| { |
| "title": "Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups", |
| "abstract": "We study the periodicity of subshifts of finite type (SFT) on Baumslag-Solitar groups.\nWe show that for residually finite Baumslag-Solitar groups there exist both strongly and weakly-but-not-strongly aperiodic SFTs. In particular, this shows that unlike , but like , strong and weak aperiodic SFTs are different classes of SFTs in residually finite BS groups. More precisely, we prove that a weakly aperiodic SFT on BS(m,n) due to Aubrun and Kari is, in fact, strongly aperiodic on BS(1,n); and weakly but not strongly aperiodic on any other BS(m,n).\nIn addition, we exhibit an SFT which is weakly but not strongly aperiodic on BS(1,n); and we show that there exists a strongly aperiodic SFT on BS(n,n).", |
| "sections": [ |
| { |
| "section_id": "1", |
| "parent_section_id": null, |
| "section_name": "Introduction", |
| "text": "The use of tilings as a computational model was initiated by Wang in the 60s [20 ###reference_b20###] as a tool to study specific classes of logical formulas.\nHis model consists in square tiles with colors on each side, that can be placed next to each other if the colors match.\nWang studied the tilings of the discrete infinite plane () with these tiles, now called Wang tiles; a similar model exists to tile the infinite line () with two-sided dominoes.\nWang realized that a key property of these tilings was the notion of periodicity.\nAt the time, it was already known that if a set of dominoes tiled , it was always possible to do it in a periodic fashion, and Wang suspected that it was the same for Wang tiles on .\nHowever, a few years later, one of his students, Berger, proved otherwise by providing a set of Wang tiles that tiled the plane but only aperiodically [5 ###reference_b5###]. Numerous aperiodic sets of Wang tiles have been provided by many since then (see for example [18 ###reference_b18###, 14 ###reference_b14###, 13 ###reference_b13###]).\nThe model of Wang tiles is actually equivalent to tilings using an arbitrary finite alphabet with adjacency rules, and it has become a part of symbolic dynamics, a more general way to encode a smooth dynamical system into symbolic states and trajectories.\nThis approach by discretization (see [10 ###reference_b10###] for a comprehensive historiography) is itself part of the field of discrete dynamical systems.\nIn this broader context, it is interesting to study the set of all possible tilings for a given finite list of symbols and adjacency rules, called a Subshift of Finite Type (SFT).\n-SFTs have been studied extensively, and many of their properties are known (see for example [15 ###reference_b15###]).\n-SFTs (for ) seem to be much more complex, as many results from the unidimensional case cannot be directly transcribed to higher dimensions.\nIn recent years, the even wider class of SFTs built over Cayley graphs of groups has been attracting more and more attention [6 ###reference_b6###, 12 ###reference_b12###, 1 ###reference_b1###].\nIn this broader setting, many questions about periodicity are still open [6 ###reference_b6###].\nFor example there are many groups for which we do not know if there exists a non-empty SFT with only aperiodic configurations (called an aperiodic SFT).\nAnother relevant question is the relation between two notions of aperiodicity, weak and strong aperiodicity: the first one requires all configurations to have infinitely many distinct translates; the second one that no configuration has any period whatsoever.\nThey are equivalent for SFTs over or , but this is not the case for other groups (notably ), and for many the relation is not even known.\nBaumslag-Solitar groups of parameters and , commonly denoted by , initially gathered interest in symbolic dynamics because of their simple description and rich properties.\nNotably, their domino problem is undecidable [3 ###reference_b3###]; and Aubrun and Kari built a weakly aperiodic SFT in order to prove it.\nTheir proof is essentially sketched in the general case, where it encodes piecewise affine maps, as it is done on in [14 ###reference_b14###].\nOnly the case is detailed in their paper; and an explicit period is provided for a given configuration, which shows that the resulting SFT, although weakly aperiodic, is not strongly aperiodic.\nIn the present paper, after a few definitions (Section 1 ###reference_###);\nwe thoroughly reintroduce Aubrun and Kari’s construction in the general case,\nand provide a precise proof of its weak aperiodicity by encoding piecewise linear maps (Section 2 ###reference_###) (although Aubrun and Kari sketched the proof for encoding piecewise affine maps, only piecwise linear ones are needed for the aperiodicity result).\nThen, we show that the resulting SFT is weakly but not strongly aperiodic on any (Section 2.2 ###reference_###) (as sketched in [3 ###reference_b3###]); except on where, with some extra work, we prove that it is strongly aperiodic (Section 2.4 ###reference_###).\nThen, using a different technique based on substitutions on words, we exhibit a weakly but not strongly aperiodic SFT on (Section 3 ###reference_###).\nFinally, by using tools from group theory and a theorem by Jeandel [12 ###reference_b12###], we build a strongly aperiodic SFT on any (Section 4 ###reference_###).\nThese new results are summarized in the following table (in bold):\nThe end result is that any residually finite Baumslag-Solitar group with or has a strongly aperiodic SFT and a weakly but not strongly aperiodic SFT. The remaining question is whether a general admits a strongly aperiodic SFT, when not in the , or cases." |
| }, |
| { |
| "section_id": "2", |
| "parent_section_id": null, |
| "section_name": "On a construction by Aubrun and Kari", |
