leastsignificant bit (located immediately to the left of the cross-propagating x -column).Hence this counter evolves as deterministically as any of its finite width cousins.The absence of a least significant bit for the counter running in the other halfplane,as well as for the unique infinite width counter in the latter case, however, allowsfor a choice of whether and on which (unique) z -row the counter value increases.This causes those counters to follow a rather different, partially non-deterministicbehavior explained in the following lemma. Lemma 5.4.
An infinite width binary counter with a least significant bit increasesits value in each step – exactly as a finite width counter – leading to at most onetotal overflow.The value of an infinite width binary counter without a least significant bit staysunchanged across its entire x x , z y -coset, unless there is a single increase produced bya partial or total overflow at a unique but arbitrary (i.e. non-deterministic) positionin its evolution.Proof. The first statement is a direct consequence of the local rule forcing a binarycounter’s least significant bit to periodically change between 0 and 1, which thentriggers the usual counting including periodic partial overflows on arbitrarily longfinite suffixes.For the second part we have to analyze the effect of a couple of local patternspossibly seen in ω P Ω on the infinite counter’s coset x x , z y y y ( y P Z ): Suppose TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 17 there is an A Count -filled site h P x x , z y y y with ω p h q P t , u but ω p x h q P t , u ,signalling an increasing digit at h . The locally admissible patterns shown in Figure 7then force ω p z ´ k h q “ ω p z ´ k x h q (until z ´ k h hits the infinite counter’s possiblyexisting most significant bit) as well as ω p z k h q “ ω p z k x h q “ k P N .As a secondary effect the same local rules then imply ω p z k x ` l h q “ ω p z k x h q and ω p z k x ´ l h q “ ω p z k h q for all l P N and all k P Z such that site z k h is part of thecounter. Hence the counter is stuck on the same value on all z -rows x z y x ´ l h below,experiences a single (infinite) partial overflow between z -rows x z y h and x z y x h ,before staying on its newly reached value for all z -rows x z y x l h above.Let us assume next that the above does not happen. The counter might stillcontain a local pattern with ω p h q “ ω p x h q “
0, i.e. a decreasing bit, atsome of its A Count -filled sites h P x x , z y y y . This however is only possible if thecounter experiences a total overflow, forcing all its digits in z -rows x z y x ´ l h to be1 and all its digits in z -rows x z y x l h to be 0 (for l P N ). Therefore the counter isat its highest value (all 1 digits) in all z -rows below and stays on the value zero forall z -rows above this unique total overflow.In the absence of both such local patterns we have ω p h q “ ω p x h q on allcounter sites h P x x , z y y y , i.e. all digits stay unchanged forever, trivially concludingthe argument. (cid:3) Once having set up those binary counters, the final task in our construction isto synchronize all counters having the same finite width 2 i ( i P N ). This is done bycontrolling the x y , z y -cosets in which those counters may overflow using superim-posed diagonal lines to communicate such an event across a Robinson tiling’s entirenet of level- i supertiles. After implementing all previous parts, there is a surpris-ingly simple solution to this, involving one last local rule: Suppose site h P H p Z q is filled with a counter-symbol ω p h q “ p b h , s h q P A Count having b h P t , u ,thus marking a cross-propagating x -column. Consider its next-to-nearest neigh-bor ω p x h q “ p b x h , s x h q P A Count , which then necessarily has b x h P t , u aswell, to see whether or not the counter experiences a total overflow. If b h “ and b x h “ this is the case (the most significant bit decreases) and we require s h “ to be the A Seg -tile containing both the diagonal as well as the forward segment,whereas in the remaining cases – either the combination b h “ b x h “ or b h “ –the symbol s h P A Seg is forced to be the blank tile not containing any line segments.We point out that this final rule in fact excludes five of the 8 A Count -symbolshaving a bold-face binary digit as its first component. Additionally we can removefrom our initially defined alphabet the two A Count -symbols combining a 0 in its firstcomponent with a tile including a forward line segment in its second component.This is due to the fact that whenever a counter is about to undergo a total over-flow none of its binary digits is 0. With these observations the cardinality of thealphabet actually used in configurations of p Ω , H p Z qq decreases to 56 ` ` “ Alignment of supertiles in the y direction and counter synchroniza-tion. We prove further rigidity results about supertiles and binary counters, effec-tively characterizing the Z -projective subdynamics seen in our construction.The next lemma shows that diagonal segments generated during a counter over-flow enforce the width of all counters whose most significant bits are located on thesame northwest-southeast pointing diagonal inside a x y , z y -coset to coincide. Lemma 5.5.
