Pointwise periodic maps with quantized first integrals

We describe the global dynamics of some pointwise periodic piecewise linear maps in the plane that exhibit interesting dynamic features. For each of these maps we find a first integral. For these integrals the set of values are discrete, thus quantized. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of open tiles of certain regular or uniform tessellations. The action of the maps on each invariant set of tiles is described geometrically.


Introduction
A pointwise periodic map is a bijective self-map in a topological space such that each point is periodic. A periodic map is a bijective self-map in a topological space such that some iterated of the map is the identity. For a periodic map F : X ÝÑ X the minimum natural number p satisfying F p " Id is called the period of F. Notice that a pointwise periodic map satisfying that the period of the points has an upper bound is periodic and its period is the least common multiple of the periods of the elements of the space.
A classical result of Montgomery establishes that any pointwise periodic homeomeorphism in an Euclidean space is periodic, [19]. Non-periodic but pointwise periodic bijective maps do exist when the continuity assumption is relaxed, see [23] for instance. In the series of papers [7,8] and [10], the authors introduce three explicit examples of pointwise periodic maps that are not periodic. The examples given by these authors in the above mentioned references belong to the family of piecewise affine maps with a line of discontinuity: Gpx, yq " py,´x´ρy`signpyqq , where signpyq " for |ρ| ă 2. In particular they correspond to the cases ρ P t´1, 0, 1u. There are other values of ρ for which there exist non-periodic points, see [9]. Notice also that maps (1) correspond to the second order discontinuous difference equations x n`2 "´x n´ρ x n`1`s ignpx n`1 q.
As we will see in next section, each map G is linearly conjugate with the piecewise rotation map F px, yq "˜c ospαq sinpαq sinpαq cospαq¸˜x´s ignpyq y¸, where ρ "´2 cospαq with α P p0, πq. Observe that the maps G with ρ "´1, 0 and 1 are conjugate with the maps F with α " π{3, π{2 and 2π{3, respectively. As we will see, the normal form F regularizes the shape of the invariant sets and keeps the same discontinuity line y " 0. These maps are included in the class of symmetric maps studied in the remarkable paper [14] together with other more general piecewise rotations, see a further comment below. As noticed in [5], they exhibit complex dynamics and they belong to the type of piecewise rotations with the same rotation angle that elude the generic dichotomy that appears in most piecewise rotations of being globally attracting or globally repelling maps, see [5,Theorem 1].
Piecewise affine maps with a line of discontinuity appear as models in many fields like in the study of mechanical systems with friction, power electronics, relay control systems or economics [2,6,24]. In fact, as is explained in [7,8,10], the three maps (1) with ρ P t´1, 0, 1u appear in the study of steady states of certain cellular neural networks.
Despite their apparent simplicity, piecewise affine maps exhibit great dynamic richness and a variety of phenomena that are characteristic of these systems, see [3,5,14,22,24] and references therein. As we will show, the examples considered in this paper are also very rich from a dynamical viewpoint, even though each orbit is periodic. In fact, one of our motivations was to highlight the beautiful features of these examples.
Recall that a first integral of a discrete dynamical system associated with a map F is a non-constant real valuated function V such that V˝F " V , which means that the level sets tV " cu, typically called the energy levels, are invariant under the action of the map. It is known that periodicity issues are related with integrability since most continuous periodic maps are completely integrable (there exist as many functionally independent first integrals as the dimension of the phase space), see [11] and [12]. In this work we consider the piecewise affine maps F with α P tπ{3, π{2, 2π{3u under the light of their properties as integrable systems. For each of these three maps, we obtain a non-trivial first integral periodic islands for a family of maps that contain the ones studied in this papers. These necklaces and the set of periods associated with their periodic orbits are characterized both analytically and geometrically, and its existence is the key to prove the boundedness of the orbits of the maps considered there. We want to point out that in our maps, all the integral's level sets are necklaces. In Remark 2 we comment the relation between both families of necklaces. In addition, as we will see, the maps F with α P tπ{3, π{2, 2π{3u have also a second continuous first integral, see Remark 23. This second first integral, however, is not useful to control the set of periods.
The results in the literature indicate that some polygonal dual billiards should also have quantized integrals, see Figure 3 in [20], Figure 2 in [13], or  and the results in [23]. We believe that the explicitness of the analytic expression of the quantized integrals with positive measure level sets for the maps (2) is quite novel in the context of discrete dynamical systems theory. It is interesting to notice the fact that the regular tessellations that we find in this paper also appear in the study of some polygonal dual billiards like the one introduced by Moser in [20] or those that appear in [23]. Observe, however, that these dual billiards are not conjugated to any map considered in our paper, because they exhibit different sets of periods.
A consequence of our results for F, when α P tπ{3, π{2, 2π{3u, is the existence of an open and dense subset U on which the dynamics of the map is strongly stable and simple.
We will see that for any x P U there exists an open neighborhood of x, say U x , and n x P N such that F nx | Ux " Id . Moreover, varying x P U, the values n x are unbounded.
We will study the three cases separately in three different sections. In a few words, the main results that we will state in detail in the next section, are: (a) We present first integrals V for each case. See Section 6 for a constructive approach for obtaining them. In the three figures the beads of a necklace have the same color. In fact, the shape and the number of beads, say M, only depend on the level set k and α. Moreover, the inter-tile dynamics can be described in a very simple way: if we collapse each of the open tiles in a point, the interior of tpx, yq : V px, yq " cu can be identified with Z M , simply following the order given by the necklace in clockwise sense, were, as usual, given q P Z, we denote by Z q the set of the residue classes induced by the congruence n " m if and only if n´m is a multiple of q with n, m P Z. Then we will prove that the dynamical system generated by F, restricted to this set, is conjugated to an affine discrete dynamical system generated by a map h : Z M Ñ Z M , where hpiq " i`upc, αq, for some upc, αq P Z M that we also determine explicitly in this paper, see Theorems A, B and C. Notice that, geometrically, F acts as a rotation among the beads of any necklace.
A similar inter-tile dynamics' description in the context of dual polygonal billiards can be found in [17], and also in the context of piecewise linear maps [14, Theorem 1 and Due to the above conjugation, the dynamics on the interior of each level set can be completely understood, see Theorems A, B and C. Roughly speaking, for each map and for each necklace (set of tiles with the same energy level), there exists a certain number k P tM, M {2u X N, that depends (explicitly) on the energy level, so that each tile is invariant by F k . Furthermore, on each tile, F k is a rotation of order p around the center of the tile, where p P t2, 4u when α " π{2, or p P t3, 6u when α P tπ{3, 2π{3u and it is determined explicitly by the energy level. As a consequence, on each tile there is a k-periodic point (the center) and the rest of points are kp-periodic. The dynamics in the necklaces is, therefore, a discrete version of an epicyclic motion around a discrete deferent which is the locus of the centers of the tiles, [21, p. 123].
As we will see, the dynamics on the boundaries of the tiles (edges and vertices) requires a little bit more elaborated description.
(c) As a consequence of the above results and the study of the dynamics on the boundary of the level sets, for each map, we easily characterize the period of every point in terms of the value of its associate first integral and obtain the global dynamics of the map. For instance in Proposition 1 we present our results in an algorithmic way when α " π{2.
In particular, the set of periods of the maps are presented.

