Truchet tilings and renormalization
Abstract.
The Truchet tiles are a pair of square tiles decorated by arcs. When the tiles are pieced together to form a Truchet tiling, these arcs join to form a family of simple curves in the plane. We consider a family of probability measures on the space of Truchet tilings. Renormalization methods are used to investigate the probability that a curve in a Truchet tiling is closed.
1. Preliminary remarks
This article was written before I was aware of the connection to the corner percolation model studied by Gábor Pete [Pet08]. The main result in this paper duplicates results in that paper, albeit by completely different methods. The author believes that the approach in this paper can be used to obtain stronger results than indicated in this paper, such as stronger information of the probability of a curve to have length for any constant . I hope to soon release a new version addressing this point of view.
The point of view of this paper also connects to rectangle exchange maps. This connection is developed in [Hoo12].
2. Introduction
The Truchet tiles are the two squares decorated by arcs as below.
We call the left tile and the right tile . The subscripts were chosen to indicate the slope of segments formed by straightening the arcs to segments.
Given a function , the Truchet tiling determined by is the tiling of the plane formed by placing a copy of the tile centered at the point for each . We denote this tiling by . Variations of these tilings were first studied for aesthetic reasons by Sébastien Truchet in the early 1700s [Tru04], and this version of tiles were first described by Smith and Boucher [SB87].
The arcs on the tiles of a Truchet tiling join to form a disjoint collection of simple curves in the plane. See Figure 1. We call these the curves of the tiling. Each curve is either closed or biinfinite. A natural question to ask is “how prevalent are the closed curves?” This was asked by Pickover for Truchet tilings which are random in the sense that for each , is determined by the flip of a fair coin [Pic89].
In this paper, we consider Truchet tilings that arise from functions defined in terms of two functions , and . These are the Truchet tilings determined by the function
(1) 
An example of a portion of such a tiling is shown in Figure 1. We will analyze these tilings with techniques coming from the theory dynamical systems. We will show that closed curves curves are highly prevalent in some families of tilings , while not so prevalent in other families. For the tilings of the form , we will show that these tilings are renormalizable in the sense of dynamical systems. Renormalization powers the strongest results in this paper. We explain the motivating connections to dynamical systems in the Motivation Section below.
To make the notion of prevalence rigorous, observe that and are naturally elements of the full twosided shift on the alphabet , denoted . The shift map is defined by . We may think of choosing our and at random according to some shiftinvariant probability measures and on . We then choose an edge of the tiling by squares centered at the integer points (e.g., take to join to ). We ask “what is where is the collection of where the curve through of the tiling is closed?” A simple argument shows that shift invariance of and guarantees that this number is independent of the choice of .
In general, we have the following result, which often prevents the tilings with a closed curve through from being full measure.
Theorem 1 (Drift Theorem).
Suppose and are shiftinvariant probability measures on satisfying
Then the measure of the set of pairs such that the curve of through edge is biinfinite is at least .
The above result is independent from our renormalization arguments. We prove it and state a stronger version of the result in section 4.4.
The main case of interest for this paper is when the measures and are the stationary (shiftinvariant) measures associated to some Markov chain with the state space . The above theorem indicates that for such measures, the probability that the curve through is closed is less than one whenever and occur with different probabilities. But, there is a oneparameter family of stationary measures associated to Markov chains with the property that and occur with equal probability. The following theorem addresses these cases.
Fix real constants and with . We describe a method of randomly choosing and . Choose randomly according to the flip of a fair coin. Define the remaining values according to the rule that for all integers , with probability and with probability . Do the same for with probability .
Theorem 2.
Let be an edge of the square tiling of the plane as above. Fix and , and define and randomly as above. With probability , the curve through of is closed.
The above theorem is proven with the renormalization techniques mentioned above. Much of this technique applies in general to pairs of shiftinvariant measures and .
We will informally explain how this notion of renormalization works. Fix and consider the tiling described by Equation 1. Generally, there is a collection of rows and columns of the tiling following statements hold, letting be the union of all rows and columns contained in the collection. (An example of consists of the union of gray squares in figure 1.)

