Pentagonal Domain Exchange

Self-inducing structure of pentagonal piecewise isometry is applied to show detailed description of periodic and aperiodic orbits, and further dynamical properties. A Pisot number appears as a scaling constant and plays a crucial role in the proof. Further generalization is discussed in the last section.

Adler-Kitchens-Tresser [1] and Goetz [10] initiated the study of piecewise isometries. This class of maps shows the way to possible generalizations of results on interval exchanges to higher dimensions [16,30]. In this paper we examine the detailed properties of the map shown in Figure 1 from an algebraic point of view. Figure 1. A piecewise rotation T on two pieces. The triangle is rotated 2π/5 around a and the trapezium is rotated 2π/5 around b. Periodic points with short periods are shown below, in two colours to illustrate that they cluster into groups, each forming a pentagon.
The goal of this paper is to see how this map is applied to show number theoretical results. First we reprove that almost all orbits in the sense of The first author is supported by the Japanese Society for the Promotion of Science (JSPS), grant in aid 21540010.
Lebesgue measure are periodic, and in addition, there are explicit aperiodic points. Second we show that aperiodic points forms a proper dense subset of an attractor of some iterated function system and are recognized by a Büchi automaton (c.f. Figure 14). The dynamics acting on this set of aperiodic points are conjugate to the 2-adic odometer (addition of one) whose explicit construction is given (Theorem 3). As a result, we easily see that all aperiodic orbits are dense and uniformly distributed in the attractor. We finally give a characterization of points which have purely periodic multiplicative coding by constructing its natural extension (Theorem 6). In doing so we obtain an intriguing picture Figure 16 that emerges naturally from taking algebraic conjugates, whose structure is worthy of further study. We discuss possible generalizations for 7-fold and 9-fold piecewise rotations in Section 3.
A dynamical system is self-inducing if the first return map to some subset has the same dynamics as the full map. The most important example is the irrational rotation, presented as exchange of two intervals. An elementary example begins with Φ shown in Figure 2. For this interval exchange, now consider the second interval B. As shown in Figure 3 this interval is translated to the left once, and to the right. Thus Φ 2 (B 1 ) is back in B, the interval B 2 requires one more step, but Φ 3 (B 2 ) also lies within B. This first return dynamics on B is therefore conjugate to the dynamics on A∪ B. Selfinducing subsystem of two interval exchange corresponds to purely periodic orbits of continued fraction expansion and they are efficiently captured by the continued fraction algorithm.
This gives a motivation to study the interval exchange transform (IET) of three or more pieces, trying to find higher dimensional continued fraction with good Diophantine approximation properties. The study of self-inducing structure of IET's was started by a pioneer work of Rauzy [26], now called Rauzy induction, and got extended in a great deal by many authors including Veech [31] and Zorich [33], see [32] for historical developments.
Self-inducing piecewise isometries emerged from dynamical systems as a natural generalization of IET [21,1,5,11,12,8,22] and the first return dynamics appears in outer billiards [28,6]. Like IET they provide a simple setting to study many of the deep and perplexing behaviors that can emerge from a dynamical system.
The self-inducing structure links such dynamical systems to number theoretical algorithms, such as, digital expansions and Diophantine approximation algorithms, and allows us to study their periodic orbits by constructing their natural extensions. This idea leads to complex and beautiful fractal behavior.
Our target is the piecewise isometry in Figure 1, but to illustrate the bridge formed between the two fields let us begin with a simple conjecture from number theory: Conjecture 1. For any −2 < λ < 2, each integer sequence defined by 0 ≤ a n+1 + λa n + a n−1 < 1 is periodic.
Since a n+2 ∈ Z is uniquely determined by (a n , a n+1 ) ∈ Z 2 , we treat this recurrence as a map (a n , a n+1 ) → (a n+1 , a n+2 ) acting on Z 2 . It is natural to set λ = −2 cos(θ) to view this map as a 'discretized rotation': with eigenvalues exp(± √ −1θ). As the matrix is conjugate to the planar rotation matrix of angle θ, putting P = 1 0 cos θ − sin θ , we have P a n+1 a n+2 = cos θ − sin θ sin θ cos θ P a n a n+1 + P 0 λa n+1 where x is the fractional part of x. Therefore this gives a rotation map of angle θ acting on a lattice P Z 2 but the image requires a bounded perturbation of modulus less than two to fit into lattice points of P Z 2 . For conjecture 1 we expect that such perturbations do not cumulate and the orbits stay bounded, equivalently, all orbits become periodic.