| "text": "In [3 ###reference_b3###], Aubrun and Kari provide a weakly aperiodic Wang tileset on Baumslag-Solitar groups, with a proof focusing on the specific case of , for which they also present a period for one specific configuration, implying that the corresponding Wang tileset is not strongly aperiodic.\nA more general version of the construction can be found in [4 ###reference_b4###]. We repeat most of it here for the sake of completeness, since we study that construction in more details to obtain additional results: we will show that it yields a weakly but not strongly aperiodic SFT on any with and , and a strongly aperiodic SFT on any ." |
| }, |
| { |
| "section_id": "2.1", |
| "parent_section_id": "2", |
| "section_name": "Aubrun and Kari’s construction", |
| "text": "Aubrun and Kari’s construction works by encoding orbits of piecewise affine maps applied to real numbers.\nWe will only apply their construction for piecewise linear maps, and begin this section with the necessary definitions." |
| }, |
| { |
| "section_id": "2.1.1", |
| "parent_section_id": "2.1", |
| "section_name": "2.1.1 Definitions", |
| "text": "Let .\nWe say that a binary biinfinite sequence represents a real number if there exists a growing sequence of intervals of size at least such that:\nNote that if is a representation of , all the shifted sequences for every are also representations of . Note that a sequence can a priori represent several distinct real numbers since different choices of interval sequences may make it converge to different points. A sequence always represents at least one real number by compactness of .\nThen, we define a generalization of piecewise linear maps: multiplicative systems.\nThe main difference with piecewise linear maps is that points may have several images as definition intervals of different pieces might overlap.\nA multiplicative system is a finite set of non-zero linear maps\nwith and closed intervals of .\nIts inverse is defined to be\nThe image of is the set\nThe -th iteration of on is then:\nNote that if none of the intervals overlap, can be represented as a piecewise linear function and the definition of inverse and iteration coincide with the usual ones.\nLet be a multiplicative system.\nThe real is immortal if for all ,\nA periodic point for this system is a point such that there exists such that\nThe level of is the set .\nWhen considering a tiling of ,\ngiven a line of tiles located between levels and , we talk about the upper side of the line to refer to level , and the lower side of the line to refer to level .\nThe height of is, for any way of writing it as a word in , its number of ’s minus its number of ’s; it is denoted as .\nSince the only basic relation in uses one and one , all writings of as a word give the same height. Furthermore, it is actually the height of all elements in its level.\nA set of tiles multiplies by if for any tile (see Fig. 1 ###reference_### for the notation),\nLet be a tileset multiplying by .\nIf we consider a line of tiles of next to each other without tiling errors (as defined in Section 1.2 ###reference_###), as left and right colors match, we can average Eq. 1 ###reference_###:\nwhere is the average of the top labels of the line and the average of the bottom ones.\nTherefore, if an infinite line has its upper side representing and its lower side representing , taking the limit of Eq. 2 ###reference_### on a well chosen sequence of intervals gives:\nHence the name of multiplying tileset for ." |
| }, |
| { |
| "section_id": "2.1.2", |
| "parent_section_id": "2.1", |
| "section_name": "2.1.2 A multiplying tileset", |
| "text": "Let us define, in a fashion similar to [3 ###reference_b3###], a couple of useful functions to build a multiplying tileset.\nLet (or just when and are clear) be defined by the recursion:\nThe map can be extended to elements of , due to the fact that for any pair of words and in : is then for any word representing in the group.\nNow, we define as follows:\nThe function can be seen as a projection of every element of on the euclidean plane .\nFinally, let be defined as\nLet and an interval, let\nThen, we define the tileset as:\n###figure_1### One can show that Eq. 1 ###reference_### holds for these tiles.\nLet , for any , is a tileset that multiplies by .\nLet us define the balanced representation of , which is the biinfinite sequence defined for any by\nNote that does not depends on , and can only take two values: or .\nLet and . The upper side of any tile in is of the form\nfor . In particular, its labels are in .\nThe lower side is of the form\nRewriting the top labels using balanced representation yields\nSince each is either or , and , one obtains labels in .\nFor the bottom side, note that , which gives labels\n∎\nFor our purpose we need to have a finite tileset, because subshifts use a finite alphabet.\nLet , for any the tileset is finite.\nProposition 2.2 ###reference_theorem2### gives us that there are finitely many top and bottom labels. It remains to prove that there are also finitely many left and right labels.\nFirst of all, one can check that , and so . Consequently, we simply have to prove that lies in a finite set.\nLet with , and write\nSince its numerator is an integer bounded by from below and from above using usual inequalities on the floor function, we have that for any , is in the finite set\n∎\nThanks to the multiplying property of , we can use it to encode multiplicative systems in such a way that non-empty tilings corresponds to immortal points of the system.\nIf we modify the tileset a little bit, we can encode multiplying systems defined on other intervals than\nLet be a multiplicative system with\nand interval with rational bounds included in some .\nWe can explicitly and algorithmically build an SFT with the following properties:\nany top of a line of tiles in a configuration represents at least one real .