Preliminaries and main results
The families of maps F and G given in (1) and (2), respectively, are linearly conjugated because F px, yq "`Q´1¨G`Q¨px, yq t˘˘t , where Q "˜1´c ospαq 0 sinpαq¸.
In this paper we will, therefore, work with the above normalized one-parameter family of maps F. Notice that each map F is bijective with inverse F´1px, yq "˜c ospαq´sinpαq sinpαq cospαq¸˜x y¸`˜s ignpsin pαq x`cos pαq yq 0¸.
Notice also that F is discontinuous in the set LC 0 " tpx, 0q : x P Ru. We will consider the critical lines LC´i " tpx, yq such that F i px, yq P LC 0 u, and also the critical set F " Ť iPN LC´i formed by all the preimages of the critical line LC 0 , where we use the notation introduced by Mira et al. in [18] (see also [1,4]). We call the open set U " R 2 zF, the zero-free set because none of the orbits starting at point in U touches the discontinuity line LC 0 , where the second coordinate of the points is zero.
Regarding the above conjugation, of course it is only defined in the case sinpαq ‰ 0 which corresponds with the cases ρ ‰˘2. In the case α " 0 (resp. α " π) the map F (resp. F 2 ) has trivial dynamics of translation type, and do not correspond to the initial family G with ρ "˘2.
For each α P tπ{2, 2π{3, π{3u we introduce some specific notations and also state our main results: Theorems A, B and C, respectively. Each one of them will be proved in a different section. The results in these theorems have the following structure: in (i) we characterize the geometry of the critical set; in (ii) we state the existence of a first integral in the non-critical set and we characterize the global dynamics in this set proving that is conjugated with the composition of two rotations; in (iii) we establish the dynamics in the critical set; in (iv) we characterize the set of periods of the maps.
While statements (iv) are known in the literature, the geometric description given in statements (i)-(iii) is, as far as we know, novel.

The case α " π{2
When α " π{2, the map F is the one in (1) with ρ " 0 and was studied in [8]. Consider F π{2 the grid formed by the straight lines x " k and y " , with k, P Z. This grid defines the square Euclidean regular tiling [15,16], also named quadrile, see Figure 1. Each (open) tile is denoted by T k, " tpx, yq; such that k ă x ă k`1 and ă y ă `1u.
The centers of each of these tiles are denoted by p k, " pk`1{2, `1{2q . We also introduce the set U π{2 " ď pk, qPZ 2 T k, " R 2 zF π{2 , and the function where Epzq " tzu is the floor function of z P R that recall gives as output the greatest integer less than or equal to z. We also define V k, " V π{2 pp k, q and denote N 0 " N Y t0u.
We prove: Theorem A. Consider the discrete dynamical system (DDS) generated by the map F given in (2) with α " π{2, F px, yq " py,´x`signpyqq. Then: (ii) The function V " V π{2 is a first integral of F on the free-zero set U " U π{2 . Each level set tV px, yq " cu X U, with c P N 0 , is a necklace formed by 4c`2 squares, see Figure   1. If we identify each square with a point (for instance the center), the DDS restricted to this set is conjugated with the DDS generated by the map h : Z 4c`2 Ñ Z 4c`2 , hpiq " i`c. As a consequence, when c is odd (resp. even), each square in this level set is invariant by F 4c`2 (resp. F 2c`1 ) and restricted to this square, F 4c`2 (resp. F 2c`1 ) is a rotation of order 2 (resp. 4), around the center of the tile. In particular, all points but the center in each of these tiles have period 8c`4.
(iii) All orbits with initial condition on F are p8n`4q-periodic for some n P N 0 , see Theorem 10 for more details.
(iv) The map F is pointwise periodic. Furthermore, its set of periods is PerpF q " t4n`1; 8n`4; and 8n`6 for all n P N 0 u . Item pivq was already proved in [8]. All the geometric description of the dynamics of F given in the other items is new.
Observe that the statement (ii) in the above result can be formalized in the following way: the dynamics of F on each necklace tV " cu X U with c P N 0 , is conjugate with the dynamics of the map where q " 2 if c is odd, and q " 4 if c is even. This map can be seen as the product of two finite order rotations, its first component gives the dynamics on the discrete deferent formed by the set of centers of tiles, and the second component gives the dynamics on a epicycle. A similar situation is described in statements (ii) of Theorems B and C.
Also notice that a simple check shows that the function V is not a first integral of F on the whole plane, since the relation V pF q " V is not satisfied for some points in F " R 2 zU.
As a consequence of the above theorem we can easily give a simple algorithm to know the period of each orbit in terms of its initial condition. Recall that given a point px, yq P R 2 , k " Epxq, " Epyq and V k, " V π{2 pp k, q. For the forthcoming cases α P t2π{3, π{3u, from our results a more complicated algorithm could be obtained, but for the sake of brevity, we do not detail it.
Proposition 1. Any point px, yq P R 2 is a p-periodic point of F , where: (a) When x R Z and y R Z and, moreover, either x´k ‰ 1{2 or y´ ‰ 1{2, then p " 8V k, `2. When x´k " 1{2 and y´ " 1{2, then p " 2V k, `1 if V k, is even, and (b) When x P Z and y R Z, if k is even, p " 8V k, `4 and if k is odd, p " 8V k´1, `4.
(c) When x R Z and y P Z, if is even, p " 8V k, `4 and if is odd, p " 8V k, ´1`4 .
(d) If x P Z and y P Z, then: (i) When k is even, p " 8V k, `4 if is even, and p " 8V k, ´1`4 if is odd.
The statement (d) in the above result is a consequence of Theorem 10. To obtain the result we will identify some tiles, that we will call perfect squares, such that their boundaries (including their edges and all the vertices in F) avoid the discontinuity effects and, therefore, the points on the boundary of such a tile have the same periodic behavior as the interior points, except their centers. To study the periodicity in the rest of the edges (without vertices) we will associate them with an appropriate tile, so that the points in the edge follow the periodic behavior of the interior points. See Section 3.3 for more details.

The case α " 2π{3
In this case, the map F in (2) is conjugate with the map G in (1) with ρ " 1, which was studied in [10].
Theorem B. Consider the discrete dynamical system generated by the map F given in (2) with α " 2π{3. Then: (i) Its critical set is F " F 2π{3 .
(ii) The function V " V 2π{3 is a first integral of F on the free-zero set U " U 2π{3 " (a) Each level set tV px, yq " cu, with c P 2N 0 even, in U is a necklace formed by 6c`2 triangles, see Figure 2. If we identify each triangle with a point (the center, for instance), the DDS restricted to this set is conjugated with the DDS generated by the map h : Z 6c`2 Ñ Z 6c`2 , hpiq " i`2c. As a consequence, each tile in this level set is invariant by F 3c`1 and restricted to this triangle, F 3c`1 is a rotation of order 3 around the center of the tile. In particular, all points but the center in each of these tiles have period 9c`3.
(b) Each level set tV px, yq " cu, with c P 2N 0`1 odd, in U is a necklace formed by 3c`1 hexagons, see Figure 2. If we identify each hexagon with a point, the DDS restricted to this set is conjugated with the DDS generated by the map h : Z 3c`1 Ñ Z 3c`1 , hpiq " i`c. As a consequence, each tile in this level set is invariant by F 3c`1 and restricted to this hexagon, F 3c`1 is a rotation of order 3 around the center of the tile. In particular, all points but the center in each of these tiles have period 9c`3.
(iii) All orbits with initial condition on F are periodic with period 9n`3 for some n P N 0 .
(iv) The map F is pointwise periodic. Furthermore, its set of periods is PerpF q " t3n`1 and 9n`3 for all n P N 0 u .
Similarly to Theorem A, item pivq was already known, see [10]. Again, all the geometric description of the dynamics of F given in the other items is new.
From statement (ii), on each necklace, F is conjugate with the product of rotations ϕ : Z 6c`2ˆZ3 ý given by ϕpi, jq " pi`2c, j`1q when c is even, and ϕ : Z 3c`1ˆZ3 ý given by ϕpi, jq " pi`c, j`1q when c is odd.