Let l P Z zt u . Whenever a valid configuration has two sites p x, p y, z qq , p x, p y ´ l, z ` l qq P H p Z q both containing the A Count -symbol p , q , the correspond-ing z -rows x z y y y x x and x z y y y ´ l x x contain binary counters of equal width.Proof. Suppose the z -row x z y y y x x contains bold-face digits with finite period 2 i ( i P N ) smaller than the (possibly infinite) width of the binary counters implementedalong x z y y y ´ l x x . Propagation of the line segments emanating from the A Count -symbol p , q seen at site p x, p y, z qq would force all counters along x z y y y x x as wellas the counter having its most significant bit at site p x, p y ´ l, z ` l qq to overflowon layer x y , z y x x . Counters of width 2 i would then repeat overflowing 2 ¨ i cosetsabove, where cross-propagation has displaced all bold-face digits to sites x z i y z z ` ¨ i ¨ y y y x x ` ¨ i “ x z i y z z y y x x ` ¨ i . Similarly, the most significant bit located at p x, p y ´ l, z ` l qq propagates along the x -direction to site h : “ ` x ` ¨ i , ` y ´ l, z ` l ` ¨ i p y ´ l q ˘˘ thus also containinga bold-face digit. Since one of the newly overflowing binary counters of width 2 i has its most significant bit stored at position ` x ` ¨ i , ` y, z ` ¨ i p y ´ l q ˘˘ P x z i y z z y y x x ` ¨ i , the diagonal line necessarily emanating from there would extend to the most signif-icant bit at h . However, due to its larger width the corresponding binary counterwould not yet experience a total overflow on layer x y , z y x x ` ¨ i , making it impos-sible to have a bold-face digit with a diagonal line segment there and contradictingour initial assumption. (cid:3) In order to prove synchronization of our binary counters we show another rigidityresult about Robinson tilings seen in x y , z y -cosets of configurations in p Ω , H p Z qq : Lemma 5.6.
In each x y , z y -coset filled with symbols from the Robinson alphabet A Rob , the additional local rules about binary counters force level- i supertiles (for all i P N ) to be also fully aligned along the y -direction.Proof. Let us start by showing that for every i P N (mis-)alignment of level- i su-pertiles along y -rows is preserved across all A Rob -filled x y , z y -cosets. This followsdirectly from the propagation of crosses constraint: Due to Lemma 5.2, the exis-tence of a single level- i supertile centered at coordinate p x, p y, z qq P H p Z q forcesthe presence of level- i supertiles centered on all sites x z i ` y z z y y x x . Suppose thatadjacent to this filled-in z -row strip of aligned supertiles there is another level- i supertile centered at p x, p y ` i ` , z ` o qq P H p Z q , which – invoking Lemma 5.2 asecond time – guarantees the appearance of another infinite z -row strip of alignedlevel- i supertiles centered on x z i ` y z z ` o y y ` i ` x x . Here o P i ` Z corresponds tomutual alignment, whereas o R i ` Z indicates a partial offset between these two z -row strips of supertiles. Moving up two x y , z y -cosets, propagation of crosses thendisplaces the centers of those level- i supertiles from sites x z i ` y z z y y x x Y x z i ` y z z ` o y y ` i ` x x ( x y , z y x x to sites x z i ` y z z ` y y y x x ` Y x z i ` y z z ` y ` i ` ` o y y ` i ` x x ` ( x y , z y x x ` . TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 19
Simplifying the z -component x z i ` y z z ` y ` i ` ` o “ x z i ` y z z ` y ` o in the secondset of coordinates establishes the preservation of the offset o across consecutive A Rob -filled x y , z y -cosets as claimed.Next we prove that the offset o indeed has to be a multiple of 2 i ` , i.e. thatarbitrary adjacent z -row strips of level- i supertiles in a fixed x y , z y -coset are in factaligned along the y -direction. Presume as before that there is an entire z -row stripof aligned level- i supertiles centered along x z y y y x x . Each of those supertiles obvi-ously constitutes one quadrant of some level- p i ` q supertile, all of which – followingfrom Lemma 5.2 – again have to align properly along the z -direction. Assumingwithout loss of generality that this infinite family of level- p i ` q supertiles extendsto the left having their central crosses placed at sites x z i ` y z z ´ i y y ´ i x x , alignmentbetween the two families of constituent level- i supertiles centered at x z i ` y z z y y x x and x z i ` y z z y y ´ i ` x x is then forced by the ordinary Z -Robinson rules. To showalignment on the opposite side, suppose there exists a level- i supertile centeredat coordinate p x, p y ` i ` , z ` o qq P H p Z q , thus giving