\nif the top of a line of tiles represents a real , then the bottom of that line represents a real in ;\nif and only if has an immortal point;\nWe build a tileset performing the computation by the linear functions ; i.e. the linear maps with bigger intervals than the ones defining .\nIn order to encode the multiplication correctly, one cannot simply take the union of all , because tiles coming from different could be mixed on a single line.\nIn order to ”synchronize” the computations on every line, we create a product alphabet with the left and right colors of the tiles, and the number of the current function being used. This ensures that one line can have tiles from only one of the .\nFormally,\nThis way, we can interpret any line of a tiling by as being ”of color” for some .\nNext, we restrict the intervals of reals that can be represented in each line of the SFT.\nLet us write , for each .\n will be with the following additional local constraints:\non each line of color , we force the upper side labels to respect\nRecall that the top labels of any line of color only has labels and by definition of .\nFrom this, we can also deduce\nSince these constraints are on or consecutive labels, and any other constraint from the ’s is on neighboring tiles, is an SFT.\nFirst, assume that is non-empty and contains some configuration .\nThen, the sequence at the top of each level represents at least one real , since it is a sequence made of at most two integers.\nThanks to the multiplying property of each (Proposition 2.1 ###reference_theorem1###), and the fact that the local rules of every line impose that any real represented belongs to an interval , is an infinite orbit of .\nIndeed, consider a real represented by the upper side of a line of color . We prove that (we drop the in this paragraph to make notations lighter).\nLet us write the intervals from Definition 2.1 ###reference_definition1### (denoted as there) on which we compute the mean for the represented real, and let denote the proportion of ’s over ’s in the line representing restricted to .\nThanks to the two conditions Eq. 3 ###reference_### and Eq. 4 ###reference_### (and the two we deduced), we have, for all in ,\nMoreover, since is the limit of the means computed on each , one can show that\nUsing the left inequality of Eq. 5 ###reference_### gives that\nSo\nSimilarly, the right inequality of Eq. 5 ###reference_### gives\nand so .\nThis notably ensures Item 1 ###reference_i1### and, by the use of the ’s, Item 2 ###reference_i2###.\nNow, assume that has an immortal point .\nThen there exists a sequence such that if we define\nthen for all .\nFor every , we place at a tile from the tileset with , with colors:\nThese tiles are obviously from the tileset . Recall that we have\nAnd therefore,\nand for any ,\nIt remains to show that the labels at the top of every line follow the conditions (3 ###reference_###) and (4 ###reference_###).\nFix some and consider the level . Fix , denote , and drop the in the other variables to simplify notations.\nFor , we write for the label at position of the considered level, with being position .\nRemark that\nholds for all thanks to Eq. 6 ###reference_###.\nAssume that .\nLet us write the number of labels that appear in the word .\nIf we sum all the labels of , we have on the one hand\nAnd on the other hand,\nTherefore,\nwhich can be rearranged to\nwhich implies (3 ###reference_###).\nIn the same way, if we assume , we can show , which is exactly (4 ###reference_###).\nThis shows Item 3 ###reference_i3###.\n∎\nInstead of considering a multiplicative system with rational bounds for the intervals, one can dilate them and encode a dilated system with integer bounds, which is equivalent to the original. For instance, instead of using a linear function on , one can consider that function multiplied by , on which has integer bounds.\nHowever, although this would yield a shorter proof, the results would only hold through some sort of conjugation, and we wanted to effectively build a Wang tileset for any multiplicative system.\nNote that we only build a multiplying tileset in the current paper.\nAubrun and Kari provided the details of the encoding of any finite set of affine maps into a tileset of in [4 ###reference_b4###]." |
| }, |
| { |
| "section_id": "2.2", |
| "parent_section_id": "2", |
| "section_name": "A weakly aperiodic SFT on", |
| "text": "The SFT previously defined on a given is also linked to the periodicity of .\nIf has no periodic point, then is a weakly aperiodic SFT.\nWe prove the contrapositive.\nAssume that has a strongly periodic configuration , i.e. with .\nIn particular, each set is finite of cardinality lesser than , and therefore for every , there exists such that .\nIf we define , we obtain that for all , .\nLet . Let , i.e. there exists some so that . Then\nwhich means that the level is -periodic.\nTherefore any level is -periodic, and consequently represents a unique rational number . Since the alphabet of is finite, there are only finitely many different such rationals. Consequently, there exist two levels and with that represent the same rational number .\nBy the multiplicative property of ,\nwhich means that is a periodic point for .\n∎\nThe consequence of this theorem is that, to obtain a weakly aperiodic SFT on , we only need to explicitly build a multiplicative system with an immortal point but no periodic points.\nFor the rest of this section, let\nand\nIt is easy to see that this system has immortal points (in fact, all points of are immortal).\nhas immortal points.\nAdditionally, since and are relatively prime:\nhas no periodic points.\nis non-empty and weakly aperiodic.\nHowever, this construction does not avoid weakly periodic configurations when and are not 1, as already remarked by Aubrun and Kari.