The case α " π{3
In this last case, the map F in (2) is conjugate with the map G in (1) with ρ "´1, which was studied in [7].
We consider F π{3 the grid formed by the straight lines y " ? 3px´2k´1q; y " ? 3 and y "´?3px´2mq, with k, , m P Z that, again, form a trihexagonal Euclidean uniform tiling which is a translation of the one that appeared in the previous case α " 2π{3, see Figure 3. The interior of each tile is defined by and´?3px´2mq ă y ă´?3px´2m´2q ( . As before, we call U π{3 the complement of this grid. Any point px, yq P U π{3 belongs (only) to the tile T k, ,m with k " Bpx, yq, " Cpyq and m " Dpx, yq, where nd now it can be seen that m " k` ´1 or m " k` or m " k` `1. In this case, • The tile T k, ,k` is a regular hexagon and its center is at the point • The tiles T k, ,k` ´1 and T k, ,k` `1 are equilateral triangles whose centers are q k, " We also introduce the following function Observe that by construction, it is constant on each tile T k, ,m . Hence we can associate to each point in this tile, the value V k, ,m " max p|k´ `m|, |k` `m`1|´1, |´k` `m`1|´1q .
Our results for this case are collected in the next theorem. We remark that the proof of item piiiq will be the more complicated part of the paper.
Theorem C. Consider the discrete dynamical system generated by the map F given in (2) with α " π{3. Then: (a) Each level set tV px, yq " cu, with c P 2N 0 even, in U is a necklace formed by 3c`2 hexagons, see Figure 3. If we identify each one of them with a point, the DDS restricted to this set is conjugated with the DDS generated by the map h : Z 3c`2 Ñ Z 3c`2 , hpiq " i`c{2. As a consequence, when c " 4j (resp. c " 4j`2), each tile in this level set is invariant by F 3c{2`1 (resp. F 3c`2 ) and restricted to this hexagon, F 3c{2`1 (resp. F 3c`2 ) is a rotation of order 6 (resp. 3) around the center of the tile. In particular, all points but the center in each of these tiles have period 9c`6.
(b) Each level set tV px, yq " cu, with c P 2N 0`1 odd, in U is a necklace formed by 6c`4 triangles, see Figure 1. If we identify each one of them with a point, the DDS restricted to this set is conjugated with the DDS generated by the map As a consequence, each tile in this level set is invariant by F 6c`4 and restricted to this triangle, F 6c`4 is a rotation of order 3 around the center of the tile. In particular, all points but the center in each of these tiles have period 18c`12.
(iii) All orbits with initial condition on F are periodic with periods 36n`6, 18n`9, 18n`15 or 108n`72, for some n P N 0 , for more details see Theorem 22.
Once more, although item pivq is known, see [7], all the geometric description of the dynamics of F given in the other items is new. From the above result, on each necklace, F is conjugate with the map ϕ : Z 3c`2ˆZq ý given by ϕpi, jq " pi`c{2, j`1q, where q " 6 when c " 0 mod p4q, and q " 3 when c " 2 mod p4q; or ϕ : Z 6c`4ˆZ3 ý where ϕpi, jq " pi`c, j`1q, when c is odd. Figure 3: Level sets of the first integral V of F for α " π{3, given in in (6). In each tile the level and the period of the center are indicated respectively between brackets.
Remark 2. As we have mentioned, in [14] an infinite number of necklaces of a family of maps that include the ones studied in this work are characterized. We want to note that Theorems A-C show that for our maps all energy levels are necklaces. In particular, in the case α " π{2 the necklaces studied in [14] correspond to the energy levels whose centers have period 4n`1 (which are those with even energy level). Let us observe that from Theorem A we know that there are other necklaces whose period is 4n`2 (those with odd energy level). In the case α " 2π{3, the necklaces in [14] cover all energy levels since all the necklaces have centers of period 3n`1. In the case α " π{3 the necklaces in [14] are those whose centers have period 6n`1. Observe that Theorem C guarantees the existence of much more necklaces.

The address of a point
We end this section with the concept of address of a point that will be used in the proof of Theorems A, B, C.
Recall that every map in the considered one-parameter family F has discontinuity line LC 0 " ty " 0u, so we introduce the sets H`" tpx, yq P R 2 : y ě 0u and H´" tpx, yq P R 2 : y ă 0u, and we call F`and F´the map F restricted to H`and H´respectively. For any point px, yq P R 2 we define its address Apx, yq as follows: Moreover for every n P N, we call the itinerary of length n of the point px, yq the sequence of n symbols I n px, yq " pApx, yq, ApF px, yqq, . . . , ApF n´1 px, yqqq.
For instance if px, yq P H`, F px, yq P H´and F 2 px, yq P H´, then the length 3 itinerary of the point is t`,´,´u, and F 3 px, yq " F´˝F´˝F`px, yq.
Lemma 3. Let J n " pi 1 , . . . , i n q be a sequence of symbols of length n with i i P t`,´u and consider the set BpJ n q " tpx, yq P R 2 such that I n px, yq " J n u. Then BpJ n q is convex.
Moreover F n restricted to BpJ n q is an affine map.
Proof. The proof of the convexity follows easily by induction. If n " 1, BpJ n q is either Hò r H´both convex sets. Assume that the result holds for sequences of length n´1 and set J n´1 " pi 1 , . . . , i n´1 q. Therefore we have This fact proves that BpJ n q is convex because it is the intersection of two convex sets. This ends the inductive proof of convexity. Furthermore, F n px, yq " F in˝Fi n´1˝¨¨¨˝F i 1 px, yq, for all px, yq P BpJ n q, showing that F n restricted to BpJ n q is an affine map.

Preliminaries
We start by determining the set of tile centers, p k, , such that V pp k, q " c for c P N 0 .
It is easy to see that these two tiles are invariant. To describe the rest of level sets, (a) tp k, : V k, " cu X Q 1 " tpk`1{2, `1{2q : l " k`c, k "´c,´c`1, . . . ,´1u. We denote by X 1 , X 2 , . . . , X c these c centers for k "´c,´c`1, . . . ,´1 respectively. Every one of them lies on the straight line y " x`c.
Proof. In order to prove paq we begin by considering the points pk`1{2, k`c`1{2q with and consequently V k,k`c " c. Clearly, k`1{2 ă 0 and k`c`1{2 ą 0, and hence the points belong to Q 1 .
To see the other inclusion take pk`1{2, `1{2q P Q 1 with V k, " c. We have to prove that " k`c and´c ď k ď´1. We know that k ă 0, ě 0 which easily implies that ą k. Hence | ´k| " ´k, and `k ă ă ´k. Then V k, " max p|k` `1|´1, |k´ |q " max p|k` `1|´1, ´kq .
Consider the following two cases: It implies that ´k " c, that is " k`c. Furthermore, since " k`c and ě 0 we get k ě´c.
The proof of statements pbq, pcq and pdq follows using the same easy arguments.
In Figure 4 we show the points p k, in the levels c " 2 and c " 3, respectively.