A nice feature of the map (a n , a n+1 ) → (a n+1 , a n+2 ) is that it is clearly bijective on Z 2 by symmetry, while under the usual round off scheme, the digital information should be more or less lost by the irrational rotation. This motivates dynamical study of global stability of this algorithm.
The conjecture is trivial when λ = 0, ±1. Among non-trivial cases, the second tractable case is when θ is rational and λ is quadratic over Q. Akiyama, Brunotte, Pethő and Steiner [3] proved: It seems hard to prove Conjecture 1 for other values. The case λ =  [2], whose proof is short but not so easy to generalize. We try to give an accessible account using self-inducing piecewise isometry in the case λ = ω =

1+
√ 5 2 , together with its further dynamical behavior. The proof in Section 1 is basically in [3]. However this version may elucidate the background idea and is directly connected to the scaling constant of self-inducing structure of piecewise isometry acting on a lozenge.
A Pisot number is an algebraic integer > 1 whose conjugates have modulus less than 1. Throughout the paper, we will see the importance of the fact that the scaling constant of self-inducing system is a Pisot number. Our all discussions heavily depend on this fact. Indeed, Pisot scaling constants often appear in self-inducing structures of several important dynamical systems, for e.g., IET and substitutive dynamical systems. We discuss this point in Section 3. It is pretty surprising that we see this phenomenon in cubic piecewise rotations as well. We hope this paper gives an easy way to access this interesting area of mathematics.
We wish to show our gratitude to P.Hubert, W.Steiner and F.Vivaldi for helpful comments and relevant literatures in the development of this manuscript.
Proof. From ω = −ζ 2 − ζ −2 , we have On the other hand Taking the complex conjugate, the same statement is valid with another basis 1, ω, ζ −1 , ωζ −1 . Thus each element in Z[ζ] has a unique expression: Denote by x the fractional part of x ∈ R. Then a small computation gives 0 ≤ a n + ωa n+1 + a n+2 < 1 a n + ωa n+1 + a n+2 = ωa n+1 and x n = ωa n . Our problem is therefore embedded into a piecewise isometry T acting on a lozenge [0, 1) + (−ζ −1 )[0, 1): The action of T is geometrically described in Figure 1. The lozenge L = [0, 1) + (−ζ −1 )[0, 1) is rotated by the multiplication of −ζ −1 and then the trapezoid Z which falls outside L is pulled back in by adding −ζ −1 . In total, the isosceles triangle ∆ is rotated clockwise by the angle 3π/5 around the origin and the trapezoid Z is rotated by the same angle but around the point 1 2 + i √ 5(5+2 √ 5) 10 ≃ 0.5 + 0.6882i indicated by a black spot, that is the intersection of two diagonals. Our aim is to show that each point x ∈ Z[ζ] ∩ L gives a periodic T -orbit.
Clearly the map T is bijective and preserves 2-dimensional Lebesgue measure µ. However the measure dynamical system (L, ν, B, T ) (with the σalgebra B of Lebesgue measurable sets) is far from ergodic. It turned out that orbits of T is periodic for almost all points but for an exceptional set of Lebesgue measure zero. Our goal is to prove that the set Z[ζ] has no intersection with this exceptional set. This is not so obvious since Z[ζ] is dense in L because Z[ω] is dense in R.
To illustrate the situation, it is instructive to describe an orbit of 1/3. See Figure 4. Later we will show that the orbit of 1/3 is aperiodic and forms a dense subset of the exceptional set of aperiodic points. Roughly speaking, our task is to show that Z[ζ] ∩ L has no intersection with the fractal set appeared Figure 4.