\nFor any , on contains a weakly periodic tiling, with period .\nTo prove it, we need the following lemma:\nLet .\nFor any , .\nSince , using the definition of it is easy to show that by recurrence on the length of .\nThen, .\n∎\nWe happen to have built a periodic configuration already: the one from the proof of Theorem 2.4 ###reference_theorem4###.\nIndeed, let be any real in , since they are all immortal for . Then there exists a sequence such that if we define\nthen for all .\nFor every , we place at a tile from the tileset with , with colors:\nWe already checked that the resulting tiling was in , see the proof of Theorem 2.4 ###reference_theorem4###.\nIt remains to show that the tiles at and at are the same for all , to conclude that , or equivalently that is a period of .\nThis is actually surprisingly easily, considering that for any , so they use the same integer and the same real ; and , see Lemma 2.11 ###reference_theorem11###. The tiles have the same labels as a consequence of this.\nThe consequently defined is -periodic. The only thing left to show is that is a nontrivial element of as long as .\nSince it is freely reduced and does not contain or as subwords, and since is a HNN extension with , we can apply Britton’s Lemma: cannot be the neutral element.\nTherefore contains a configuration with a nontrivial period, and consequently is not strongly aperiodic.\n∎" |
| }, |
| { |
| "section_id": "2.3", |
| "parent_section_id": "2", |
| "section_name": "A deeper understanding of the configurations", |
| "text": "We first present additional results on this tileset of . Most of the ideas present in this section were already present in [11 ###reference_b11###] in the context of tilings of the plane.\nFor a given line , we define the sequence to be the sequence of digits on the line (its origin depending on ).\nLet be the following bijective continuous map:\nis strongly related to because for any , for any ,\ndue to the fact that for our particular .\nAn easy consequence is the following:\nLet . Let .\nLet be a real represented by ; then represents (which means either or if ).\nrepresents at least one such because of Item 1 ###reference_i1### of Theorem 2.4 ###reference_theorem4###. The rest is due to Item 2 ###reference_i2### of Theorem 2.4 ###reference_theorem4### and Eq. 7 ###reference_###.\n∎\nIt turns out that with our choice of multiplicative system , any line can represent only one real number. We need several lemmas to prove this; all inspired by [11 ###reference_b11###].\nLet be defined as follows:\nis a correctly defined mapping that conjugates the dynamical systems and , where and are the usual topologies on the considered sets.\nSince the action considered on is , one only needs to check that is bijective and continuous to conclude that it yields a conjugation.\nis clearly continuous everywhere except on ; there, one can check that the left and right limits both lead to which is correctly defined.\nis defined by except with collapsed images for and .\n∎\nThe map can be considered a rotation of irrational angle when identifying and .\nFor every ,\nSimilarly, for every , one has\nFinally, .\n∎\nWe now have the tools to prove the following key lemma:\n(Uniqueness of representation)\nFor any , for any , the sequence represents a unique real number.\nAssume that represents two distinct reals and . They cannot be and because uses only digits in or in . Therefore they also are distinct reals in .\nFor any , notice that and same for , from Lemma 2.13 ###reference_theorem13###.\nWe will study the behavior of and under iterations of .\nThe angle , of which is a rotation by Lemma 2.14 ###reference_theorem14###, is irrational. As a consequence, the sets and are both dense in .\nWe introduce for , where is the only real in congruent to mod .\nWe call the oriented arc distance (measured counterclockwise) between two elements of . It is not a distance per se since it is not symmetric and has no triangular inequality, but its basic properties will suffice here.\nSince is a rotation, it is easy to check that it preserves . Hence we have that is constant equal to some . Up to considering instead, and doing the following reasoning by swapping and , we can assume that .\nLet us split between , , and .\nWe want to show that there is some for which and .\nBy density of , there exists some such that and . We cannot have without contradicting the previous inequality, hence it is either in or in . But if it was in , then the arc from to would contain all of . This is not possible because .\nHence there exists such that and .\n###figure_2### Since , and considering the definitions of and , and . This would cause to be represented by a sequence of ’s and ’s (with an infinite number of ’s) and by a sequence of ’s and ’s.\nHowever, the SFT is built such that a line contains only elements in or , but not both (see proof of Theorem 2.4 ###reference_theorem4###): this is a contradiction.\nTherefore, and must be equal, hence the uniqueness of the real number represented by a given sequence.\n∎\nUsing previous results, we are now able to prove for that the real represented by the sequence only depends on , its “depth” in the Cayley graph.\nLet , and the identity of .\nLet be the unique real represented by the sequence .\nThen for every , represents (with a choice between and , possibly different for different ’s, if the resulting value is ).\nWe prove the result by reasoning on words , by induction on their length. Note that we have no need of proving that different ’s representing the same yield the same result, since this is guaranteed by Lemma 2.15 ###reference_theorem15###.\nThe result is true for .\nSuppose the result is true for words of length . Let be a word of length . Then:\nand represent the same real as since they are the same sequence up to an index shift;\nrepresents due to Lemma 2.12 ###reference_theorem12### and the induction hypothesis, which is ;\nsuppose represents ; then represents due to Lemma 2.12 ###reference_theorem12###. Then we have, by induction, .\n∎\nThe previous proof heavily relies on the fact that is a bijection on , and that we do not have to differentiate between and there." |