Proof of items (i) and (ii) of Theorem A: dynamics on the zero-free set
Recall that in this case, F px, yq " py,´x`signpyqq, F`px, yq " py,´x`1q and F´px, yq " py,´x´1q. We will split our proof of items piq and piiq of Theorem A in several lemmas and propositions.
We start facing the dynamics of the center points of the tiles, or in other words, the dynamics among the beads of each necklace, that as we will prove will be invariant under the map F.
The proof for i " c`2, c`3, . . . , 2c`1 is done in a similar way, and also for the rest of values of i " 2c`2, . . . , 4c`2, but taking into account that in these cases F pX i q " F´pX i q.
As a consequence of Lemma 5 the center points of a level set form an invariant set and we can prove that the function V π{2 defined in (3) is a first integral of F.
Proof of the first part of item piiq of Theorem A. We start proving that the function V " Consider a point px, yq P U, then px, yq P T k, for a certain k, and by definition we know that V px, yq " V pp k, q. From Lemma 5 we know that F pp k, q " pk ,¯ with V pp k, q " V ppk ,¯ q.
On the other hand, since each tile is entirely contained in H`zty " 0u or in H´, F pT k, q " F`pT k, q or F pT k, q " F´pT k, q. Since F`and F´are rotations (thus isometries) we get that F sends tiles to tiles. In particular F pT k, q " Tk ,¯ and hence V px, yq " V pp k, q " V ppk ,¯ q " V pF px, yqq.
Now we are able to describe the dynamics of the center points and, in particular, to prove that they are periodic.
Proposition 6. Every center p k, is a periodic point of F. Furthermore, setting V k, " c we have that when c is even (resp. odd), then p k, has period 2c`1 (resp. 4c`2).
Proof. Fix a level tV " cu with c P N. From Lemma 4 we know that on tV " cu there are 4c`2 different centers. From Lemma 5, we know that F sends centers to centers, that is, the set tX 1 , X 2 , . . . , X 4c`1 u is invariant by F. Hence, given a center X i 1 of the previous set we can study the sequence Since the orbit of every center has a finite number of elements and, since F is a bijective map and therefore the orbit of X i 1 can not be preperiodic, we get that X ip " X i 1 for a certain p, and therefore it is periodic.
Clearly the period must be less or equal to 4c`2.
From Lemma 5, the map F restricted to tX 1 , X 2 , . . . , X 4c`1 u is conjugate to the map Assume that c " 2k is an even number. Then 2kp " np8k`2q ô kp " np4k`1q. It implies that p is a multiple of 4k`1 " 2c`1. Since p ď 4c`2 we get that p " 2c`1 or p " 4c`2. But we observe that the orbit of X i only contains some points X j with j having the same parity of i. Hence, we get two different periodic orbits, each one of them of period 2c`1.
We introduce now the concept of itinerary map associated with a center.
Definition 7. Fix c P N and consider one of the centers of the tiles X j for some j " on whether c is even or it is odd, if we consider its itinerary of length p: I p " pi 1 , i 2 , . . . , i p q we have that F p pX j q " F ip˝Fi p´1˝¨¨¨˝F i 1 pX j q " X j . We denote this composition by I j " F ip˝Fi p´1˝¨¨¨˝F i 1 and we call it the itinerary map associated with X j .
For instance, if c " 2 then the center X 1 is 5-periodic and its orbit is This can be easily obtained using the formula (7) in Lemma 5 (see also Figure 4). Hence I 5 pX 1 q " p`,`,`,´,´q and its itinerary map is I 1 " F 2˝F 3 .
When c " 3, then using again formula (7) in Lemma 5 (see again Figure 4) we get: and hence the itinerary map of X 1 is I 1 " F 2˝F 2˝F 3˝F 2˝F 2˝F 3 .
Lemma 8. Fixed c P N, consider the centers X 1 , X 2 , . . . , X 4c`2 lying in the level set tV " cu. Then for all j " 1, 2, . . . , 4c`2, the itinerary map I j is a rotation centered at X j of order 4 if c is even (angle π{2), and of order 2 if c is odd (angle π). In particular X j is an isolated fixed point of I j .
Proof. We already know that I j pX j q " X j . We write If c " 2k, then by using Proposition 6 we have that X j is p2c`1q-periodic, hence, using also that A 4 " Id we obtain Hence I j is a rotation of order 4 centered at X j . Since it has a unique fixed point (as RankpA´Idq " 2) then the center of this rotation is X j .
If c " 2k`1, then X j has period 4c`2, hence using that A 2 " R π "´Id, we have: for a certain w j P R 2 . By using the same argument as before, I j is a rotation of order 2 centered at X j .
To end the technical results we establish the next lemma which ensures the all the points in a tile have the same itinerary of arbitrary length: Lemma 9. All the points in a given tile T k, have the same itinerary of length n P N for every n P N.
Proof. Fix n P N, and suppose that there exist two points p and q in T k, with different itinerary of length n and let j ď n´1 the first time that ApF j ppqq ‰ ApF j pqqq. That is p and q have the same itinerary of length j but F j ppq and F j pqq have different addresses. From Lemma 3 we have that all the points in the segment p q have the same itinerary of length j and therefore F j restricted to p q is continuous. Since F j ppq and F j pqq have different addresses it follows that there exists a point r P p q such that F j prq P LC 0 . A contradiction because since T k, is also convex, r P T k, and must be zero-free.
We can now prove item piq, that is, the zero-free points are exactly the points U " U π{2 , which belong to the tiles.
Proof of item piq of Theorem A. We have already noticed that the zero-free set is included in U π{2 . Now we are going to see that the boundaries of the tiles are formed by points which are not zero-free. Consider a point p " pk, yq where ď y ď `1 for a certain k, P Z.
Then p belongs to the right-boundary of T k´1, and to the left boundary of T k, . Consider also the segment rs where r " pk´1{2, yq and s " pk`1{2, yq. From Lemma 9 the itinerary of any length of r coincides with the itinerary of the same length of p k´1,l and the itinerary of any length of s coincides with the corresponding itinerary of p k,l . Since p k´1,l and p k,l have different infinite itineraries, there exists j such that I j prq " I j psq but ApF j prqq ‰ ApF j psq. Now from Lemma 3 it follows that there exists t P rs such that F j ptq P LC 0 . Clearly this point must be p.
If we consider a point which belongs to a horizontal boundary of two consecutive tiles, then its iterate belongs to a vertical boundary of two consecutive tiles and then we can apply the above result.
Continuation of the proof of item piiq of Theorem A. Consider the tile T k, . Let X j be its center. By Proposition 6 it is a p-periodic point, and by Lemma 9 all the points in the tile have the same itinerary of length p, hence F p | T k, " I j . Moreover, by Lemma 8, on each tile I j is a rotation centered at X j of the order established in the statement.
Assume that V k, " c with c an even number. We have already proved that each center X j in this level set has period 2c`1 (see Proposition 6). The points in the orbit of X j are points X i with i having the same parity of j (see Lemma 5). Hence we have two orbits, the first one formed by X 1 , X 3 , . . . , X 4c`1 and the second one formed by X 2 , X 4 , . . . , X 4c`2 , and therefore we know the period of the centers of the tiles.
The period of all the points of the tiles T k, but the centers is a consequence that we have proved that F p | T k, " I p k, , where I p k, is the itinerary map of p k, , which is a rotation of order 4, see again Lemma 8.
When V k, " c with c an odd number the proof is similar.
3.3 Proof of item piiiq of Theorem A: dynamics on the non zero-free set Following the notation introduced in Lemma 4, for any fixed energy level c of V, there are 4c`2 tiles with centers X 1 , X 2 , . . . , X 4c`2 . Let us denote T j to the tile with center X j .
Also, for a fixed energy level c, we denote by Q j the closed square formed by the tile T j and its boundary, that is Q j " T j Y BT j .
For each energy level c even, we will call the squares Q 1 , Q 3 , . . . , Q 4c`1 perfect squares because, as we will see, these closed squares evolve avoiding the discontinuity effects of F.
Clearly every edge of a square is also an edge of the consecutive square. The perfect squares are positioned as Figure 5 displays, the perfect squares being the red ones.  is a p8c`4q-periodic point.
(b) If c is an odd number, then when j is odd (resp. even) the two horizontal (resp. vertical) edges of Q j , without the vertices, are formed by p8c`4q-periodic points.
Prior to proving the result we stress the following fact: Remark 11. On every point in H`the map F " F`. Hence, for all j " 1, 2, . . . , 2c`1, we have that F pQ j q " F`pQ j q " Q i with i " j`c mod 4c`2 (see Lemmas 5 and 9).
Analogously, for i " 2c`3, . . . , 4c`1 we have F pQ j q " F´pQ j q " Q i with i " j`c mod 4c`2, since these squares are contained in H´. Observe, however, that the situation for the squares Q 2c`2 and Q 2c`4 is quite different because on the top edge of these two squares F " F`while in the rest of the square F " F´. In Figure 6, we display the position of the tiles corresponding with the centers X 1 , X 2c`1 , X 2c`2 and X 4c`2 with respect to the discontinuity line LC 0 . Figure 6: Position of the squares Q 1 , Q 2c`1 , Q 2c`2 and Q 4c`2 , together with its centers, for any level set V " c.
Proof of Theorem 10. Consider the squares Q 1 , Q 3 , . . . , Q 4c`1 for c even, that is, the perfect squares on this level. The only squares in this particular collection Q j with j odd which intersect y " 0 are Q 1 and Q 2c`1 . But from Remark 11 we know that F pQ j q " Q k with k " j`c mod 4c`2 for all j odd, including the cases with j " 1 and j " 2c`1. In particular this implies that this set of squares is invariant. In consequence, by continuity, the points in the boundary of Q j inherit the dynamics of the points in T j zX j and, therefore, they are periodic with period 4p2c`1q. Furthermore, F 2c`1ˇQ j is a rotation of order 4. Now assume that c is odd. We notice that the squares Q 1 , Q 3 , . . . , Q 4c`1 (resp. Q 2 , Q 4 , . . . , Q 4c`2 ) share every vertical (resp. horizontal) edge with an edge of a perfect square, which we already know is periodic. Hence we need to follow the dynamics of their horizontal (resp. vertical) edges or, in other words, the dynamics of r Q j " Q j zt its vertical edges, including the verticesu (resp. r Q j " Q j ztits horizontal edges, including the verticesu).
Since now X 1 , X 2 , . . . , X 4c`2 belong to the same periodic orbit, the set of corresponding squares contains r Q 2c`2 and r Q 4c`2 . The result will be proved if we can ensure the invariance of the set of squares r Q j . In order to do this, we must ensure that the edges we are studying are not pre-images of the top edges of the squares Q 2c`2 and Q 4c`2 . So, first, we study for which values of p, F p pX j q " X 2c`2 or F p pX j q " X 4c`2 .
• From Lemma 5 we have F p pX j q " X 2c`2 if and only if j`pc " 2c`2 mod 4c`2, that is if there exists n P N such that j`pc " 2c`2`np4c`2q. Hence j`pc is an even number and since c is odd we get that p and j have the same parity.
• Analogously, F p pX j q " X 4c`2 if and only if j`pc " 0 mod 4c`2, which means that there exists n P N such that j`pc " np4c`2q. As before p and j have the same parity.
Assume that j is odd, and let the p-iterate of Q j be the first one that reaches Q 2c`2 (or Q 4c`2 ). Since it is the first time that the images of Q j intersect ty " 0u, we can still apply the arguments in the proof of Lemma 8 and therefore F p | Q j is an even-order rotation. Thus, since p is also odd, the horizontal edges of Q j are mapped via F p to the vertical edges of Q 2c`2 (or Q 4c`2 ). Therefore F p p r Q j q " r Q 2c`2 (or r Q 4c`2 ). It implies that for all j odd, The same arguments work when j is even: p is even too and F p sends the vertical edges of Q j to the vertical edges of Q 2c`2 (or Q 4c`2 ). So also F p r Q j q " r Q j`c for every j even.
The condition F p r Q j q " r Q j`c implies that, by continuity, the points in the edges under study of Q j inherit the dynamics of the points in T j zX j , which are periodic with period 2p4c`1q.
Consider the square Q k, " T k, Y BT k, , and set c " V k, ; then, we will say that the square has odd label (resp. even label) if Q k, " Q j for an odd value j (resp. even) in the order introduced in Lemma 4. Observe that, in particular, with the proof of Theorem 10 we also have proved the following result that gives the dynamics of F on all the points in F.