rise to an entire fam-ily of level- i supertiles with their central crosses spaced 2 i ` -periodically at sites x z i ` y z z ` o y y ` i ` x x . Looking one x y , z y -coset above, the z -row x z y y y ` i ` x x ` isthus filled with segments of binary counters of width 2 i ` having their most signifi-cant bit stored at x z i ` y z z ` y ` o y y ` i ` x x ` . Using the result about the persistenceof the offset o across successive x y , z y -cosets obtained at the beginning, we mayassume without loss of generality that the binary counter whose most significantbit is located at ´ x ` , ´ y ` i ` , z ` y ` o ´ i ` ¨ Q o i ` U¯¯ experiences a total overflow. The A Count -symbol seen at this location is thus a bold-face digit decorated with both line segments. The forward segment then stretchesacross x z y y y ` i ` x x ` making all other binary counters along its way overflow aswell, whereas the diagonal segment extends along all sites !´ x ` , ´ y ` i ` ´ l, z ` y ` o ´ i ` ¨ Q o i ` U ` l ¯¯ : l P Z ) ( H p Z q . Recall from Section 3 that the level- p i ` q supertiles centered at x z i ` y z z ´ i y y ´ i x x have their crosses placed at x ( ˆ y ´ ¨ i ` j ` j ` k : 0 ď k ă i ` ´ j ( ˆ ` z ´ ¨ i ` j ` j ` Z ˘ for all j P t , , . . . , i ` u . The propagation of crosses constraint then forces bold-face binary digits at sites x ` ( ˆ y ´ ¨ i ` j ` j ` k : 0 ď k ă i ` ´ j ( ˆ ` z ` y ´ ¨ i ` ` j ` Z ˘ for all j P t , , . . . , i ` u . Those sites notably comprise the contiguous y -segment x ` ( ˆ y ´ k : 0 ď k ă i ` ( ˆ z ` y ` i ` ( which is hit by the diagonal line forced by the binary counter undergoing a totaloverflow. The site of intersection ` x ` , ` y ` o ´ i ` ¨ Q o i ` U , z ` y ` i ` ˘˘ then has to be filled with another bold-face digit decorated with both line seg-ments. This implies that the corresponding binary counters along x z y y y ` o ´ i ` ¨ r o i ` s x x `
10 AYS¸E A. S¸AH˙IN, MICHAEL SCHRAUDNER, AND ILIE UGARCOVICI are overflowing as well and according to Lemma 5.5 have to have the same width2 i ` as the ones along x z y y y ` i ` x x ` . Hence y ` o ´ i ` ¨ P o i ` T “ y , the uniquecoordinate of a z -row having the correct period of crosses, and we conclude o P i ` Z . (cid:3) Remark . Together Lemmata 5.2 and 5.6 exclude any misalignment betweensupertiles. The only Robinson tilings appearing in the A Rob -filled x y , z y -cosets inour H p Z q -SFT Ω thus have their level- i supertiles repeated fully periodically alongboth the y - and the z -direction for each i P N . This means that whenever a site p x, p y, z qq P H p Z q contains the central cross of some level- i supertile the centers ofall (other) level- i supertiles are located at ď k P Z x y i ` , z i ` y z z ` k ¨ y y y x x ` k . Moreover, any pattern seen in such a fully aligned Robinson tiling is already asubpattern of some level- i supertile for sufficiently large i P N . Hence it occurs insideevery Robinson tiling and the family of tilings seen in x y , z y -cosets thus comprisesexactly the unique minimal subsystem contained inside the Z -Robinson SFT [12]. Corollary 5.8.
Given a valid configuration in p Ω , H p Z qq , all binary counters hav-ing the same (finite) width undergo a total overflow on the same x y , z y -cosets.Proof. Note that for i P N the binary counters of width 2 i have their most signifi-cant bit stored along the x -columns propagating the central crosses of level- p i ´ q supertiles. (Here we explicitly include the elements in the 2 Z ˆ Z coset filled bythe alternating crosses constraint as (centers of) level-0 supertiles.) According toRemark 5.7 those crosses appear exactly at Ť k P Z x y i , z i y z z ` k ¨ y y y x x ` k for some p x, p y, z qq P H p Z q . Consequently the most significant bits of width 2 i counters ina fixed but arbitrary A Count -filled coset x y , z y x x ` k ` ( k P Z ) are stored at sites x y i , z i y z z `p k ` q¨ y y y x x ` k ` forming a doubly-periodic lattice. The total overflowof one of those binary counters – indicated by the presence of a bold-face digit ac-companied by the A Seg -tile with both line segments – triggers the total overflowof all binary counters along its own z -row (propagation of the forward segment) aswell as along its x y ´ z y -diagonal (propagation of the diagonal segment) hitting andthus enforcing a total overflow of all binary counters along the remaining z -rows(propagation of corresponding forward segments). This collective overflow thenrepeats periodically on every 2 ¨ i th x y , z y -coset above and below. (cid:3) A strongly aperiodic shift of finite type.
