Critical curves of rotations

In rotations with a binary symbolic dynamics, a critical curve is the locus of parameters for which the boundaries of the partition that defines the symbolic dynamics are connected via a prescribed number of iterations and symbolic itinerary. We study the arithmetical and geometrical properties of these curves in parameter space.


Introduction
We consider a rotation g on the circle (unit interval): (1) g : [0, 1) → [0, 1) where θ is the angle of rotation and {•} denotes the fractional part.The partition of the circle (2) 1), defines a symbolic dynamics in two letters a and b. (For background on symbolic dynamics, see [1,5,12]).
The parameter space of this system is the closed unit square [0, 1] 2 of all pairs ζ = (θ, ρ), and a critical point is a pair ζ for which the equation (3) iθ = j + ρ has a solution z = (i, j) ∈ Z 2 .This means that at a critical point there is a critical orbit of g containing both boundary points 0 and ρ of the partition (2).The shortest portion of orbit connecting such boundary points, in some order, will be called the centre of the orbit, with the convention that it includes the initial boundary point but not the final one.We then consider the symbol sequence w of the centre.If θ is irrational, then the boundary points are visited only once and in a defined order, and w is determined by ζ.If θ is rational, then the orbit is periodic, and we may choose either 0 or ρ as the initial point of the centre.Hence when θ is rational, ζ determines two centres and their corresponding words w, typically of different length.A critical word w is a word constructed in this fashion, and the critical curve C w is the set of critical points which share the same critical word w.
Even though critical curves of rotations are fairly basic objects, to the best of our knowledge they have not been considered explicitly.It is known that critical points determine the complexity of rotational words -the symbol sequences generated by the rotation (1) with partition (2).At a critical point with irrational θ, a critical word of length n is a Date: April 3, 2023.
factor (finite sub-word) of an infinite word with complexity K(t) (number of factors of length t) equal to (4) K(t) = 2t t n t + n t > n, see [1, theorem 10].The case n = 1 is the much studied Sturmian words [12, section 6], while n = ∞ corresponds to non-critical parameters -the boundary points are not on the same orbit.Note that, due to the minimality of irrational rotations, the collections of all factors of an infinite word (the symbolic language) does not depend on the initial condition [12, p. 105].An irrational critical point can therefore be characterised in terms of the complexity of the word of any orbit.
Critical curves are relevant to piecewise-defined dynamical systems supporting rotational motions, such as the much-studied family of planar maps (5) F(x, y) = (ax − y, x) x 0 (bx − y, x) otherwise, where a and b are real parameters [4,6,7,[9][10][11].Since F sends rays to the origins to themselves, part of its dynamics is described by a circle map with well-defined rotation number θ(a, b) which, when irrational, allows a topological conjugacy between the circle map and the rigid rotation (1).Each orbit of F applied to a ray from the origin then corresponds to acting on the ray a growing product of the two matrices A and B defined by (5).This product corresponds to a growing word in the symbols A, B, which follows the rotational word in a, b from the induced circle map.In this way, F can be interpreted [9][10][11] as the discrete Schrödinger equation on the one-dimensional lattice with a two-valued potential sequence generated by a rotational word.Such models with quasiperiodic twoletter words developed via two-letter substitution rules have received much attention [2,3].
In [10, theorem 2.2] it was shown that for any parameter pair (a, b) for which the partition boundaries (the positive and negative ordinate semi-axes) belong to the same F-orbit, invariant curves exist, consisting of the union of arcs of conic sections.In our terminology, these are critical points in parameter space, each with an associated finite symbol sequence.The locus of critical parameters in (a, b) space with the same sequence defines an algebraic curve, examples of which were first given (not under the name of critical curves) in [10, examples 4.1-3].Critical curves have also appeared implicitly in the works [7,[14][15][16] on mode-locking and bifurcations in piecewise-linear maps, where they are used to characterise sequences of mode-locking regions near a bifurcation point.
In [13], we investigated some properties of the critical curves of the map F, and we found structures of considerable complexity.We believe that understanding these curves in the simpler -yet non-trivial-case of rotations is an essential pre-requisite for the development of a theory of these dynamical objects in a general setting.This is the motivation for the present paper.