| }, |
| { |
| "section_id": "2.4", |
| "parent_section_id": "2", |
| "section_name": "A strongly aperiodic SFT on", |
| "text": "If or is equal to 1, then the previous weak period of Proposition 2.10 ###reference_theorem10### does not work anymore – it is a trivial element.\nIn fact, we prove in this section that for , is strongly aperiodic.\nOne key property of is that there is a simple quasi-normal form for all its elements.\n(Quasi-normal form in )\n\nFor every , there are integers and such that .\nFrom the definition of , we have that (1), (2), (3) and (4). Consequently, taking an element of as a word written with and , we can:\nMove each positive power of to the right of the word using (1) and (2) repeatedly;\nMove each negative power of to the left of the word using (3) and (4) repeatedly;\nso that we finally get a form for the word which is: with and .\n∎\nA general normal form – the same, with imposed to be minimal – can be obtained from Britton’s Lemma. The form obtained here is not unique ( for instance), but we use it because it admits a simple self-contained proof, and it is enough for what follows: the sum is constant for all writings of a given group element, hence we name it “quasi-normal”.\nIndeed, suppose we have . Then\nHence we get .\nSince it is clear that if and only if in , we obtain which is what we wanted.\n∎\nThis quasi-normal form is the only thing that missed to prove the following.\nFor every , the Baumslag-Solitar group admits a strongly aperiodic SFT.\nLet , and .\nUsing Lemma 2.18 ###reference_theorem18###, we can write with .\nLet be the real represented by .\nBy Lemma 2.16 ###reference_theorem16###, represents . Since , and so by the uniqueness of the representation from Lemma 2.15 ###reference_theorem15###. The aperiodicity of then implies that .\nLet us assume .\nThen and .\nWe can reduce to using the relation .\nMore generally, we notice that for any positive integer , iterating the process times, we obtain that .\nSince for all , , we can obtain a contradiction with an argument similar to Prop 6. of [3 ###reference_b3###].\nWe have for any . This means that hence is a -periodic sequence. We have a finite number of said sequences, since they can only use digits among .\nConsequently, there are such that the two levels and read the same sequence (up to index translation).\nThese two levels represent respectively and due to Lemma 2.16 ###reference_theorem16###, and since the two sequences on these levels are the same, .\nThis equality contradicts the fact that has no periodic point, since we had .\nAs a consequence, any non-trivial cannot be in , and we finally get that : is strongly aperiodic.\n∎\nFollowing Theorem 2.20 ###reference_theorem20###, a question remains: is the strong aperiodicity of Aubrun and Kari’s SFT a property of the group itself, or does it only arise on carefully chosen SFTs, as ? Is this because behaves like and all its weakly aperiodic SFTs are also strongly aperiodic, or does Aubrun and Kari’s construction happen to be “too much aperiodic”?\nIt turns out that the latter is the correct answer, as we build in the following section an SFT on that is weakly but not strongly aperiodic." |
| }, |
| { |
| "section_id": "3", |
| "parent_section_id": null, |
| "section_name": "A weakly but not strongly aperiodic SFT on", |
| "text": "Our weakly but not strongly aperiodic SFT will work by encoding specific substitutions into .\nIndeed, the Cayley graph of is very similar to orbit graphs of uniform substitutions (see for example [9 ###reference_b9###, 2 ###reference_b2###] for a definition of orbit graphs and another example of a Cayley graph similar to an orbit graph).\nIn this section, we find a set of substitutions that are easy to encode in (Section 3.1 ###reference_###), and show how to do it (Section 3.2 ###reference_###)." |
| }, |
| { |
| "section_id": "3.1", |
| "parent_section_id": "3", |
| "section_name": "The substitutions", |
| "text": "Let . For , let be the following substitution:\nWe may also write and call the other ones the shifts of .\nNote that, for and , if and only if and (starting to count from the indices of the word ).\nAll are cyclic permutations of the same finite word. Denote the shift action on a biinfinite word , i.e. , as a way to write the action of on .\nFor any biinfinite word , any and ,\nFor , depends on the letter of at position only, that is (See Fig. 6 ###reference_###), hence\nSimilarly, the letter does not depend on the totality of but only on : it is the th letter of .\n∎\n###figure_3### For any ,\nLet . Let and .\nConsidering that if , , we conclude that we always have , and so .\n∎\nFor , has a unique fixpoint.\nFor , has no fixpoint but has two fixpoints.\nProposition 4 from [19 ###reference_b19###] characterizes biinfinite fixpoints of substitutions. In the present case of , [19 ###reference_b19###] states that if and only if with and with and , , .\nNotice that and , for , so the only choice for and is .\nThen has a fixpoint that is and which is unique.\nFor the same reasoning concludes that has no fixpoint. However, since and , the same reasoning also yields that has two fixpoints that are and .\n∎\nFor every and every , the fixpoints of are aperiodic.\nTo prove the aperiodicity of a fixpoint of (in the case where such a fixpoint exists), we follow a proof from [17 ###reference_b17###], simplified for our specific case.\nFirst, let us show that the two subwords and can be found in .\nFor ,\nlet us define .