Corollary 12.
Consider the square Q k, " T k, Y BT k, , and set c " V k, , then: (a) If c is even, and the square has odd label then Q k, is invariant under the action of the map F 2c`1 , and F 2c`1ˇQ k, is a rotation of order 4 centered at p k, ; as a consequence the edges of these Q k, are formed by 4p2c`1q´periodic points.
(b) If c is odd, and the square has odd label (resp. even label) then the horizontal (resp. vertical) edges (excluding the vertices) are invariant under the action of the map F 4c`2 which is also a rotation of order 2 centered at p k, on that edges (excluding the vertices); as a consequence the edges of these squares which are not edges of a perfect square are formed by 2p4c`2q´periodic points.
Remember that if the square has even energy level and even label we treat their boundaries as being part of the boundary of the adjacent odd-energy level tile.

Proof of item pivq of Theorem A
The proof simply follows by collecting the results of the previous items.

Proof of Theorem B 4.1 Preliminaries
For each tile T k, ,m , we start determining m in terms of k and .
Lemma 13. Any point px, yq P U " U 2π{3 belongs to a tile T k, ,m where either, m " k` or m " k` `1 or m " k` `2.
In Figure 7, we can see that the parallelogram has been partitioned into three sets: two triangles and one hexagon. Thus, we have proved the following lemma, where we use the notation introduced in Section 2.2.
Lemma 14. Let T k, ,m be one tile of U 2π{3 " U.
(a) If m " k` or m " k` `2, then T k, ,m is a triangle whose center is either q k, or r k, , respectively.
(b) If m " k` `1 then T k, ,m is a hexagon whose center is p k, .

Proof of items piq and piiq: dynamics on the zero-free set
As in the case α " π{2 we split the proof of these two items into several lemmas and propositions. Here there is an added difficulty, there are tiles with hexagonal shape and others with triangular shape. We study them separately.

Dynamics on the hexagonal tiles. The case m " k` `1
From Lemma 14 the tile T k, ,k` `1 is a regular hexagon. Set V k, for the value of V on T k, ,k` `1 . Then V k, " V k, ,k` `1 " max p|2k`1|, 2|k` `1|´1, |2 `1|q . The next two results characterize the set of centroids of the hexagons, that is their number and geometric locus, for this case.
(b) The set tp k, : V k, " cu has 3c`1 points. In particular there are 3c`1 hexagons T k, ,k` `1 in this energy level.
(c) The points p k, with V k, " c lie in the irregular hexagon determined by the intersection of the straight lines y " ? 3px˘cq, y "˘?3c{2 and y " ? 3p´x˘p1`cqq, see Figure   8. Proof. Since V k, depends on the signs of 2k`1, 2 `1 and k` `1, we are going to consider the different cases.
The Lemma follows from the above case-by-case study.
Consider an odd energy level V k, " c; we will label the center points p k, analogously as in the case α " π{2: we denote by X 1 the point on the corresponding irregular hexagon defined by the lines in the above lemma, which belongs to H`and its first component is the smallest one; that is, X 1 " p´c`1{2, ? 3{2q. After we denote by X 2 , X 3 , . . . X 3c`1 the consecutive points on the hexagon turning clockwise (see Figure 8 for instance). The set of center points in such a level set is invariant under the action of F and its dynamics is given in the next result: Proposition 16. Assume m " k` `1. Fixed V k, " c an odd number, consider the points X 1 , X 2 , . . . , X 3c`1 introduced above. Then (a) For all i " 1, 2, . . . , 3c`1 , F pX i q " X j with j " i`c mod p3c`1q.
Proof. To prove statement paq we are going to consider the points that are on each of the six sides of the irregular hexagon delimited by the straight lines in Lemma 15.
Since the distance between two consecutive points is constant and F`is an isometry we get that X 2 , X 3 , . . . , X pc`1q{2 are mapped to X c`2 , X c`3 , . . . , X p3c`1q{2 respectively.
In order to prove pbq we proceed as in the proof of Proposition 6. We use that the map F restricted to tX 1 , X 2 , . . . , X 3c`1 u is conjugated to h : Z 3c`1 ÝÑ Z 3c`1 defined by hpiq " i`c.
Then F p pX i q " X i ô h p piq " i ô i`cp " i mod p3c`1q ô Dn P N : cp " np3c`1q.
This implies that p must be a multiple of 3c`1, and since p ď 3c`1 we get that the minimal period is p " 3c`1 as we wanted to see.