The following two propositionslet us conclude the existence of strongly aperiodic shifts of finite type on the discreteHeisenberg group.
Proposition 5.9.
The H p Z q -SFT Ω constructed above is non-empty.Proof. Non-emptiness of Ω basically follows from Lemma 5.1 plus the fact thatcounter-cosets can be filled in without violating any of the later rules. For anexplicit configuration ω P Ω take the particular Z -Robinson tiling ρ P p Ω Rob , Z q used in that lemma and as before let ω ` p x, p y, z qq ˘ : “ ρ p y, z q for all x P Z .The crosses appearing in ρ and thus in ω still enforce the presence of bold-face A Bin -digits precisely at all sites p x, p y, z qq P H p Z q with x P Z ` p y, z q P B “ p , q ( Y ď i P N ` p i ´ ` i Z q ˆ i Z ˘ . TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 21
Note that this set comprises the entire y -row x y y x ´ and define ω ` p´ , p y, z qq ˘ : “ , ˘ iff p y, z q P B ` , ˘ iff p y, z q R B (the forward and diagonal lines emanating from x y y x ´ cover the entire x y , z y -coset).Hence ω has all its binary counters – independent of their width – overflowingsimultanously on layer x y , z y x ´ , thus resulting in ω ` p , p y, z qq ˘ “ , ˘ iff p y, z q P B ` , ˘ iff p y, z q R B (all blank tiles forced as no counters overflow). Deterministic evolution of thefinite as well as infinite binary counters on the other A Count -cosets then dictatesall remaining symbols: Given k P Z and i P N , the value of all counters of width 2 i on layer x y , z y x k ` is k mod 2 i . (The value of an infinite width counter on layer x y , z y x k ` is k , applying the usual convention of representing negative numbers inbase 2 by an infinite prefix of 1s.) For k ` ” i , all counters of width 2 j with j ď i overflow, triggering the occurrence of A Seg -tiles with a forward segmentat all sites p k ` , p y, z qq P H p Z q with p y, z q P Ť j ď i ` p j ´ ` j Z q ˆ Z ˘ and adiagonal segment at all sites with p y, z q P Ť j ď i tp j ´ ` j ¨ m ´ l, l q : l, m P Z u , henceresulting in the presence of A Seg -tiles with both line segments at all coordinates inthe intersection, i.e. with p y, z q P Ť j ď i ` p j ´ ` j Z q ˆ j Z ˘ and blank tiles at allremaining coordinates (i.e. in the complement of the union of both sets). (cid:3) Proposition 5.10.
The H p Z q -SFT Ω constructed above is strongly aperiodic.Proof. Let ω P Ω be an arbitrary configuration and let a, b, c P Z be integers forwhich p : “ p a, p b, c qq P Stab H p Z q p ω q . The periodicity σ p p ω q “ ω then implies thatwhenever ω sees a particular symbol at site h P H p Z q , the same symbol has toappear at site h ¨ p . Select i P N sufficiently large to satisfy 2 i ą max t | a | , | b | , | c | u and take x, y, z P Z such that ω ` p x, p y, z qq ˘ P A sees the central cross of somelevel- i supertile. Lemma 5.2 guarantees the existence of infinitely many alignedlevel- i supertiles centered on all sites x z i ` y z z y y x x thus forcing crosses on the z -row x z y y y x x to be spaced periodically with distance 2 i ` . Note that all other z -rows x z y y y ` k x x with ´ i ă k ă i , k ‰ p forces crosses along x z y y y x x ¨ p “ x z y y y ` b x x ` a to occur with the same period, butat sites x z i ` y z z ` c ` bx y y ` b x x ` a . Now propagation of crosses along the x -directionallows us to deduce what happens along z -row x z y y y ` b x x : there are crosses exactlyat sites x z i ` y z z ` c ` bx ´ a p y ` b q y y ` b x x . Due to the bound | b | ă i this z -row is still included in the strip occupied by theinfinite family of aligned level- i supertiles forced by the central cross at p x, p y, z qq ,all of whose z -rows except x z y y y x x contain crosses at smaller periods. Hence weconclude b “
0, which locates both z -rows x z y y y x x and x z y y y x x ¨ p within the same x x , z y -coset. Consequently the value of the synchronized binary counters of width2 i ` seen along x z y y y x x ` would have to coincide with the value seen in the width2 i ` binary counters along x z y y y x x ` ¨ p “ x z y y y x x ` a ` . Since | a | ă i ! i ` (the counter’s period) this only happens for a “