\nThen, by definition, (by convention if ).\nWe are going to prove that always contains a . As a consequence, contains because .\nSuppose .\nIf , it means that , but then so this is impossible.\nIf , then so the only way to have is to have , but again .\nIf , let us define . With this notation, . The assumption causes .\nHowever, this is impossible since has no antecedent by . Therefore must contain a and we can find in .\nFor , the only way for not to contain is to be of the form , or .\nBut it is clear that , and hence none of them can be fixpoints.\nHence and can also be found in since .\nFrom this, we build by induction infinitely many words with two possible right extensions. We have ; consider the largest prefix on which they agree, call it , with . Then both and can be found in . Hence and can also be found in .\nWe have ; consider the largest prefix on which they agree, call it , with . Then both and can be found in . Hence and can also be found in .\nBy induction, we can build subwords of as large as we want that have two choices for their last letter. Hence the factor complexity of is unbounded, and so is aperiodic (see Section 1.3 ###reference_###).\n∎" |
| }, |
| { |
| "section_id": "3.2", |
| "parent_section_id": "3", |
| "section_name": "Encoding substitutions in", |
| "text": "We now show how to encode such substitutions in SFTs of the group given by a tileset.\nWe define the tileset on , to be the set of tiles shown on Fig. 7 ###reference_### for all and . Remark that a tile is uniquely defined by the couple .\n###figure_4### This tileset will be the weakly but not strongly aperiodic tileset we are looking for.\nLemmas 3.3 ###reference_theorem3### and 3.4 ###reference_theorem4### study the words that can appear on levels of the tiling, by looking at the fixpoints of .\nThey prove that no biinfinite word can be both a fixpoint for the ’s and a periodic word, forbidding one direction of periodicity for any configuration we will encode with our tileset.\nThis naturally leads to the following proposition:\nNo configuration of can be -periodic for any .\nSuppose that there is a configuration of such that for any , (-periodicity).\nCall the biinfinite word based on level . is -periodic by -periodicity of the configuration . But is also -periodic. Hence is -periodic. Indeed, by construction, when applying the correct substitution to and , one obtains the words and which are one and the same by -periodicity of . Since there is only one preimage for a word by , .\nBy the same argument, one can show that for any integer , must be -periodic. However, these biinfinite sequences only use digits among so there is a finite number of such sequences. In particular, two of these sequences are the same. Since one is obtained from the other by applying the correct succession of ’s, we get a periodic sequence that is a fixpoint of some for some . This contradicts Lemma 3.4 ###reference_theorem4###.\n∎\nThere exists a weakly periodic configuration in for .\nWe define the unique fixpoint of obtained thanks to Lemma 3.3 ###reference_theorem3###.\nLet be the function that maps to the quotient in the euclidean division of by and its remainder.\nWe also define and . This means that , , but also , and consequently ." |
| }, |
| { |
| "section_id": "3.2.x", |
| "parent_section_id": "3.2", |
| "section_name": "is nonempty", |
| "text": "We define a configuration describing which tile (a tile being uniquely defined by such a couple) is assigned to , i.e. , using the quasi-normal form .\nThen, we check that does verify the adjacency rules.\nDefine by\nRemember that Lemma 2.18 ###reference_theorem18### states that any can be written . Suppose it has a second form with up to exchanging the notations (were they equal, it is easy to prove the two forms would be the same).\nThen , that is, .\nThis means that and .\nWith that, we prove our is well-defined. causes in order to have . Consequently,\nwith a variation on the second to last line if : we have .\nNow, we prove that . Let .\nIf , we have\nIf , we have\nLet . We have\nConsequently, describes a valid configuration of : all adjacency conditions are verified." |
| }, |
| { |
| "section_id": "3.2.x", |
| "parent_section_id": "3.2", |
| "section_name": "is -periodic", |
| "text": "With the definition of , it is easy to check that for any , . Hence it is a weakly periodic configuration.\n∎\nWe can now obtain our second main theorem:\nThe tileset forms a weakly aperiodic but not strongly aperiodic SFT on .\nFirst, in the case, there is a weakly periodic configuration in , see Lemma 3.6 ###reference_theorem6###. Hence it is not a strongly aperiodic SFT.\nIn the case, we define and the two fixpoints of (Lemma 3.3 ###reference_theorem3### again) and remark that and . We define a configuration by:\nand we use the same notations as in the proof of Lemma 3.6 ###reference_theorem6###. The reasoning is also the same, except instead of using an alternation appears between and in all the equations. As a consequence, the configuration is -periodic instead of . Once again, is consequently not strongly aperiodic.\nNow, using Proposition 3.5 ###reference_theorem5###, and since all powers of \nare of infinite order in , we get that for any valid configuration of , , for any . Hence no configuration of is strongly periodic, and so the SFT is weakly aperiodic.\n∎" |
| }, |
| { |
| "section_id": "4", |
| "parent_section_id": null, |
| "section_name": "A strongly aperiodic SFT on", |