4.2.2
Dynamics on the triangular tiles. The cases m " k` and m " k` `2 For the triangular tiles, a result analogous to Lemma 15 is the following. We omit all the details.
(b) The set tq k, : V pq k, q " cu has 3c`1 elements. In particular there are 3c`1 triangles T k, ,k` in this energy level.
(c) The points q k, : V pq k, q " c lie in the irregular hexagon determined by the intersection of the six straight lines y " ? 3px`c´1{3q, y " ? 3p3c`1q{6, y " ?
(b) The set tr k, : V pr k, q " cu has 3c`1 elements. In particular there are 3c`1 triangles T k, ,k` in this energy level.
(c) The points r k, : V pr k, q " c lie in the irregular hexagon determined by the intersection of the six straight lines y " ? 3px´c`1{3q, y " ? 3p3c´1q{6, y " ?
The proof follows exactly by the same arguments involved in the proofs of Lemma 5 and Proposition 6. The next corollary simply consists of gluing (in a suitable way) the two sets given in the previous proposition, to form a single necklace with 6c`2 triangular beads.
Proof of item piq of Theorem B. Following the spirit of Definition 7, we can introduce the concept of itinerary map for the centers p k, , r k, and q k, in an analogous way. Then, the proof is exactly the same proof as for item piq of Theorem A. It is based on the fact that all the points in the same tile have the same itineraries of arbitrary length (a result analogous to Lemma 9) and also on Lemma 3.
Proof of item piiq of Theorem B. We start proving that V " V 3π{2 is a first integral. As in the proof of item piiq of Theorem A, we notice that since the tiles are completely contained in H`zty " 0u or H´and the maps F˘are rotations, then F sends tiles to tiles. Remember that by its definition V is constant on each tile, and in particular takes the value attained at the center point. The result follows now from the fact that in each level set, the set of centers is invariant, see Propositions 16 and 18.
Similarly that in the proof of Theorem A we consider the tile T k, ,m . We know that all the points in the tile have the same itinerary than its center which, by Propositions 16,18 and Corollary 19 give the discrete dynamical systems generated by the functions h given in the statement of Theorem B between the corresponding Z M . Moreover, we know that the centers are periodic with period 3c`1. Hence, if I is the itinerary map associated with the center point, that is I " F 3c`1ˇT k, ,m , we have that IpT k, ,m q " T k, ,m . Writing F px, yq " A¨px´signpyq, yq t where A " R 2π{3 , we have I " A 3c`1`v " A`v for a certain v P R 2 , v ‰ 0, which implies that I is a rotation with a unique fixed point, hence it is the center point. Furthermore I 3 " Id because I 3 " A 3`p A 2`A`I dqv " Id, since A 2`A`I d " 0. In summary, I is a rotation of angle α " 2π{3 which implies that the points px, yq P T k, ,m which are not centers are 3-periodic for F 3c`1 and consequently, they are p9c`3q-periodic.

Proof of item piiiq of Theorem B: dynamics in the non zero-free set
From the previous results, we know that the non zero-free set F is formed by the borders of the tiles, both hexagons and triangles.
Consider an energy level c P N 0 . Assume that c is an odd number, then the level set tV " cu is formed by 3c`1 hexagonal tiles, whose centers X 1 , X 2 , . . . , X 3c`1 form a periodic orbit. Denoting by H i the closure of this hexagon we also know that H 1 and H 2c´1 intersect y " 0 at the bottom edge while H 2c and H 3c`1 intersect y " 0 at the top edge. When c is even, we have the points Y 1 , Y 2 , . . . , Y 3c`1 (respectively, Z 1 , Z 2 , . . . , Z 3c`1 ). Each Y i (resp. Z i ) is the center of an upward (resp. downward) facing triangle; its closure intersects y " 0 only when i " 1 and i " p3c`2q{2 (resp. i " p3c`2q{2 and i " 3c`1), see Figure 10. We are going to call perfect triangles the ones corresponding to Y 1 , Y 2 , . . . , Y 3c`1 . As for perfect squares, we will prove that these figures will evolve avoiding the discontinuity of F.
They are positioned as the Figure 11 shows, the perfect triangles being the red ones, which are precisely the ones pointing upwards. The blue ones correspond to Z 1 , Z 2 , . . . , Z 3c`1 . Proof of item piiiq of Theorem B. First, we observe that the borders of the perfect triangles (including the vertices) have the same period as the interior points which are not centers.
Indeed, set an even number c P N 0 and denote by T i the closed triangle (i.e. including the boundary with vertices) which contain the point Y i . For i ‰ 1, 3c{2`1, the triangle T i does not intersect y " 0, hence F pT i q " T i`c . For i " 1 or i " 3c{2`1, F pT i q " F`pT i q which also is T i`c . Therefore, by continuity, the points in the boundary of T i are periodic with period 3p3c`1q " 9c`3, as for the points in the interior of T i . Now take c odd and let H i be the closed hexagon which contains X i in its interior.
Looking at Figure 11 we see that H i has three edges which also are edges of a perfect triangle; if we call these edges the perfect edges, we consider r H i " H i zperfect edges. (the motivation for this name is similar that the ones of perfect triangles, or squares, and will be apparent later It remains to consider the edges of the triangles which are not perfect. But, as can be seen in Figure 11, all these edges are also the edges of the contiguous hexagons, which we have already proved that all of them are periodic. Observe also that all the vertices belong to perfect triangles.

Proof of item pivq of Theorem B
As in the case α " π{2, the proof follows by replacing the value of V by 2n or 2n`1, for n P N 0 , in the results of the previous items. We re-obtain the results of [10].
5 Proof of Theorem C

Preliminaries
As in the case studied in the previous section, for each tile T k, ,m , the values k, , m are not independent. Here, either m " k` ´1 or m " k` or m " k` `1.
Lemma 20. Let T k, ,m be one tile of U π{3 " U.
(a) If m " k` then T k, ,m is a hexagon whose center is p k, "`2k` `1{2, ?
(b) If m " k` ´1 or m " k` `1, then T k, ,m is a triangle whose center is either q k, "`2k` ´1{2, ?