We summarise the contents and main results of this paper.In the next section we establish terminology and notation, and introduce the notion of a chain, the set of solutions of equation (3) for fixed (i, j).In section 3 we show that a chain is partitioned by Farey fractions (see [8, chapter III]) into a sequence of critical curves (open segments) separated by degenerate curves called Farey points, and that along a chain the code changes from a n to b n , or vice-versa (theorem 1).Furthermore, the identification of a Farey point on a chain is a relative concept: every such point belongs to a critical curve transversal to the given chain (corollary 4).
In section 4 we describe all curves through a critical rational point, for which equation (3) has infinitely many solutions.We show that in general every such point ζ is an interior point of exactly two curves -the dominant curves of ζ-as well as the common Farey point of four infinite pencils of curves (theorem 5).The symbolic dynamics of these curves is determined in theorem 6.
In section 5 we consider the geometric figure determined by the six dominant lines of a rational critical point and two points closest to ζ which share the value of θ.We show that these lines form two triples concurrent to the two triple points of ζ, whose rotation numbers are convergents of the continued fraction expansion of θ (theorem 7).
Acknowledgements.This research was supported by the Australian Research Council grant DP180100201 and by JSPS KAKENHI Grant Numbers JP16KK0005 and JP22K12197.

Basic properties
The parameter space for the symbolic dynamics of the map g of ( 1) is the set of pairs ζ = (θ, ρ) ∈ [0, 1] 2 .From (2) we find that the values ρ = 0 and ρ = 1 correspond to the trivial symbolic dynamics built from the single letter b and a respectively.For θ = 0 we have the identity map, while θ = 1 is included for consistency with Farey sequences1 .
We begin by taking a closer look at equation (3).For given ζ, every solution (i, j) correspond bi-uniquely to an orbit segment of length |i| having 0 and ρ as end-points, in the order prescribed by the sign of i.We first deal with the associated symbolic dynamics.
Def.A critical orbit of ( 1) is an orbit containing both boundary points 0 and ρ, and a boundary word is the symbolic dynamics of a finite section of a critical orbit connecting such boundary points, in some order, in such a way that the initial point is included and the final one is not.A critical word is a boundary word of minimal length, that is, the symbolic dynamics of the centre of the orbit.(Thus every boundary word contains a critical word as a prefix.)If ρ = 0, then equation (3) has the trivial solution z = (0, 0) for every θ, the centre of the critical orbit is empty, and the critical word is the empty word w = ε.There are also non-trivial solutions for rational θ -with associated boundary words-which we shall consider below.Likewise, for ρ = 1 we get the trivial solution z = (0, −1) as well as nontrivial solutions.
Assume for the moment that ρ = 0, 1.Then at a critical point, θ and ρ are both rational or both irrational -indeed they belong to the same number field.Suppose that θ is irrational.Then the points x = 0 and x = ρ appear only once in the doubly infinite orbit through 0, and hence (3) has only one solution z = (i, j).We now keep this solution fixed and regard (3) as an equation for (θ, ρ), subject to the constraint (θ, ρ) ∈ [0, 1] 2 (the condition ρ = 0, 1 was needed to determine (i, j) from ζ, and is no longer required -see below).We obtain a line segment of critical points -called a chaingiven by ( 6) Note that i = 0 by assumption and that (i, j) is also a solution of equation ( 3) when ρ = 0, 1, namely at (θ −s , 0) and (θ s , 1), where s = sign(i).
From the above and (3) one verifies that for any integer i, a pair (i, j) ∈ Z 2 corresponds to a chain if j is subject to the bounds ( 9) A pair of integers (i, j) satisfying the above conditions will be called the affine parameters of the chain L i,j .They are the solution of (3) shared by all points on the chain.
We are interested in the critical words w = w(θ, ρ(θ)) of the points of L i,j of ( 6) and (8).The following definition relates points and words.
Def.Let L i,j be a chain, let ζ ∈ L i,j , and let w be the critical word at ζ (not necessarily of length |i|).The critical curve C w on L i,j containing ζ is the set of points of L i,j sharing the same critical word.If such a set reduces to a point, we shall speak of a Farey point B w .Thus a chain is partitioned into critical curves and Farey points.In particular, each of the chains (8) has empty word as critical word, and therefore consists of a single critical curve C ε , and no Farey points.
Def.A non-empty boundary word w is positive (negative, respectively) if the first symbol of w is a (b, respectively).The empty word is both positive and negative.Likewise, the sign of the affine parameters z = (i, j) is that of i if i = 0, and if i = 0 then z is both positive and negative.