A curve of the tiling is contained entirely in if and only if the curve closes up after visiting only four squares.

All other curves of the tiling intersect both and its compliment.

When a curve enters from the left (respectively, the right) through a vertical edge in , it exits through the nearest vertical edge in directly to the right (resp. the left) of .

When a curve enters from below (resp. above) through a horizontal edge in , it exits through the nearest horizontal edge in directly above (resp. below) .
Assuming these statements, we can then form a new tiling by collapsing all the columns contained in , identifying the pairs of edges mentioned in statement (3). Then we collapse the rows in , identifying the pairs of edges mentioned in statement (4) in a similar manner. So from we have obtained a new Truchet tiling, say . Now let be a curve in the tiling which intersects both and its compliment. Consider the collection of arcs of which are contained in the compliment of . Because of statements (3) and (4), the collapsing process takes these arcs and reassembles them to make a new connected curve in the tiling . The first two statements imply that we have removed all loops of length , and reduced the length of all closed loops. So, a loop is closed if and only if it vanishes under some finite number of applications of this collapsing process.
Under certain natural assumptions on and , the resulting tiling is determined by another pair and . This map is well defined on some Borel subset , and we call the collapsing map. The collapsing map acts on shift invariant probability measures via .
We can immediately observe for instance that a.e. curve is closed (in the sense of the above theorems), if and only if a.e. curve is closed. Typically much more than this is true. We define by . Assuming that for all integers , the measures and never has an atom at , there is a limiting formula for the probability that a curve is closed in terms of the measures and and the behavior of a cocycle acting on a function space. (See Corollary 24.) In the case of the stationary measures associated to a Markov chain as implicitly discussed in Theorem 2, the action of this cocycle leaves invariant a finite dimensional subspace. Understanding the action of this cocycle restricted to this subspace allows us to prove this Theorem 2.
2.1. Motivation
The original motivation for studying Truchet tilings here arose from connections between Truchet tilings and certain low complexity dynamical systems, such as interval exchange maps and polygon exchange maps. In the paper [Hoo12], the connection between Truchet tilings and a family of polygon exchange maps will be discussed in depth. We will briefly explain this connection here because it motivated this work.
A polygon exchange map is a map , where is a union of polygons, which is piecewise continuous on polygonal pieces, acts as a translation on each piece, and has an image of full area in . (There may be some ambiguity of definition on the boundaries of the pieces.)
For a more specific example, let be a finite set, and consider the product which consists of copies of . A polygon exchange map on , , is a decomposition of into polygonal pieces , a choice of elements and a choice of elements so that the image of the map
has full area. We define the subgroup . Fixing a and an , observe that the orbit of a point is contained in the set . Given any , we define an embedding via . For generic , we can pull back the action of to an action on . We define by . The arithmetic graph associated to and , is the graph whose vertices are the points in , and for which an edge is drawn between and for all . Note that the curve of the arithmetic graph through represents the orbit of the point under . So for instance, the point is periodic if and only if the curve passing through in is closed.
The arithmetic graph and similar constructions have been a useful tool for proving results about low complexity dynamical systems which require a detailed understanding orbits. For instance, in [PLV08, Proposition 13] it is shown that the analog of an arithmetic graph for a certain interval exchange consists of only finitely many curves. In [Sch07] Schwartz coined the term arithmetic graph, and used the arithmetic graph to resolve a longstanding question of Neumann, “are there outer billiards systems which have unbounded orbits?” In later work, Schwartz showed that a certain first return maps of outer billiards in polygonal kites is related to polyhedral exchanges by a dynamical compactification construction [Sch09, The Master Picture Theorem]. This construction relates the arithmetic graph used in outer billiards with the arithmetic graph of a polygon exchange mentioned above. The arithmetic graph also played a primary role in [Sch10].
A powerful tool for understanding the dynamical systems mentioned above has been renormalization. For a polygon exchange map , for instance, we can construct a new polygon exchange map by considering the return map to a polygon or union of polygons. Repeatedly applying this trick can be useful for proving results about the long term behavior of the system. This is commonly used to answer ergodic theoretic questions about interval exchange maps [MT02]. For a specific polygon exchange map, renormalization has been used to show that the set of points with periodic orbits have full measure [AKT01]. This is similar to our goals here.
This paper came out of an attempt find a simple example where renormalization can be understood in terms of the arithmetic graph. The polygon exchange maps appearing in [Sch09] appear complicated, but the associated arithmetic graphs have many nice properties. For instance, the connected components of these arithmetic graphs form a collection of simple curves. Truchet tilings represent a ways to force this behavior independent of constructions involving outer billiards. Moreover, we can obtain many Truchet tilings as instances of arithmetic graphs of polygon exchange maps [Hoo12]. In these cases, the renormalization of tilings described in the introduction corresponds to a renormalization (first return map) of the associated polygon exchange map. This paper came out of the realization that, in this particular case, by understanding the action of renormalization on the arithmetic graph, we can generalize the renormalization of a family of polygon exchange maps to a renormalization scheme applicable to a more general class of dynamical systems. These are the systems described in Section 4.