Proof of items piq and piiq of Theorem C: dynamics on the zero-free set
These results can be proved by the same arguments that we have used in the proofs of Theorems A and B, in Sections 3 and 4. Although we will not give all the details of their proofs, we want to highlight the main features and results that allow to give the dynamics in this case.
Consider an even number c. Then, by Lemma 20, the tiles on the level set V " c are hexagons whose centers are some of the points p k, for some k, . It can be proved that there are 3c`2 centers in this level. This centers lie in certain hexagons. We denote them by tX 1 , X 2 , . . . , X 3c`2 u labeling them as in the case α " 2π{3, see the Figure 12. In this case F restricted to tX 1 , X 2 , . . . , X 3c`2 u is conjugated to From this equality, and using that 3c "´2 mod 3c`2, one easily gets that when c{2 is even then the minimal period is 3c{2`1, and that when c{2 is odd, then the minimal period is 3c`2. In the first case we get two periodic orbits tX 1 , X 3 , . . . , X 3c`1 u and tX 2 , X 4 , . . . , X 3c`2 u while in the second one all the points X i , i " 1, 2, . . . , 3c`1 belong to the same periodic orbit.
To study the periodicity of the points in the hexagonal tile different from its center, for each X j we consider its itinerary map I j .
• When c{2 is even, I j has the form I j " A 3c{2`1`v (where v " X j´A 3c{2`1 X j ) and 3c{2`1 " 3¨2n`1 " 6n`1 for some n P N 0 . Hence, since A 6 " Id, it holds that I j " A`v. Therefore I j restricted to the hexagon which contains X j , is a rotation of angle π{3 centered at X j and every point in the hexagon is a 6-periodic point for I j .
It implies that these points are 6p3c{2`1q " 9c`6 periodic points for F.
• When c{2 is odd, I j " A 3c`2`v and 3c`2 " 3¨2p2n`1q`2 " 6p2n`1q`2 for some n P N 0 . Thus I j " A 2`v , using again that A 6 " Id . This implies that every point in the hexagon is a 3p3c`2q " 9c`6 periodic point for F. Now let c be an odd number. Then the tiles in tV " cu are triangles whose centers are either the points q k, or the points r k, introduced in Lemma 20, for some k, P Z. The points q k, lie in some lines that define a hexagon, as do the points r k, . But now all these centers belong to the same periodic orbit. To prove this, as usual, we label these points in a clockwise direction, as Figure 13 shows for the case c " 3: the red points are the points q k, P tV " 3u while the blue ones are r k, P tV " 3u. Figure 13: Position of the centers in the level c " 3. Observe that all the points belong to the same orbit. As shown in Section 6, the lines joining the centers play a role in the determination of the first integral V and do not represent two different orbits.
With this labeling it can be proved that F pX i q " X j with j " i`c (mod 6c`4) which implies that the minimal period is p " 6c`4. To see the periodicity of the points in the triangles different from its center we consider the itinerary function of the center X j which has the form I j " A 4`v . Then I 3 j " A 12`p A 4`A2`I dqv " Id . Arguing as before we get that each point in the triangle different from its center is a 3p6c`4q " 18c`12 periodic point.

Proof of item (iii) of Theorem C: dynamics on the non zero-free set
In this case, the dynamics of the points on the edges and vertices of the tiles is more complicated than the ones found in the cases α " π{2 and α " 2π{3, so we are going to give the details.

Perfect edges and vertices
We begin by considering the levels c " 4k, k P N. We already know that in these levels there are 3c`2 " 12k`2 centers, X 1 , X 2 , . . . , X 12k`2 and F pX i q " X j where j " i`2k mod p12k`2q. Also X 1 , X 3 , . . . , X 12k`1 form a periodic orbit of period 6k`1, as does X 2 , X 4 , . . . X 12k`2 . Let H j be the hexagon such that X j P H j including its boundary (hence also its vertices). Then the hexagons that meet y " 0 are H 1 , H 6k`1 (its bottom edge is contained in y " 0) and H 6k`2 , H 12k`2 (its top edge is contained in y " 0). See Figure 14. Clearly for all j " 1, 2, . . . , 6k`1, F pH j q " F`pH j q " H j`2k . while for j " 6k`3, 6k4 , . . . , 12k`1 also F pH j q " F´pH j q " H j`2k mod p12k`2q . But the hexagons H 6k`2 , H 12k`2 do not satisfy this property because on the top edge of these hexagons F " F`. Then we easily get: Lemma 21. Assume that c " 4k and consider the (closed) hexagons H 1 , H 3 , . . . , H 12k`1 .
Then for all j " 1, 3, . . . , 12k`1 every point in H j different from its center is periodic of period 36k`6.
Proof. For j " 1, 3, . . . , 12k`1, the hexagons satisfy F pH j q " H j`2k mod p12k`2q , hence it is easy to observe that their images are never the hexagons H 6k`2 and H 12k`2 . Then, by continuity, every point on the boundary of H j has the same periodicity as the points inside the hexagon (except the center). In particular, H 1 , H 3 , . . . , H 12k`1 form an invariant set.
As in the above sections we call H 1 , H 3 , . . . , H 12k`1 perfect hexagons and their edges and vertices behave as the corresponding interior points, apart from the centers, that is they are p36k`6q´periodic. Also we will call non-perfect edges or vertices those which do not collide with a perfect hexagon.

Non-perfect edges
We continue the study considering the even levels of the form c " 4k`2. We know that in these level sets there are 3c`2 " 12k`8 centers and that all of them belong to the same periodic orbit. The hexagons which meet y " 0 are H 1 , H 6k`4 , H 6k`5 and H 12k`8 , see  We are going to follow the dynamics of the interior of the bottom edge of H 1 , that we will denote as L (for simplicity we will use the term edge although the two boundary points are not included). This dynamics is, by far, the most complex of those we have studied in this paper. Since the argument is long, we first briefly summarize it: we will show that every point in L is p108k`72q-periodic. The edge is rigidly mapped by by iterating F into the edges of the hexagons in the level 4k`2, but also into the edges of the triangles in the levels 4k`1 and 4k`3.
Indeed, after the first iteration, F pLq is the edge of H 2k`2 obtained after rotating L an angle equal to π{3, because F pX 1 q " X 2k`2 (remember that from (9), F pX i q " X i`2k`1 mod p12k`8q .q We continue iterating until we find the hexagon H 6k`5 . To compute how many iterations we need for X 1 to reach X 6k`5 we ask for the minimal positive number p such that F p pX 1 q " X 6k`5 . That is, F p pX 1 q " X 6k`5 , or equivalently, 1`pp2k`1q " 6k`5 mod p12k`8q. Thus, p " p2k`1q´1p6k`4q " p6k`1qp6k`4q " 36k 2`3 0k`4 " 6k`4 mod p12k`8q.
That is F 6k`4 pX 1 q " X 6k`5 . Now F 6k`4 pLq is the edge of H 6k`5 after rotating L an angle equal to 4π 3 . Hence we follow iterating until we arrive to X 12k`8 , that is, three iterates more: F 3 pX 6k`5 q " X 12k`8 . This implies that F 6k`7 pLq is the edge of H 12k`8 obtained after rotating L an angle equal to π{3. Now we ask for the minimal p such that F p pX 12k`8 q " X 6k`5 . The computation gives that p " 12k`5. Hence we can write and following the edge L we have that after 18k`15 iterates the initial edge L of H 1 is the top edge of H 12k`8 , which is the bottom edge of the triangle T 1 in the level 4k`3.
Next, we follow the orbit of the centers of the triangles in the level c " 4k`3. Let Y 1 , Y 2 , . . . , Y 24k`22 be the centers of the triangles T 1 , T 2 , . . . , T 24k`22 . All of them form a unique periodic orbit and F pY i q " Y j where j " i`4k`3 mod p24k`22q, remember that in the triangles F pY i q " Y i`c mod p6c`4q . The triangles with edges in the critical line are displayed in the Figure 17. Taking into account that p4k`3q´1 " 18k`15 in Z 24k`22 and solving the corresponding congruences we find that: Then we see that the bottom edge of T 1 is transformed into the top edge of T 12k`12 after 36k`33 iterates. This one also is the bottom edge of the hexagon H 6k`4 in the level c " 4k`2, see again Figure 15, and also Figure 3.
Following the same procedure it can be seen that F 6k`1 pX 6k`4 q " X 6k`5 and using the calculations made before we obtain Hence, the bottom edge of H 6k`4 is transformed into the top edge of H 6k`5 after 18k`9 iterates.
The top edge of H 6k`5 is also the bottom edge of one triangle whose center belongs to the level set 4k`1. In this level set there are 24k`10 centers of triangles, that we denote by Z 1 , Z 2 , . . . , Z 24k`10 , and we know that F pZ i q " Z j with j " i`4k`1 mod p24k`10q.
We call T 1 , T 2 , . . . , T 24k`10 these triangles. Specifically, the top edge of H 6k`5 is the bottom edge of T 12k`5 , see  Using that in Z 24k`10 , p4k`1q´1 " 6k`1 and solving the corresponding congruences we have that It follows that after 36k`15 iterates, the bottom edge of T 12k`5 is transformed in the top edge of T 24k`10 .
But this top edge of T 24k`10 is exactly the edge L. Hence summing up the involved iterates we have that every point in L is a 108k`72 periodic point. Also the same holds for all the points belonging to the 108k`72 edges obtained iterating L. In other words, we get a periodic orbit of edges of period 108k`72 and, of course, the points of L are mapped to themselves after these iterations.