The key to the proof is a self-inducing structure with a scaling constant ω 2 . We consider a region L ′ = ω −2 L and consider the first return map For any x ∈ L ′ , the value m(x) = 1, 3 or 6. We can show that for x ∈ L. The proof is geometric, shown in Figure  Note that the equation is valid for all x ∈ L ′ . This makes the later discussion very simple. Unfortunately this is not the case for other quadratic values of γ and we have to study the behavior of the boundary independently, see [3].
Let U be the 1-st hitting map to L ′ for x ∈ L, i.e., U (x) = T m(x) (x) for the minimum non-negative integer m(x) such that T m(x) (x) ∈ L ′ . Note that U is a partial function, i.e., U (x) is not defined when there is no positive integer m such that T m (x) ∈ L ′ . Since it is easy to make the map U explicit: where P 0 is the largest open pentagon and P 1 and P 2 = P 1 /ζ are two second largest closed pentagons in Figure 6.
to use later. We introduce a crucial map S which is the composition of the 1-st hitting map U and expansion by ω 2 , i.e. S(x) = ω 2 U (x). Denote by π(x) the period of T -orbits of x ∈ L and put π(x) = ∞ if x is not periodic by T . (We easily see π(x) = 5 in P 0 and π(x) = 10 in P 1 ∪ P 2 unless x is the centroid of the pentagon.) Then if π(x) and π(S(x)) are defined and finite, then we see that π(S(x)) < π(x) which is a consequence of Equation (1). Therefore if π(x) is finite then we have a decreasing sequence of positive integers. This shows that there exists a positive integer k such that S k (x) is not defined. In this case we say that S-orbit of x in finite. We easily see that if S-orbit of x ∈ L is finite, then clearly π(x) is finite by Equation (1). Thus we have a clear distinction: By the assumption S k (x) is defined for k = 1, 2, . . . and stays in L. Consider the conjugate map φ which sends ζ → ζ 2 . As φ(ω) = −1/ω, we have ) and their complex conjugates are bounded by a constant which does not depend on k. This implies that the sequence (S k (x)) k must be eventually periodic.
Summing up, for a point x in Z[ζ], its S-orbit is finite or eventually periodic. When it is finite then its T -orbit is periodic and when its S-orbit is eventually periodic then T -orbit is aperiodic. Thus we have an algorithm for x ∈ Z[β] ∩ L to tell whether its T -orbit is periodic or not. Since for any positive ε, the right hand side is bounded by ε + A ω 2 − 1 for a sufficiently large k. This means that under the assumption that there is an infinite S-orbit, the set for a sufficiently small ε. Since there are only finitely many candidates in B, we obtain an algorithm to check whether an element x ∈ Z[ζ] ∩ L with π(x) = ∞ exists. In fact, all elements in B gives a finite S-expansion, we are done.
The same algorithm applies to 1 M Z[ζ] with a fixed positive integer M . In this way, we can also show that points in 1 2 Z[ζ] are periodic. We can find aperiodic orbits in 1 3 Z[ζ]. For example, one can see that 1/3 has an aperiodic T -orbit because its S-orbit: It is crucial in the above proof that the scaling constant of the self-inducing structure is a Pisot number. Scaling constants of piecewise isometries often become Pisot numbers, moreover algebraic units. We discuss these phenomena in Section 3.

Coding of aperiodic T -orbits
Denote by A the set of all T -aperiodic points in L. By the proof of the previous section, we have We also have S(A) ⊂ A. This means that for Conversely a sequence {m i } i=1,2,... defines a single point of Y . Therefore A must be a subset of the attractor Y of the iterated function system (IFS): an approximation of which is depicted in Figure 7  assert that 2-dimensional Lebesgue measure of aperiodic points in L must be zero, because ω 4 ≃ 6.854 · · · > 6. We notice that the digits in Q are not arbitrarily chosen because the image of S must be in T (Z). Thus the digits d 1 and d 4 appears only at the beginning in the expression of Equation (2). Therefore it is more suitable to study A ∩ T (Z). The attractor is depicted in Figure 7(b). This iterated function system satisfies OSC by a pentagonal shape K with whose vertices are Figure 8. We confirm that the pieces K m = ζ m ω 2 K + d m do not overlap. We consider the induced system of (L, B, ν, T ) to T (Z). Denote by T the first return map on T (Z). Then the induced system (T (Z), T ) is the domain exchange of two isosceles triangle A and B depicted in Figure 9. We see Again we find self-inducing structure with the scaling constant ω 2 : for all x ∈ T (Z). This can be seen in Figure 10 with Figure 10. Self Inducing Structure of (T (Z), T ) the set A.