| "text": "This section is a mere assembly of known results, that we think are worth gathering in the context of the current paper.\nIt uses a theorem from [12 ###reference_b12###] seen as an extension of the construction presented in [14 ###reference_b14###]. The idea behind that theorem is that admits a strongly aperiodic SFT as soon as can encode piecewise affine functions. This is reflected by the condition described in [12 ###reference_b12###] and restated below.\nLet . Let be a finite set of piecewise affine rational homeomorphisms, where each and is a finite union of bounded rational polytopes of . Let be the common domain of all functions of and their inverses.\nLet be the closure of the set under composition. We define , the group .\nA finitely generated group is -recognizable if there exists a finite set of piecewise affine rational homeomorphisms such that:\n(A) ;\n(B) .\nIf is an infinite finitely generated -recognizable group, then admits a strongly aperiodic SFT.\nWe need two additional propositions to obtain the desired result on :\nIf is a finitely generated group and is a finitely generated subgroup of of finite index, then we have the following:\nadmits a weakly aperiodic SFT admits a weakly aperiodic SFT\nadmits a strongly aperiodic SFT admits a strongly aperiodic SFT.\nThe following proposition is known, but we include a self-contained proof.\nadmits as a subgroup of finite index, where is the free group of order .\nLet be the subgroup of generated by .\nFirst, is normal in . We prove that by verifying it on its generators: the only verification needed is . Similarly, ; and finally, (same for ) since .\nSecond, is isomorphic to through the following isomorphism (denoting the generators of and its identity):\nIt is a morphism by construction, which is correctly defined since the only basic relation of , that is , is preserved in : .\nSaid morphism is surjective, because is generated by and .\nFinally, it is also injective: let , with where the are in .\nThis form is a canonical form in : any word in can be uniquely written as such. Indeed, any word in is a succession of generators of it, and . But commutes with all the other generators due to the relation of , so such a form is always attainable. To prove it is unique, it is enough to prove it for : suppose we have some . First, realize that no relation in allows to reduce the total power of in a word, causing necessarily. Then, consider the resulting word in : it cannot be reduced in since all powers of between two ’s are of absolute value smaller than .\nAs a consequence, the previous equality is true only when and . Hence the injectivity of the map.\nMoreover, any element of can be written in a form that much resembles the one mentioned above:\nwith . To do so, first move all ’s in the rightmost power of in the word, to the leftmost part of the word. Ensure that , the remaining power, is in . Then force to appear on the left of the itself to the left of , and call the remaining power of (it is in up to moving another to the leftmost part of the word) before another to the left. Repeat this operation until there is no to the left of the power of you consider, and split this final into .\nAs a consequence, .\nHence is of finite index in .\n∎\nFor every , admits a strongly aperiodic SFT.\nFirst, finitely generated subgroups of compact groups of matrices on integers are -recognizable (see [12 ###reference_b12###], Proposition 5.12). , the free group of order , is isomorphic to a subgroup of (see [8 ###reference_b8###, Lemma 2.3.2]), hence it is -recognizable.\nIt is also known (see [8 ###reference_b8###, Corollary D.5.3]) that is a subgroup of ; so it is isomorphic to a subgroup of and -recognizable too.\nTherefore by Theorem 4.1 ###reference_theorem1### admits a strongly aperiodic SFT.\nUsing Proposition 4.2 ###reference_theorem2### and Proposition 4.3 ###reference_theorem3###, we obtain that admits a strongly aperiodic SFT.\n∎" |
| } |
| ], |
| "appendix": [], |
| "tables": { |
| "1": { |
| "table_html": "<figure class=\"ltx_table\" id=\"Sx1.4\">\n<div class=\"ltx_inline-block ltx_align_center ltx_transformed_outer\" id=\"Sx1.4.4.4\" style=\"width:433.6pt;height:61.2pt;vertical-align:-0.7pt;\"><span class=\"ltx_transformed_inner\" style=\"transform:translate(-105.6pt,14.7pt) scale(0.672370698383973,0.672370698383973) ;\">\n<table class=\"ltx_tabular ltx_guessed_headers ltx_align_middle\" id=\"Sx1.4.4.4.4\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"Sx1.4.4.4.4.5.1\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_l ltx_border_r ltx_border_t\" id=\"Sx1.4.4.4.4.5.1.1\" style=\"padding:3pt 2.0pt;\">Group</th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t\" id=\"Sx1.4.4.4.4.5.1.2\" style=\"padding:3pt 2.0pt;\">Strongly aperiodic SFT</th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t\" id=\"Sx1.4.4.4.4.5.1.3\" style=\"padding:3pt 2.0pt;\">Weakly-not-strongly aperiodic SFT</th>\n</tr>\n<tr class=\"ltx_tr\" id=\"Sx1.1.1.1.1.1\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_l ltx_border_r ltx_border_t\" id=\"Sx1.1.1.1.1.1.1\" style=\"padding:3pt 2.0pt;\"></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t\" id=\"Sx1.1.1.1.1.1.2\" style=\"padding:3pt 2.0pt;\">Yes (Berger <cite class=\"ltx_cite ltx_citemacro_cite\">[<a class=\"ltx_ref\" href=\"https://arxiv.org/html/2004.02534v4#bib.bib5\" title=\"\">5</a>]</cite>)</th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t\" id=\"Sx1.1.1.1.1.1.3\" style=\"padding:3pt 2.0pt;\">No (Folklore)</th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"Sx1.2.2.2.2.2\">\n<td class=\"ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t\" id=\"Sx1.2.2.2.2.2.1\" style=\"padding:3pt 2.0pt;\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"Sx1.2.2.2.2.2.2\" style=\"padding:3pt 2.0pt;\">\n<span class=\"ltx_text ltx_font_bold\" id=\"Sx1.2.2.2.2.2.2.1\">Yes, adapted from Aubrun-Kari</span> (<a class=\"ltx_ref\" href=\"https://arxiv.org/html/2004.02534v4#S2.SS4\" title=\"2.4 A strongly aperiodic SFT on 𝐵𝑆(1,𝑛) ‣ 2 On a construction by Aubrun and Kari ‣ Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups\"><span class=\"ltx_text ltx_ref_tag\">Section</span> <span