Non-perfect vertices
And what about the vertices? As we will see in the proof of Theorem 22, we only need to prove the periodicity of the vertices in y " 0. Observe that if such a vertex belongs to a perfect hexagon, then we already know that it is periodic with the same period as the interior points. If it is non-perfect, then either paq it is mapped to a vertex colliding from the top with a triangle of level V " 4k`1 and a hexagon of level V " 4k`2, both in H`, as the solid-circle point in Figure 19; or (b) it collides from the top with a hexagon of level V " 4k`2 and triangle of level V " 4k`3, both in H`, as the box-shaped point in Figure 19. To study the dynamics of the non-perfect vertices in y " 0, we will use the following notation: given a hexagon H, we label its vertices as V i pHq with i " 1,¨¨¨, 6 starting from the left-bottom vertex and in clockwise sense, see Figure 20. Let H 1 be the non-perfect hexagon at level V " 4k`2 in Q 1 , whose intersection with y " 0 is its bottom edge. Then: paq We will follow the orbit of the point V 6 pH 1 q (the blue point in Figure 19) by using the results found in Section 5.3.2. In particular, we know that F 3 pH 1 q " H 6k`4 , easily find hence the point V 6 pH 1 q is p18k`9q-periodic.
(b) We will pursue the orbit of the point V 1 pH 1 q (the box-shaped point in Figure 19).
hence the point V 1 pH 1 q is p18k`15q-periodic.  (c) If px, yq is a non-perfect vertex then it is periodic with period 18k`9 or 18k`15 for some k P N 0 .
Observe that any non zero-free point belongs to one of the above cases.
Proof. We already know that the set of the non zero-free points is formed by the edges and the vertices of the hexagons and triangles introduced before.
Consider the points in one edge. Then after a finite number of iterates this edge is transformed in one edge contained in y " 0. If this edge correspond to an edge of a perfect hexagon with center belonging to the level 4k, every point will be 6p6k`1q´periodic. If not, it will be an edge of a polygon with center belonging to the level either 4k`1, 4k`2 or 4k`3. From the discussion above we know that every point will be p108k`72q´periodic.
With respect to the vertices, observe that since any vertex belongs to F, after some iterates it will be mapped to a vertex point in y " 0. Hence there are three possibilities: it is mapped to a perfect vertex of a perfect hexagon in H`, which has energy level V " 4k (and in this case it is periodic of period 36k`6); or it is mapped to a vertex colliding from the top with a triangle of level V " 4k`1 and a hexagon of level V " 4k`2 (and in this case it is periodic with period 18k`9); or it is mapped to a vertex colliding from the top with a hexagon of level V " 4k`2 and a triangle of level V " 4k`3. In this case it is periodic with period 18k`15.
The proof of item pivq is a straightforward consequence of all the previous results.

Obtaining the first integrals
We have intuited the expressions of the first integrals after several simulations. For completeness we present in detail a three-step constructive procedure that allows to obtain the first integrals of F given in (3) corresponding to α " π{2. For the other two cases the line of argument is the same, and the details are analogous, and we only give some comments.
Step 1. By displaying some preimages of the critical line LC´i we realize that the zero-free set is formed by open tiles of a regular or uniform tessellation of R 2 . This fact is trivial in this case where α " π{2 but, a priori, it was not so obvious in the cases α " 2π{3 and α " π{3 studied in Sections 4 and 5. The normal form of F given in (2) regularizes the tesselation.
Step 2. From preliminary numerical explorations we also realize that the centers of some tiles form an invariant set under the dynamics of F. In the case α " π{2 these centers were located in the lines y " x`c, y "´x`c`1, y " x´c and y "´x´c´1 for a certain fixed value c P N 0 , depending on the quadrant where the center points are located (see Lemma 4 and Figure 4, and see Lemmas 15 and 17 for the case α " 2π{3).
Step 3. Isolating the value in the expression of the lines linking the centers we obtain that c " y´x for px, yq P Q 1 , c " y`x´1 for px, yq P Q 2 , c " x´y for px, yq P Q 3 and c "´x´y´1 for px, yq P Q 4 , where recall that Q j , j " 1, 2, 3, 4, are the four quadrants of R 2 . From these expressions and taking into account that c P N 0 and that given a zero-free point px, yq the center point of its associated tile is pEpxq`1{2, Epyq`1{2q, we arrive to the expression of the first integral V π{2 px, yq " max p|Epxq`Epyq`1|´1, |Epxq´Epyq|q .

Final comments.
We have proved that for α P tπ{3, π{2, 2π{3u, the corresponding zero-free sets U are the union of a countable number of open sets (the tiles), hence the associated critical sets F " R 2 zU are closed sets. In consequence, for any point px, yq P R 2 , the distance dist ppx, yq, Fq is well defined. Since F is also invariant, we have: Remark 23. Any map (2) with α P tπ{3, π{2, 2π{3u has the non-quantized continuous first integral W px, yq " dist ppx, yq, Fq.
We believe that the only pointwise periodic cases for the maps F with α P p0, 2πq, are the ones studied in this work as well the cases α P t4π{3, 3π{2, 5π{3u (recall that we where motivated by the study of the maps G in (1) with |ρ| ă 2, which are conjugated with the maps F in (2) with α P p0, πq). In these later cases we have observed that the quantized first integrals given in this paper are also integrals of these maps. In particular: V π{2 , V 2π{3 and V π{3 are first integrals of F when α " 3π{2, 5π{3 and 4π{3 respectively. However notice that none of these last maps are conjugated to the maps considered in this work: for example, note that the maps F with α P t4π{3, 3π{2, 5π{3u do not have fixed points since the centers of the rotations are virtual.
The maps F belong to the class of symmetric maps studied in the relevant paper [14].
We refer the reader to this reference to learn about the general properties of the maps F with α a general value in r0, 2πqztπ{3, π{2, 2π{3, 4π{3, 3π{2, 5π{3u. For instance, in that paper it is proved that for any α ‰˘π being a rational multiple of π there exists a sequence of open invariant nested necklaces, that tend to infinity, each one of them being similar to the level sets of our quantized first integrals, whose beads are polygons, and where the dynamics of F is given by a product of two rotations. Remarkably, although the adherence of the union of all these invariant necklaces does not fill the full plane, it allows to prove that all orbits of F are bounded.