Finally, it should be noted that the original motivation for studying Truchet Tilings by Truchet [Tru04] and Smith [Tru04] was aesthetic. These motivations continue today. See for instance, [LR06], and [Bro08]. More general families of curves associated to tilings have been considered. For instance, [OC99] considers similar curves in Penrose tilings.
Aside from the above motivations, Truchet tilings have also played role in understanding a variant of the cellular automata known as Langton’s Ant [GPST95].
2.2. Outline
In Section 3, we explain how to think of the space of Truchet tilings as a dynamical system. In Section 4, we restrict attention to the case of tilings determined by a pair of elements of the shift space as in Equation 1. We also provide the necessary background on shift spaces and shift invariant measures here. In Section 4.4, we prove the Drift Theorem and a stronger variant. Section 5 states and proves the renormalization results which apply to many pairs of shiftinvariant measures. This culminates in Section 5.3, which explains the renormalization process and constructs the cocycle alluded to in the introduction. Finally, Section 6 develops these renormalization theorems in the context of the measures relevant to Theorem 2. Section 6.3 does the necessary calculation to prove this theorem.
3. Dynamics on Truchet tilings
Consider the unit square centered at the origin with horizontal and vertical sides. An inward normal to the square is a unit vector which is based at a midpoint of an edge and is pointed into the square. See below
We do not keep track of the location at which the normal is places, so the four inward normals are just the vectors , and . We use to denote the collection of inward normals.
Let be the collection of maps . We will define a dynamical system on . Choose . The inward normal is a vector based at a midpoint of the square at the origin pointed inward. The Truchet tiling determined by places the tile at the origin. We drag the vector inward along an arc of this tile keeping the vector tangent to the arc. After a quarter turn, we end up as a vector pointed out of the square centered at the origin. So, the vector points into a square adjacent to the square at the origin. We translate the whole tiling so that this new square becomes centered at the origin.
Formally, this is the dynamical system given by
(2) 
where and is the translation of the plane by the vector .
4. Truchet tiling spaces from shift spaces
In this paper we will concentrate on Truchet tilings which arise from a pair of biinfinite sequences of elements of the set . As in the introduction, we will use notation from the world of shift spaces to denote these biinfinite sequences. Namely, an element is a biinfinite sequence of elements of . For , we use to denote the th element of the sequence .
Given , we obtain a function as in equation 1 of the introduction. In this paper, we will be interested in studying the dynamics of on the collection of all . This collection of tilings is invariant and admits a natural renormalization procedure as we explain in section 5. In this section, we reveal some of the more basic structure of the map restricted to these types of tilings.
4.1. Background on shift spaces
We briefly recall some basic facts about twosided shift spaces here. For further background see [LM95], for instance.
Let be a finite set called an alphabet. The set is called the full (twosided) shift on .
For integers , let be an arbitrary function. The cylinder set determined by is the set
(3) 
We equip with the topology generated by the cylinder sets. This is the coarsest topology which makes each cylinder set open. Observe that the cylinder sets are also closed. An equivalent description of this topology is obtained by considering elements of as functions . From this point of view, this is the topology of pointwise convergence on compact subsets of , where is given the discrete topology. With this topology, the set is homeomorphic to a Cantor set.
The shift map is the homeomorphism of the full shift space defined by
(4) 
A shift space is a closed, shiftinvariant subset of a full shift space. We endow with the subspace topology.
A shiftinvariant measure on a shift space is a Borel measure satisfying
Full shift spaces admit a plethora of shiftinvariant probability measures.
4.2. Tiling spaces from shift spaces
As before, we let denote the full shift space on the alphabet . Given a pair of elements , we obtain a map as in equation 1. This function in turn determines a tiling as described in the introduction. Note that the map is twotoone, with , where denotes the element of given by .
The collection is easily seen to be translation invariant, and therefore the set is invariant under . There is a natural lift of the action of on this set to an action on the set given by
(5) 
with . (Here, denotes the inverse of the shift map defined by .) We call a lift of because if then . Observe that the inverse of is given by
(6) 
We now make some preliminary observations about .
Proposition 3.
Let be shift spaces. The set is a closed invariant subset of .
The above proposition trivially follows from the definitions.
Proposition 4.
Suppose and are shift invariant probability measures on and , respectively. Let be the discrete probability measure on so that for each . Then is a invariant probability measure on .
Proof.
It is sufficient to show invariance of on sets of the form , where and and are Borel sets chosen so that the product is independent of the choice of and . Then by definition of , shift invariance of the measures and , and the permutation invariance of ,
∎
4.3. Periodic orbits
Suppose satisfies . We say has an stable periodic orbit of period if is the smallest number for which there are open neighborhoods and of and respectively for which each satisfies . The following proposition characterizes the points with stable periodic orbits.
Proposition 5 (Stability Proposition).
Let , and use to denote . The following statements hold.