class=\"ltx_text ltx_ref_tag\">2.4</span></a>)</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"Sx1.2.2.2.2.2.3\" style=\"padding:3pt 2.0pt;\">\n<span class=\"ltx_text ltx_font_bold\" id=\"Sx1.2.2.2.2.2.3.1\">Yes, using substitutions</span> (<a class=\"ltx_ref\" href=\"https://arxiv.org/html/2004.02534v4#S3\" title=\"3 A weakly but not strongly aperiodic SFT on 𝐵𝑆(1,𝑛) ‣ Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups\"><span class=\"ltx_text ltx_ref_tag\">Section</span> <span class=\"ltx_text ltx_ref_tag\">3</span></a>)</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"Sx1.3.3.3.3.3\">\n<td class=\"ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t\" id=\"Sx1.3.3.3.3.3.1\" style=\"padding:3pt 2.0pt;\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"Sx1.3.3.3.3.3.2\" style=\"padding:3pt 2.0pt;\">\n<span class=\"ltx_text ltx_font_bold\" id=\"Sx1.3.3.3.3.3.2.1\">Yes, using a theorem by Jeandel</span> (<a class=\"ltx_ref\" href=\"https://arxiv.org/html/2004.02534v4#S4\" title=\"4 A strongly aperiodic SFT on 𝐵𝑆(𝑛,𝑛) ‣ Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups\"><span class=\"ltx_text ltx_ref_tag\">Section</span> <span class=\"ltx_text ltx_ref_tag\">4</span></a>)</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"Sx1.3.3.3.3.3.3\" style=\"padding:3pt 2.0pt;\">Yes (<a class=\"ltx_ref\" href=\"https://arxiv.org/html/2004.02534v4#S2.SS2\" title=\"2.2 A weakly aperiodic SFT on 𝐵𝑆(𝑚,𝑛) ‣ 2 On a construction by Aubrun and Kari ‣ Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups\"><span class=\"ltx_text ltx_ref_tag\">Section</span> <span class=\"ltx_text ltx_ref_tag\">2.2</span></a>, Aubrun-Kari <cite class=\"ltx_cite ltx_citemacro_cite\">[<a class=\"ltx_ref\" href=\"https://arxiv.org/html/2004.02534v4#bib.bib3\" title=\"\">3</a>]</cite>)</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"Sx1.4.4.4.4.4\">\n<td class=\"ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r ltx_border_t\" id=\"Sx1.4.4.4.4.4.1\" style=\"padding:3pt 2.0pt;\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t\" id=\"Sx1.4.4.4.4.4.2\" style=\"padding:3pt 2.0pt;\">?</td>\n<td class=\"ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t\" id=\"Sx1.4.4.4.4.4.3\" style=\"padding:3pt 2.0pt;\">Yes (<a class=\"ltx_ref\" href=\"https://arxiv.org/html/2004.02534v4#S2.SS2\" title=\"2.2 A weakly aperiodic SFT on 𝐵𝑆(𝑚,𝑛) ‣ 2 On a construction by Aubrun and Kari ‣ Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups\"><span class=\"ltx_text ltx_ref_tag\">Section</span> <span class=\"ltx_text ltx_ref_tag\">2.2</span></a>, Aubrun-Kari <cite class=\"ltx_cite ltx_citemacro_cite\">[<a class=\"ltx_ref\" href=\"https://arxiv.org/html/2004.02534v4#bib.bib3\" title=\"\">3</a>]</cite>)</td>\n</tr>\n</tbody>\n</table>\n</span></div>\n</figure>", |
| "capture": "Figure 1: A Wang tile of " |
| } |
| }, |
| "image_paths": { |
| "1": { |
| "figure_path": "2004.02534v4_figure_1.png", |
| "caption": "Figure 1: A Wang tile of BS(m,n)𝐵𝑆𝑚𝑛BS(m,n)italic_B italic_S ( italic_m , italic_n )", |
| "url": "http://arxiv.org/html/2004.02534v4/x1.png" |
| }, |
| "2(a)": { |
| "figure_path": "2004.02534v4_figure_2(a).png", |
| "caption": "Figure 2: Illustration of the neighbor rules for BS(2,2)𝐵𝑆22BS(2,2)italic_B italic_S ( 2 , 2 ).", |
| "url": "http://arxiv.org/html/2004.02534v4/x2.png" |
| }, |
| "2(b)": { |
| "figure_path": "2004.02534v4_figure_2(b).png", |
| "caption": "Figure 2: Illustration of the neighbor rules for BS(2,2)𝐵𝑆22BS(2,2)italic_B italic_S ( 2 , 2 ).", |
| "url": "http://arxiv.org/html/2004.02534v4/x3.png" |
| }, |
| "3": { |
| "figure_path": "2004.02534v4_figure_3.png", |
| "caption": "Figure 3: How a substitution is applied to a biinfinite word: y=s(x)𝑦𝑠𝑥y=s(x)italic_y = italic_s ( italic_x ) with s𝑠sitalic_s a uniform substitution of size n𝑛nitalic_n", |
| "url": "http://arxiv.org/html/2004.02534v4/x4.png" |
| }, |
| "4": { |
| "figure_path": "2004.02534v4_figure_4.png", |
| "caption": "Figure 4: One tile from the tileset τq,Isubscript𝜏𝑞𝐼\\tau_{q,I}italic_τ start_POSTSUBSCRIPT italic_q , italic_I end_POSTSUBSCRIPT.", |
| "url": "http://arxiv.org/html/2004.02534v4/x5.png" |
| }, |
| "5": { |
| "figure_path": "2004.02534v4_figure_5.png", |
| "caption": "Figure 5: Preservation of the oriented arc distance darcsubscript𝑑𝑎𝑟𝑐d_{arc}italic_d start_POSTSUBSCRIPT italic_a italic_r italic_c end_POSTSUBSCRIPT by r𝑟ritalic_r and intersection of the arc (rl∘ϕ(x),rl∘ϕ(z))superscript𝑟𝑙italic-ϕ𝑥superscript𝑟𝑙italic-ϕ𝑧\\left(r^{l}\\circ\\phi(x),r^{l}\\circ\\phi(z)\\right)( italic_r start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT ∘ italic_ϕ ( italic_x ) , italic_r start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT ∘ italic_ϕ ( italic_z ) ) and the boundary between A𝐴Aitalic_A and B𝐵Bitalic_B.", |
| "url": "http://arxiv.org/html/2004.02534v4/x6.png" |
| }, |
| "6": { |
| "figure_path": "2004.02534v4_figure_6.png", |
| "caption": "Figure 6: Illustration of Lemma 3.1.", |
| "url": "http://arxiv.org/html/2004.02534v4/x7.png" |
| }, |
| "7": { |
| "figure_path": "2004.02534v4_figure_7.png", |
| "caption": "Figure 7: Tiles of τσsubscript𝜏𝜎\\tau_{\\sigma}italic_τ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT: left and right colors are identical and equal to i𝑖iitalic_i, top color is c𝑐citalic_c and bottom colors are equal to σi(c)0,…,σi(c)n−1subscript𝜎𝑖subscript𝑐0…subscript𝜎𝑖subscript𝑐𝑛1\\sigma_{i}(c)_{0},\\dots,\\sigma_{i}(c)_{n-1}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_c ) start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_c ) start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT.", |
| "url": "http://arxiv.org/html/2004.02534v4/x8.png" |
| } |
| }, |
| "validation": true, |
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| "url": "http://arxiv.org/html/2004.02534v4", |
| "new_table": {} |
| } |