Readers may notice that we can find a self-inducing structure by smaller scaling constant ω in Figure 7(b) by taking two connected pieces. However this choice of inducing region is not suitable because the self-inducing relation (with flipping) is measure theoretically valid, but has different behavior on the boundary.
Let us introduce two codings. First is the coding of T -orbits of a point x in L in two symbols {0, 1}: For e.g., the d(  b a a a b a a a b a b a b a b . . . Hereafter we discuss the coding d. Observing the trajectory of the region ω −2 (∆) and ω −2 (T (Z) \ ∆) by the first return map by the iteration of T to the region ω −2 T (Z), it is natural to introduce a substitution σ 0 : on {a, b} * and we have d(ω −2 x) = σ 0 ( d(x)) for x ∈ T (Z). More generally, following the analogy of the previous section, the first hitting map to the region ω −2 (T (Z)) provide us an expansion of a point x ∈ T (Z) exactly in the same form as (2) with restricted digits {d 0 , d 2 , d 3 , d 5 }. One can confirm that where ⊕ is the concatenation of letters. Defining conjugate substitutions by We say that an infinite word y in {a, b} N is an S-adic limit of σ i (i = 0, 1, 2, 3) if there exist y i ∈ {a, b} N for i = 1, 2, . . . such that This shows that d(x) is an S-adic limit of σ i (i = 0, 1, 2, 3).
Note that from the definition (7) of σ i , for a given S-adic limit y there is an algorithm to retrieve uniquely the sequence (σ m i ) i . Checking first four letters of y, we know the first letter of y 1 and to determine m 1 we need first 6 letters. We can iterate this process easily.
Summing up, we embedded the set A ∩ T (Z) into the attractor Y ′ of an IFS (3) and succeeded in characterizing the coding of T -orbits of points in this attractor as a set of S-adic limits on {σ 0 , σ 1 , σ 2 , σ 3 }. However recalling that points in closed pentagons P 1 and P 2 are T -periodic and Y ′ is a nonempty compact set, we see from Figure 7 We wish to characterize the set of aperiodic points in Y ′ and its coding through d. Recalling the discussion in the previous section, if x ∈ T (Z) has periodic T -orbits if and only if there exists a positive integer k such that S k (x) ∈ P 0 ∪ P 1 ∪ P 2 . The equivalent statement in the induced system (T (Z), T ) is that x ∈ T (Z) is T -periodic if and only if there exists a positive integer k such that S k (x) ∈ P 0 ∪ P 1 . Note that we have: 10 ) is the center of P 0 (resp. P 1 ) and consequently T 5 (x) = x holds for x ∈ P 0 ∪ P 1 . If x ∈ T (Z) and x is T -periodic, then there exist x i ∈ T (Z) such that x ℓ ∈ P 0 ∪ P 1 and with m i ∈ {0, 2, 3, 5}. Thus the set of T -periodic points in T (Z) consists of all the pentagons of the form with j = 0, 1 and m j ∈ {0, 2, 3, 5}. From the self-inducing structure (5), it is easy to see that if two points x, x ′ are in the same pentagon of above shape and none of them is the center, then they have exactly the same periods. Moreover two T -orbits keeps constant distance, i.e.,  Clearly all regular pentagons of the shape (7) are subtracted by this iteration and we obtain This observation allows us to symbolically characterize aperiodic points in Y ′ . First, every point x of Y ′ has an address d m 1 d m 2 · · · ∈ {d 0 , d 2 , d 3 , d 5 } N by the expansion (2). The address is unique but for countable exceptions. The exceptional points forms the set of cut points of Y ′ having the eventually periodic expansion: in the suffix of its address, which is understood by Figure 12 where K mn = ζ m ω 2 ( ζ n ω 2 K + d n ) + d m . Note that if a point x in T (Z) is periodic, then there exists a non-negative integer k such that S k (x) ∈ P 0 ∪ P 1 . Moreover, if x ∈ Y ′ ∩ T (Z), then there exists a non-negative integer k such that S k (x) ∈ ∂(P 1 ), because it can not   Figure 14. Here the double bordered states in Figure 14 are final states. Each infinite word produced by the edge labels {d 0 , d 2 , d 3 , d 5 } on this directed graph is accepted, because it visits infinitely many times the final states. We do not give here the exact shape of its complement. It is known that complementation of a Büchi automaton is much harder than the one of a finite automaton, because the subset construction does not work (c.f. [29,23]). Now consider the topology of {a, b} N induced from the metric defined by 2 − max x i =y i i for x = x 1 x 2 . . . , y = y 1 y 2 · · · ∈ {a, b} N . Take a fixed point w = (w i ) i=0,1,2,... ∈ {a, b} N with σ 0 (w) = w. This is computed for e.g., by lim n σ n 0 (a). The shift map V is a continuous map from {a, b} N to itself defined by V ((w i )) = (w i+1 ). Letting X σ 0 be the closure of the set {V n (w) | n = 0, 1, . . . }, we can define the substitutive dynamical system (X σ 0 , V ) associated with σ 0 . Since σ 0 is primitive the set X σ 0 does not depend on the choice of the fixed point and (X σ 0 , V ) is minimal and uniquely ergodic (see [9]). Let τ be the invariant measure of (X σ 0 , V ). On the other hand, for the attractor Y ′ there is the self-similar measure ν, i.e., a unique probability measure (c.f. Hutchinson [13]) satisfying

H H H H H H H H H
Theorem 3. The restriction of T to Y ′ is measure preserving and (Y ′ , B Y ′ , ν, T ) is isomorphic to the 2-adic odometer (Z 2 , x → x + 1) as measure dynamical systems: is almost one to one and measure preserving, which will be made explicit in the proof. Moreover the map from Z 2 to itself gives a commutative diagram: The above theorem may be read that (Y ′ , B Y ′ , ν, T ) gives a one-sided variant of numeration system in the sense of Kamae [15].
for ℓ = 1, 2, . . . . This gives a multiplicative coding d ′ : X σ 0 → {σ 0 , σ 1 , σ 2 , σ 3 } N . Let A ′ be the points of X σ 0 whose multiplicative coding does not end up in an infinite word produced by reading the vertex labels of Figure 15. Let us associate to z a 2-adic integer ι(z) = − i=0 κ(y i )2 2i ∈ Z 2 . The map ι is clearly bijective bi-continuous and the value ι(z) is also called the multiplicative coding of z. We write down first several iterates of V on the fix point of σ 0 , to illustrate the situation: One can see that the following commutative diagram (11) holds. (11) Therefore (X σ 0 , V ) is topologically conjugate to the 2-adic odometer (Z 2 , x → x + 1). Here the consecutive digits {0, 1} in Z 2 are glued together to give {0, 1, 2, 3} = {0, 1} + 2{0, 1}. Indeed, σ 0 satisfies the coincidence condition of height one in the sense of Dekking [25,9] and above conjugacy is a consequence of this. (Z 2 , x → x + 1) is a translation of a compact group Z 2 which is minimal and uniquely ergodic with the Haar measure of Z 2 . Moreover one can confirm that ι preserves the measure and (X σ 0 , V ) and (Z 2 , x → x + 1) are isomorphic through ι as measure dynamical systems. In view of (6), we define Then η is clearly surjective, continuous, and measurable because both τ and ν are Borel probability measures. Since the set of points with double addresses is on the open edge, the map η is bijective from A ′ to A ∩ T (Z).