If has a stable periodic orbit of period , then it is also (least) period under in the usual sense.

has a stable periodic orbit if and only if the curve of the tiling passing through the normal to the square centered at the origin is closed. In this case, the period of is the number of squares the associated closed curve of the tiling intersects, counting multiplicities.

has a stable periodic orbit of period if and only if is the smallest positive integer for which both and .
The multiplicities mentioned in the proposition deal with the fact that curves may (a priori) intersect the same square twice. (In fact, no curve intersects a square twice. This follows from Lemma 8, below.)
Proof.
For let and . By induction using equation 5, we observe that and . Now suppose is period . Note that equals the vector sum in statement (3). We see and . Therefore if , we see that must be period under the shift map. This is not an open condition, so cannot have a stable periodic orbit of period . This is similarly true if . Extending this argument, we observe that and for integers . Therefore, cannot have a stable periodic orbit of any period. Conversely, if both and , then we let and . Define and similarly. Then consider the open sets
Then observe that for and we have . Therefore, has a stable periodic orbit of period . Finally, observe that the condition that and is equivalent to the statement that the curve of the tiling passing through is closed. ∎
Remark 6.
Not all periodic orbits are stable. When and for all , we have for all , but is not an stable periodic orbit for any .
Corollary 7.
Let and be shiftinvariant measures on . Let be the set of all with stable periodic orbits of period . Fix an edge of the tiling of the plane by squares centered at the integer points as in the theorem of the introduction. Then, is equal to the measure of those so that the curve of the tiling through is closed and visits squares (counting multiplicities).
The proof follows from the Stability Proposition together with the observation that both quantities are translation invariant. The fact that the horizontal or vertical orientation of is irrelevant follows from the fact that curves of the tiling alternate intersecting horizontal and vertical edges. We omit a detailed proof of this corollary.
4.4. An invariant function and drift
In this section, we prove the Drift Theorem. Ideas for this result come from the drift vector of an interval exchange transformation. See [PLV08], for instance.
The first observation of this section is that there is a simple invariant function on .
Lemma 8 (Invariant function).
Let . This is a invariant function from to .
Sketch of proof.
We partition the space into subsets according to choices of and . These groups are defined
Write for the set of these 16 subsets. Let be the strongest equivalence relation on for which whenever intersects . The equivalence classes can be computed by drawing the graph where the nodes are elements of and the arrows are drawn from to whenever intersects ; the equivalence classes are then the connected components of this graph. One of the two maximal equivalence classes is shown below.
{xy}(10,20)*+G(1,1,(1,0))=”a”; (50,20)*+G(1,1,(0,1))=”b”; (90,20)*+G(1,1,(1,0))=”c”; (130,20)*+G(1,1,(0,1))=”d”; (10,0)*+G(1,1,(0,1))=”ap”; (50,0)*+G(1,1,(1,0))=”bp”; (90,0)*+G(1,1,(0,1))=”cp”; (130,0)*+G(1,1,(1,0))=”dp”; \ar@¿ ”a”;”b”; \ar@¿ ”b”;”c”; \ar@¿ ”c”;”d”; \ar@¡ ”ap”;”bp”; \ar@¡ ”bp”;”cp”; \ar@¡ ”cp”;”dp”; \ar@¡¿ ”a”;”ap”; \ar@¡¿ ”b”;”bp”; \ar@¡¿ ”c”;”cp”; \ar@¡¿ ”d”;”dp”; \ar@¿@/_1.5pc/ ”d”;”a”;\ar@¿@/_1.5pc/ ”ap”;”dp”;
Note that on this equivalence class. The function takes the value on the eight remaining subsets. ∎
The following is a restatement of the Drift Theorem in the introduction. Equivalence follows from the Corollary 7.
Theorem 9 (Restated Drift Theorem).
Suppose and are shiftinvariant probability measure on satisfying
Then the measure of the set of all without stable periodic orbits is at least .
Proof.
Let for . We would like to compute the integral
with the integral taken over all Let denote those with . Then,