0, which leaves us with the occurrenceof shifted crosses at sites x z i ` y z z y y x x ¨ p “ x z i ` y z z ` c y y x x . Comparing those to the position of the unshifted crosses at x z i ` y z z y y x x , thebound | c | ă i implies c “ H p Z q p ω q “ p , p , qq ( is trivial and since ω was arbitrary p Ω , H p Z qq is indeed strongly aperiodic. (cid:3) Overflow coordination – making our H p Z q -SFT almost minimal Since we explicitly control both alphabet and local rules of our strongly aperiodic H p Z q -SFT, we might wonder, whether our construction already forces minimalityof Ω. Unfortunately this is not the case for two distinct reasons, one of which isnot too hard to circumvent.The rather obvious obstacle preventing minimality is that thus far we do notcontrol the offsets between x y , z y -cosets on which binary counters of different widthexperience a total overflow. For example our SFT Ω contains two configurations ω, ω P Ω, both having counter symbols on all sites in x x , y , z y x , but such that in ω counters of all widths overflow simultaneously on the coset x y , z y x ´ – this is thecase for the particular configuration constructed in Proposition 5.9 – whereas in ω only counters of width at least 2 “ x y , z y x ´ , while all countersof width 2 “ x y , z y x ` Z . It is not hard tosee that the (disjoint) H p Z q -orbits of configurations ω and ω are separated bya positive distance . Hence neither of them is dense in our present construction,contradicting minimality of Ω.In order to fix this issue of distinct-width overflow non-coordination we slightlyenlarge our alphabet A Count and specify a couple of additional local rules forcingoverflows on larger width counters to induce simultaneous overflows on counters ofall smaller widths. To do so, let us introduce a single new square tile , replacingsome of the occurrences of , while keeping the tile unchanged in other places.Hence from here on we consider r A Seg : “ , , , , ( , forming r A Count : “ A Bin ˆ r A Seg just as before. A first supplementary constraintnevertheless forces the new overflow-coordination tile to be admissible as secondcomponent in a symbol of r A Count only if the A Bin -digit in its first component isa bold-face . Thus effectively adding only one symbol to A Count , the total ofoccurring symbols (a proper subset of r A Count ) increases from 9 to 10 fixing thecardinality of the alphabet actually used in our modified H p Z q -SFT at 66.Next construct r Ω ( ` A Rob \ r A Count ˘ H p Z q by literally applying all constraintsimposed on our previously defined H p Z q -SFT Ω ( ` A Rob \ A Count ˘ H p Z q , exceptfor adding the possibility of having a symbol ` , ˘ whenever we had a symbol ` , ˘ before. This increased flexibility, apparently veering us even further awayfrom a minimal system, is then reduced by invoking the following pair of local rules.For every ω P r Ω and every site h P H p Z q with ω p h q P r A Count we enforce: Patterns of support p x, p y, z qq : x P t , u ^ y, z P t , , . . . , u ( ( H p Z q occuring somewherein ω are disjoint from patterns of the same support appearing in ω . TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 23 (OC1) If either ω p h q “ ω p h q “ ω p x h q “
1, i.e. there is no overflow onthe corresponding binary counter’s digit, having ω p y h q “ ` , ˘ forces ω p y ´ h q “ ` , ˘ and symmetrically having ω p y ´ h q “ ` , ˘ forces ω p y h q “ ` , ˘ .(OC2) If either ω p h q “ ω p x h q “
0, i.e. there is an overflow on the corre-sponding binary counter’s digit, or ω p h q “ ` , ˘ , i.e. the entire counteroverflows, the presence of a bold-face binary digit in the r A Count -symbol atsite y h forces ω p y h q “ ` , ˘ , while the presence of a bold-face binarydigit in the r A Count -symbol at y ´ h forces ω p y ´ h q “ ` , ˘ .The global effect of those local constraints is that each overflow occurring oncertain digits of a binary counter with a larger width triggers a total overflowof each smaller width counter whose most significant bit is aligned along the y -direction with one of those overflowing digits. The horizontal line segment seenin the newly introduced square tile visually propagates the necessary informa-tion about such an event uninterruptedly across larger distances in the y -direction.Figure 8 illustrates this counter overflow coordination in two distinct x y , z y -cosetpatterns occurring in configurations in r Ω.
111 0
111 0 ùñ ùñ Figure 8.
Partial overflows – shown as lightgray shading – triggerthe new overflow-coordination tile . (For better visibility r A Seg -symbols are only drawn on most significant bits.) The pattern onthe right appears 2 ¨ “ x y , z y -cosets above the one on the left.The first consequence of the new rules is the following lemma. Lemma 6.1.
In a valid configuration in `r Ω , H p Z q ˘ no y -row can contain both r A Seg -tiles and .
Proof.