Since d(T (x)) = V ( d(x)), we have a commutative diagram: From Figure 14, it is easy to see that the set P of T -periodic points in Y ′ is measure zero by ν, i.e., ν(A ∩ T (Z)) = ν(Y ′ ∩ T (Z)) = 1, because the number of words of length n in Figure 14 is O(2 n ). Similarly as the Perron-Frobenius root of the substitution σ 0 is 4 and the number of words of lengths n in Figure 15 are O(2 n ), we see that τ (A ′ ) = τ (X σ 0 ) = 1. From (13) the pull back measure ν • η −1 of X σ 0 is invariant by V , we have τ = ν • η −1 by unique ergodicity. Therefore by taking φ = η • ι, we have the commutative diagram (8) with measure zero exceptions. Let V ′ be a map from X σ 0 to itself which acts as the shift operator on the multiplicative coding d ′ , i.e., (d ′ (V ′ (z)) = σ n 2 σ n 3 . . . for d ′ (z) = σ n 1 σ n 2 . . . . Then we see that and the commutative diagram (9) is valid but for measure zero exceptions. Proof. In the proof of Theorem 3 the map η is bijective form A ′ to A∩T (Z).
Therefore if x ∈ A ∩ T (Z), then there exists a unique element in z ∈ X σ 0 with η(z) = x. Therefore there exist an element z 0 ∈ Z 2 such that φ(z 0 ) = x.
The Haar measure µ 2 on Z 2 is given by the values on the semi-algebra: is uniquely ergodic, the assertion follows immediately from the commutative diagram (8).
Not all points in Y ′ gives a dense orbit as we already mentioned that A ∩ Y ′ is a proper dense subset of Y ′ . There are many periodic points in Y ′ as well. This gives a good contrast to usual minimal topological dynamics given by a continuous map acting on a compact metrizable space.
Proof. It is clear from the fact that (T (Z), T ) is the induced system of (L, T ).
One can construct a dual expansion of the non-invertible dynamics (Y ′ , S) by the conjugate map φ : ζ → ζ 2 in Gal(Q(ζ)/Q) and then make a natural extension: an invertible dynamics which contains (Y ′ , S). The idea comes from symbolic dynamics. We wish to construct the reverse expansion of (12) to the other direction. To this matter, we compute in the following way: Therefore it is natural to introduce a left 'expansion': with i k ∈ {0, 2, 3, 5}. As this expression does not converge, we take the image of φ because φ(ω) = −1/ω. Let us denote by u i k = φ(d i k ). Then the expansion converges and the closure of the set of such expansions gives a compact set Y depicted in figure 16.
This is an analogy of the results [14] for β-expansion. The proof below is on the same line.

Proof.
As ω is an algebraic unit and d i ∈ Z[ζ], the denominator of g i (y) is the same as that of y for i = 0, 1, 2, 3. Therefore the module y ∈ M = 1 M Z[ζ] is stable by g i for some positive integer M . Note that points y ∈ M with (y, φ(y)) ∈ Y ′ × Y is finite, because y, φ(y) and their complex conjugates are bounded in C. One can confirm that the mapŜ becomes surjective from M to itself. For a finite set, surjectivity implies bijectivity. Therefore a point y ∈ M with (y, φ(y)) ∈ Y ′ × Y produces a purely periodic orbit. On the other hand if x has purely periodic multiplicative coding, it is easy to see (y, φ(y)) ∈ Y ′ × Y.

Other self-similar systems
Pisot scaling constants appear in several important dynamics. For irrational rotations (2IET), it is well known that scaling constants of selfinducing systems must be quadratic Pisot units. A typical example Figure 2 was shown in the introduction. They are computed by the continued fraction algorithm as fundamental units of quadratic number fields. Poggiaspalla-Lowenstein-Vivald [24] showed that the scaling constant must be an algebraic unit for self-inducing uniquely ergodic IET. When the scaling constant of self-inducing IET is a cubic Pisot unit, we have further nice properties [4,19,20].
A necessary condition that 1-dimensional substitutive point sets give point diffraction is that the scaling constant is a Pisot number [7]. Suspension tiling dynamics of such substitution is conjectured to have pure discrete spectrum if the characteristic polynomial of its substitution matrix is irreducible. For higher dimensional tiling dynamics the Pisot (or Pisot family) property is essential to have relatively dense point spectra, see for e.g. [27,17].
Pisot scaling properties seem to extend to the case of piecewise isometries. To conclude we present some examples, though we do not make a systematic study.