with the integral take over all pairs with fixed by the sum. The appears because of the removal of . Consider the case . Note that , so that for this term
Similarly, in the case , we have , so again
Similar analysis holds for the cases and show that the total integrals is given by .
Let denote the set of all which have stable periodic parameters. This set is invariant, and the proposition above guarantees that
Also note that for any that if is the component of then is one of the four vectors of the form . Therefore, for any invariant set with
we have and . Applying this to the invariant set , we see
so that , as desired. ∎
We get a stronger result using the ergodic decomposition. Let us briefly recall the statement in this context. Let denote the collection of shiftinvariant probability measures on , and denote those measures which are ergodic. For any shift invariant probability measure , there is unique probability measure defined on so that and for all continuous we have
Now let be the characteristic function on the set of all without stable periodic orbits. Let be a shiftinvariant probability measure as above. And be the measure obtained from the ergodic decomposition. Then, applying the Drift Theorem to the ergodic measures yields
The following is what is weaker than the above argument gives.
Corollary 10.
Let and be the measures obtained from the ergodic decomposition applied to and , respectively. If the measure of the set of all without stable periodic orbits is zero, then for a.e. (and a.e) we have .
5. Renormalization of Truchet tilings
In this section, we will describe the renormalization procedure for the dynamical system . Informally, this procedure can be described in terms of tilings as in the introduction. Given a tiling , we renormalize to obtain a new tiling with and . The function is called a collapsing map and is defined in Subsection 5.2. The renormalized tiling is obtained from the original by collapsing some rows and columns of tiles to lines. This is explained in the following subsection.
5.1. Notation for words
A word in is an element of a set for some , called the length of . We use to denote the collection of all words. We write with to denote a word, and to denote the unique word of length . To simplify notation of the elements in , we use to denote and to denote . So the word where and can be simply written as .
Adjacency indicates the concatenation of words; if then
For an integer , the expression indicates the nfold concatenation , with appearing times.
If and with , then we can consider the function given by . By equation 3, this determines a cylinder set , which we also denote by , with the hat indicating that represents the value of the zeroth entry of the words in the cylinder set.
5.2. The collapsing map on shift spaces
The idea of the collapsing function mentioned at the beginning of this section is to removed any substrings of the form and then slide the remaining entries together. For example,
where underlined entries have been removed. There are two potential reasons why may not be well defined. First, the zeroth entry might be removed by this process. Second, the remaining list may not be biinfinite.
Formally, we define to be the union of two cylinder sets Given , we set to be
(7) 
We say that is unboundedcollapsible if is unbounded in both directions, zerocollapsible if , and collapsible if it is both unbounded and zerocollapsible. We use to denote the collection of collapsible elements of . If , then there is a unique order preserving indexing denoted so that and . We use to denote the collapse of , which we define by . So, the collapsing map is a map .
In the remainder of this subsection, we investigate properties of this map.
Proposition 11.
The collapsing map is surjective.
To prove this proposition, we explicitly construct the inverse images. For this, we define a process we call insertion. An insertion rule is determined by a function . Let . From , we determine a strictly increasing biinfinite sequence of integers inductively according to the following two rules.