Since both tiles and are always paired with a bold-face A Bin -digit , the resulting r A Count -symbols only appear on x y , z y -cosets where at least somebinary counters – those of width 2 – experience a total overflow. Due to the completealignment of Robinson supertiles (see Remark 5.7) the bold-face binary digits seenin an arbitrary r A Count -filled y -row x y y h ( h P H p Z q ) group together in contiguous y -segments of length 2 i ´ i P N Y t8u . Those y -segments areinevitably separated by a width 1 gap containing a non bold-face digit. (Herethe infinite length case allows for either two half-row segments separated by awidth 1 gap or for an entire y -row filled with bold-face digits.) The second part ofRule (OC2) excludes the coexistence of symbols ` , ˘ and ` , ˘ within a singlesuch y -segment, whereas Rule (OC1) forbids adjacent segments to change type. (cid:3) Lemma 6.2.
Valid configurations in `r Ω , H p Z q ˘ obey overflow coordination: forevery i P N , a total overflow of width i counters on some x y , z y -coset forces allwidth j counters with j ă i to also undergo a total overflow on this particularcoset. Moreover, the occurrence of a (partial, respectively, total) overflow of theunique infinite width counter not having a least significant bit forces a simultaneoustotal overflow of all finite width counters as well as a total overflow of the possiblyexisting second infinite width counter on the same x y , z y -coset.Proof. Choose h P H p Z q such that x x , z y h is filled with width 2 i counters with theirmost significant bits stored in the x -columns of x x , z i y h and which periodicallyexperience total overflows on all z -rows in x x i ` , z y h . The alignment of Robinsonsupertiles – given by Lemma 5.2 and Lemma 5.6 – forces all sites in the contiguous y -segment S “ y k z i ´ h : k P t , , . . . , i ´ ´ u ( to contain most significant bitsof binary counters of smaller width. Thus those positions are filled with r A Count -symbols having a bold-face digit as their first component. The overflowing counterof width 2 i sees a binary digit 1 at z i ´ h and a binary digit 0 at x z i ´ h . Accordingto Rule (OC2) this then triggers r A Count -symbols at all sites in S to be ` , ˘ whichin turn forces all binary counters running in x x , z y -cosets x x , z y y k h with 1 ď k ď i ´ ´ z -row x z y y k h . Note that atleast one of those cosets is filled with a width 2 j counter for every j P t , , . . . , i ´ u ,finishing the proof for the first part.For the second statement, let h P H p Z q such that x x , z y h is the (unique) excep-tional coset and suppose that the infinite width counter not having a least significantbit sees an overflowing digit at site h . This immediately implies that the digits at allpositions in t z l h : l P N u overflow as well, which in turn forces the spread of sym-bols ` , ˘ across arbitrarily long y -segments S i “ y k z z i h : k P t , , . . . , i ´ u ( ( i, z i P N with z i ď i ) causing the simultaneous total overflow of all finite widthcounters on layer x y , z y h . Recall from Lemma 5.4 that this situation occurs at mostonce in the infinite width counter’s entire evolution.Now consider the possibly existing second infinite width counter whose leastsignificant bit on z -row x z y h we assume to be stored at position z ´ z h for some z P N .Suppose this counter does not experience a total overflow on z -row x z y h . Hencethere certainly exists i P N such that site z ´ z ´ i ` h does not contain an overflowingdigit. However, according to Lemma 5.4, an overflow on this digit then has to occur As discussed in Remark 5.3, an infinite width counter having a least significant bit might runin the left half of the unique exceptional x x , z y -coset. TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 25 in the next 2 i ´ z -row x z y x l h with 1 ď l ď i ´
1. ByRule (OC2) this overflow necessarily triggers overflow-coordination symbols ` , ˘ along an entire y -segment extending across neighboring x x , z y -cosets x x , z y y k h with1 ď k ď i ´
1. In particular this would cause a total overflow of width 2 i counterson layer x y , z y x l h . By the previous argument we know those counters also undergoa total overflow on layer x y , z y h . This conflicts with their values repeating exactlywith periodicity 2 i , concluding our argument. (cid:3) The results we have established allow us to now show that r Ω is an almost in-vertible SFT-extension of a sofic subshift inside Ω.
Proposition 6.3.