It is already observed in [16,3] that Pisot scaling constants appear in our problem if θ is the n-th root of unity for n = 4, 6, 8, 10, 12 in the same way as we did in n = 5 but in a more involved manner. In each case they are quadratic Pisot units. What about if λ = −2 cos(θ) is cubic? In this case, the dynamics of Conjecture 1 are embedded into the piecewise affine mapping acting on (R/Z) 4 which is harder to visualize. Instead let us consider formal analogies of piecewise isometries generated by cubic n-th fold rotation in the plane. At the expense of losing connection to Conjecture 1, we find many Pisot unit scaling constants! Being an algebraic unit is natural and may be explained from invertibility of dynamics. However we have no idea why the Pisot numbers turn up or even how to formulate these phenomena as a suitable conjecture.
3.1. Seven-fold. We start with 7-fold case. Both pieces are rotated clockwise by 4π/7 as in Figure 17. The triangle is rotated around A and the trapezium around B. The first return map to a region and a smaller region with the same first return map (up to scaling) are described. Unlike the five fold case, returning to the subregion does not cover the full region. A simple consequence is that there are infinitely many possible orbit closures for non-periodic orbits in the system. The scaling constant α ≈ 5.04892 is a Pisot number whose minimal polynomial is x 3 − 6x 2 + 5x − 1. Figure 18 shows how this remaining space can be filled in. As this region is already a little small we will zoom in and now consider just this induced sub-system in Figure 19. The smaller substitutions are easier to see as there are two scalings giving the same dynamics (A and B). The scaling constant β ≈ 16.3937 for these subregions is the Pisot number associated to x 3 − 17x 2 + 10x − 1. The proof that the remaining substitutions work is shown in Figure 20. The first return map to the two lower triangles is shown. The same dynamics occur on a smaller region. The orbit of the smaller region covers all the regions left out of Figure 19 and so the substitution rule from that figure is now complete. The scaling constant for this triangle is α. This gives an example of recursive tiling structure by Lowenstein-Kouptsov-Vivaldi [18]. Knowing that every aperiodic orbits are in one of the above self-inducing structures, we can show that Theorem 7. Almost all points of this 7-fold lozenge have periodic orbits.
The fundamental units of the maximal real subfield Q(cos(4π/7)) of the cyclotomic field Q(ζ 7 ) are given by b and b − 1 where b = 1/(2 cos(3π/7)) ≈ 2.24698. Here b is the Pisot number satisfying x 3 − 2x 2 − x + 1. We see that α = b 2 and β = b 4 /(b − 1) 2 and thus α and β generates a subgroup of fundamental units of Q(cos(4π/7)). Note that both √ α = b and √ β = b 2 /(b−1) are Pisot numbers but b−1 is not. Our piecewise isometry somehow selects Pisot units out of the unit group! 3.2. Nine-fold. The next example is 9-fold case in Figure 21. Both pieces are rotated anti-clockwise by 4π/9, the triangle around A and the trapezium around B. The first return map (△) to the triangle is also shown. In addition the same dynamics are found on a smaller piece of the map. Like the 7-fold shown in Figure 17 this does give a full description of the dynamics, but it is △ 2 not △. The scaling constant γ ≈ 8.29086 is a Pisot unit defined by x 3 − 9x 2 + 6x − 1. Unfortunately in this case we were not able to find a complete description of the scaling structure.
First return to region Self-inducing map A B A B Figure 17. A seven-fold piecewise isometry.
The fundamental units of Q(cos(4π/9)) are b and b 2 − 2b − 1 where b = 1/(2 cos(4π/9)) ≈ 2.87939 is a Pisot number given by x 3 − 3x 2 + 1. We have γ = b 2 and are expecting to find another Pisot unit b 2 /(b 2 −2b−1) ≈ 5.41147 (or its square) as a scaling constant in this dynamics, which would give an analogy to the seven-fold case. Figure 18. The regions remaining from the self-similarity shown in Figure 17 A B Figure 19. The substitution rule of the induced subsystem shown in Figure 18.
First return map to lower two triangles Self-similar return map to smaller triangle