.

For all , then .
The insertion function determined by , evaluated at is given by the following rules.

if for some .

if for some .
Proposition 12.
Let . For all , we have
Proof.
Suppose . We will show that is an element of the set on the right hand side of the equation. Consider the set indexed by as in the definition of the collapsing map. By definition of , for all we have , and . This proves that with for all . Finally, by definition of , when we must have or . This is equivalent to the statement that whenever and .
Conversely, suppose with as above. Define . Also define as in the definition of insertion function applied to . Let . We will show that . Then it follows from the definitions of and that as desired. It is clearly true that , because every inserted word is of the form for some . Now we show that . Suppose . Since , we note that for some . Therefore by the definition of insertion function and the fact we only insert words in , we have that the word
for . Observe by definition of that and therefore , which is a contradiction. ∎
Suppose with and . Recall, we use to denote the cylinder set consisting of those for which for all satisfying .
Corollary 13 (Preimages of cylinder sets).
Let denote the set of unbounded collapsible elements of . Suppose and . Then is given by
where the union is taken over all choices of integers such that if and , and otherwise. The word if and otherwise. The word if and otherwise.
Corollary 14.
The collapsing map is continuous.
Proof.
By the above corollary, the inverse image of a cylinder set is a union of cylinder sets intersected with and therefore open in . Since the cylinder sets form a basis for the topology, the inverse image of any open set is open in . Therefore, is continuous. ∎
Proposition 15.
If is a shift space and the alternating element defined by is not an element of , then every element of is unboundedcollapsible.
Proof.
Suppose is not unboundedcollapsible. Then, for all or all for some . Therefore, can be obtained as a limit of shifts of . ∎
We have the following analog for measures.
Proposition 16.
Suppose is a shiftinvariant probability measure on , and that
Then .
Lemma 17.
Suppose is shiftinvariant, then so is . If is a shift space and , then is a shift space.
Proof.
To prove the first statement we will show that if with then . Consider the elements , as defined in the definition of the collapsing map. We have . Since , all elements in are not zerocollapsible. So, is the intersection of with two cylinder sets. So, is closed. Therefore is closed by the continuity of and compactness of . ∎
Proposition 18.
Let be a shift invariant measure on a shift space . Then, is a shift invariant measure on .
Proof.
The content of this proposition is that is shift invariant. Let be a Borel set and . Recall the definition of used in the collapsing map. Let denote the smallest positive entry of . For let