There exists a -block code from r Ω into Ω which is at most -to- and whose range consists of exactly those configurations in Ω which respect theoverflow coordination condition described in Lemma 6.2.Proof. Define a 1-block code φ : r Ω Ñ Ω induced by a local map Φ acting as theidentity on all symbols from A Rob as well as A Count while sending the new symbol ` , ˘ back to its original version ` , ˘ . By definition of r Ω, the replacementsinduced by φ respect all local rules of Ω. Since φ does not affect counter values,configurations in r Ω (whose counters are overflow coordinated by Lemma 6.2) thennecessarily map to configurations in Ω obeying the same overflow coordination con-straint. To show that the range of φ indeed coincides with this particular subshiftwe construct possible preimages.Let ω P Ω be a configuration whose binary counters are overflow coordinated.Note that by definition Φ ´ p ω p h qq is uniquely determined for all sites h P H p Z q except for those containing a symbol ` , ˘ . Consider an inclusion-maximal (finiteor infinite) y -segment S Ď x y y h ( h P H p Z q ) filled with A Count -symbols whose firstcomponents are bold-face digits. If any of the sites in S sees a symbol different from ` , ˘ no site in the preimage of ω | S is allowed to contain the symbol ` , ˘ byRule (OC2). Therefore the entire preimage Φ ´ p ω | S q is again uniquely determinedby definition of the local map Φ. Next we assume ω p s q “ ` , ˘ for all s P S . If S “ t y k h : a ă k ă b u with a ă b P Z is a finite y -segment of length b ´ a ´
1, thenone of its two neighboring x x , z y -cosets, x x , z y y a h , respectively x x , z y y b h , containsbinary counters of width 2 p b ´ a q . Overflow coordination forces the value of thosecounters seen on z -row x z y y a h , respectively x z y y b h , to equal 2 p b ´ a q ´ p b ´ a q . Hence their p b ´ a q th binary digit, one of which is just located at site y a h ,respectively y b h , undergoes an overflow. Rule (OC2) then forces the preimage of ω | S to be entirely filled with symbols ` , ˘ . If S is an infinite y -segment, theassumption ω p s q “ ` , ˘ for all sites s P S implies that all finite width countersexperience a total overflow on layer x y , z y h . Since this event can occur at most onceacross the entire configuration ω and since each x y , z y -coset can at most contain one y -row containing an infinite y -segment S , Lemma 6.1 establishes an upper boundof 2 for the number of possible preimages of ω . The two choices of filling the entiresegment S with ` , ˘ or ` , ˘ in fact appear in exactly two situations; namelywhenever S “ x y y h is an entire y -row or if S is a half-row such that the binarydigit of the infinite width counter seen at the (unique) site adjacent to S along the y -direction does not overflow. In all other cases the φ -preimage is unique showingthat φ is invertible on a dense subset of image configurations. (cid:3) Finally we prove that r Ω is almost conjugate to a minimal system.
Proposition 6.4. r Ω factors onto a strongly aperiodic, minimal (sofic) H p Z q -shift.Proof. Define a 1-block code φ : r Ω Ñ ` t A, C u \ A Bin ˘ H p Z q induced by a localmap Φ sending cross tiles to the letter C , arm tiles to the letter A , and projectingall r A Count “ A Bin ˆ r A Seg symbols onto their first component. As an immediateconsequence of the propagation of crosses constraint along each x -column, letters C always alternate with bold-face A Bin -digits, whereas letters A always alternatewith non bold-face A Bin -digits. Hence in φ p r Ω q the knowledge of all symbols on asingle A Bin -coset is sufficient to recover all symbols on each x y , z y -coset filled by thetwo new letters A and C . Applying the same proof as for Proposition 5.10 lets usconclude that φ p r Ω q is still strongly aperiodic.To prove minimality of the image system, let ω P φ p r Ω q be an arbitrary con-figuration and consider its restriction p : “ ω | F to an arbitrary but finite support F “ supp p p q ( H p Z q . Without loss of generality, assume ω | x y , z y to be filled with A Bin -symbols and choose k P N such that F Ď Ť k l “ Q k x l , where Q k : “ p , p y, z qq : | y | ă k ^ | z | ă k ( ( H p Z q . We now have to distinguish two cases.First suppose the base square Q k intersects only z -rows containing binary coun-ters of finite width. Hence in all z -rows x z y y y with | y | ă k , bold-face digits appearwith periodicity some (bounded) power of 2. Therefore the pattern ω | Q k extendsto a complete counter array , i.e. there is a rectangular patch R i : “ p , p y, z qq : | y | ă i ´ ^ | z | ď i ´ ( ( x y , z y of shape p i ´ q ˆ p i ` q for some i ą k and an element h P x y , z y such that Q k Ď R i h and ω p sh q P t , u for all s P p , p y, z qq : | y | ă i ´ ^ | z | “ i ´ ( ( R i , while ω p sh q P t , u for all s P p , p , z qq : | z | ă i ´ ( ( R i . Overflow coordinationthen implies that the pattern ω | R i h is completely characterized by the value of itslargest binary counter, namely a natural number between 0 and 2 i ´
1, stored in the z -segment p , p , z q : ´ i ´ ď z ă i ´ ( h ( R i h . Due to the complete alignmentof supertiles forcing Robinson tilings used in configurations of r Ω to come froma minimal subsystem (see Remark 5.7), the t A, C u -pattern ω | x R i h has to appearin every t A, C u -layer of every configuration in φ p r Ω q . Taking such an occurrence ω | p x R i h q g “ ω | x R i h ( g P H p Z q ) in an arbitrary configuration ω P φ p r Ω q the pattern ω | p R i h q g is necessarily another complete counter array of shape p i ´ q ˆ p i ` q uniquely determined by some value between 0 and 2 i ´
1. Moving up through thestack of A Bin -patterns ` ω | p R i hg q x l ˘ i ´ l “ this (increasing) value eventually has tomatch the value given by ω | R i h . Hence we are sure to find a copy of ω | R i h inside ω . Deterministic evolution of finite width counters then assures to see the entirepattern p “ ω | F inside ω .In the remaining case the base square Q k intersects a z -row, say x z y g for some g P Q k , contained in the unique exceptional x x , z y -coset of ω . Since x z y g sees atmost one bold-face A Bin -digit we cannot extend the support of the pattern ω | Q k toobtain a complete counter array. Instead we use that Q k zx z y g can be covered by arectangular patch R i ` h of shape p i ` ´ qˆp i ` ` q for some i ą k and h P x z y g , TRONGLY APERIODIC SFT FOR THE DISCRETE HEISENBERG GROUP 27 such that each restriction ω | R i z ˘ i ´ y ˘ i ´ h to one of R i ` h ’s four corner rectanglesof shape p i ´ q ˆ p i ` q is a complete counter array, while neither of the twosymbols ω p z ˘ i h q P A Bin is a bold-face digit. Since x z y h is part of the exceptional x x , z y -coset of ω , z -row x z y y j h is filled with binary counters of finite width 2 j ` foreach j P N . Choosing j ą i there exists a site h P t y j h, z j y j h u for which ω | R i ` h sees bold-face digits in the same locations as ω | R i ` h . Moreover staying within thesame x y , z y -coset guarantees ω | p R i ` zx z yq h “ ω | p R i ` zx z yq h (four coinciding completecounter arrays). Recall that we assumed the support F of our initial pattern p to be contained within the stack Ť k l “ Q k x l . Hence the choices of j ą i ą k and h ensure a counter suffix (i.e. segment of least significant digits) of no less than2 i of the 2 j ` binary digits to lie outside of p R i ` X x z yq h . This yields enoughroom to accommodate whatever behaviour – either staying constant or showing asingle increase instantly triggering a total overflow of all finite width counters asdescribed by Lemmas 5.4 and 6.2 – the non-deterministic infinite width counterrunning in ω | x x , z y h might show inside Ť k l “ Q k x l . Climbing up through x y , z y -layersby multiples of 2 ¨ i keeps all values of counters with widths at most 2 i unchanged,so that ω | p R i ` zx z yq h x n “ ω | p R i ` zx z yq h for all n P i ` Z . Each such successive jumphowever increases the value of the width 2 j ` counters stored in z -rows x z y h x n by 2 i modulo 2 j ` , thus producing in one periodic cycle all binary values onthe counters’ 2 j ` ´ i most significant bits. Choosing the correct value of n wetherefore reencounter a shifted version of the base pattern ω | R i ` h x n “ ω | R i ` h . Asthis new copy is located entirely in the complement of ω ’s exceptional x x , z y -coset,we are back in case 1, concluding our minimality argument just as before. (cid:3) Remark . We point out that the factor map φ : r Ω ։ φ p r Ω q constructed inProposition 6.4 is again almost invertible, producing unique preimages φ ´ p ω q onthe dense subset of configurations ω P φ p r Ω q not containing an exceptional coset.Here we finally touch upon the second, maybe less obvious reason for the non-minimality of Ω and thus also r Ω. In fact, given any configuration ω P φ p r Ω q , thepropagation of crosses constraint in itself is only strong enough to uniquely re-construct all Robinson tiles located inside some finite-level supertile. (Similarlycounter values determine the eliminated r A Seg -tiles.) Nevertheless there is someambiguity in recovering Robinson symbols, namely arm tiles pointing in the y -direction, seen outside the union of all finite-level supertiles. To obtain minimalitythose would have to be restricted further, mimicking the implicit (non-local) rulesforced on their behaviour inside finite-level supertiles by the propagation of crossesconstraint. Consequently it seems rather difficult to transform r Ω itself into a min-imal H p Z q -SFT by adding more symbol decorations and further local rules.Hence the existence of an explicit construction of a strongly aperiodic, minimal H p Z q -SFT remains an open problem. It seems worth pointing out that none ofthe commonly used strongly aperiodic Z -SFTs (Robinson, Kari-Culik etc.) areminimal and that despite the known existence of such subshifts it is rather difficultto track down literature with an explicit construction, even for Z . References [1] Nathalie Aubrun and Jarkko Kari,
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Department of Mathematics, Wright State University, Dayton, OH 45435, USA
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