We see that a non-empty word is positive (negative) if the corresponding centre starts at 0 (ρ).Let ρ = 0, 1.Because -as noted above-to an irrational critical points ζ there corresponds unique affine parameters, the point ζ has a well-defined sign and so does the unique critical curve that contains it.By the same criterion, rational critical points are both positive and negative.
Let n = |i|.At all irrational points on L i,j the critical word w = w 0 • • • w n−1 has length n, but this is not necessarily the case if ζ is rational.Indeed in this case the orbit is periodic and, being also critical, both points 0 and ρ are visited infinitely often.As a result, equation ( 3) has a doubly-infinite set of solutions (i t , j t ), which include (i, j), and to each solution there is an associated boundary word.If there is a t such that i t and i have the same sign and |i t | < n, then the centre of the critical orbit is shorter than n, and therefore w is not a critical word.In what follows, when we speak of the boundary words of a chain L i,j , we will always refer to words of length |i|.
From the above discussion we conclude that a rational critical point should be regarded as being both positive and negative, and this duplicity is reflected in the sign of the boundary words at that point.Such a correspondence however fails at the rational points of the chains (8), where the (non-empty) boundary words assume only one sign, since one element of the partition (2) is empty.Specifically, at the rational points (p/q, 0), equation (3) has the solutions (i, j) = (tq, tp), t ∈ Z, assuming both signs.However, for t = 0 the corresponding boundary words b |tq| are negative.There is an analogous discrepancy at (p/q, 1), where all boundary words are positive.

Boundary and critical words on a chain
In this section we consider the decomposition of a chain into critical curves and Farey points, as defined in section 2, as well as the associated symbolic dynamics.Figure 1 serves as an illustration of the items we shall be dealing with.
Our first result describes all the boundary words on a chain.Theorem 1.Let L i,j be a chain, let n = |i| 1 and let F n be the nth Farey sequence.Then i) The set of rotation numbers ii) The boundary words on L i,j are computed recursively as follows.Assume first that L i,j is positive, and let w = w 0 • • • w n−1 be the positive word of length n at θ = p/q ∈ F. Finally, let w ± be the words of the adjacent curves to the right (+) and left (−) (w ± is missing at θ = θ ± ).Then, for k = 1, . . ., n − 1 the following holds: 1) At θ = θ − we have w = b n , and w + k = a iff k ≡ 0 (mod q).This holds also for k = 0. 2) At θ = θ + we have w = a n , and The corresponding statements for negative chains are obtained from the above by exchanging all as and bs.
The statement of the theorem excludes the case i = 0. We remark that the critical word for the chains ( 8) is ε by definition, while the boundary words at the rational points are given in parts ii) 1,2).
Proof.i) Along the chain (6), the point x t (θ), t = 0, . . ., n of the centre are affine functions of θ with slope t if w is positive, and slope −n + t if w is negative.If for some θ in the range (7) two such functions coincide, then the map g is periodic with period not exceeding n, that is, θ ∈ F. Conversely, any θ ∈ F corresponds to a periodic orbit of g of period not exceeding n.Note that θ ± are the only elements of F whose denominator is divisible by n, because the numerators of θ ± are consecutive integers.
Assume first that θ = p/q ∈ F \ {θ − , θ + }.Then 1 < q < n, because q does not divide n.Periodicity implies that x 0 (p/q) = x q (p/q), and the intersection of x 0 and x q is transversal because the slopes of the two functions are different.It follows that the qth symbol of the critical word w changes at p/q, which is therefore the common end-point of two adjacent segments, hence a Farey point.Since θ ± are necessarily Farey points, and all rationals with denominator not exceeding n have been accounted for, we have shown that the elements of F partition the chain L i,j into |F| − 1 curves, as desired.
ii) Let w be positive.1) At θ = θ − we have ρ = 0; then, according to (2), the partition element I a is empty, so the code is b n .We have x k = 0 iff k ≡ 0 (mod q), and these are precisely the values of k (which include k = 0) for which x k ∈ I a for θ > θ − , that is, w + k = a.
α) The critical curve property together with q-periodicity implies that x k = x n = ρ iff k ≡ n (mod q).The proper symbol w k at θ is b, and since k < n, the slope of x k is smaller than that of x n , so that β) q-periodicity implies that x k = x 0 = 0 iff k ≡ 0 (mod q).The proper symbol w k at θ is a, and since k > 0, the slope of x k is greater than that of x 0 , so that The proof of ii) is complete.
If w is negative, the argument develops in a symmetrical manner.At θ + we have ρ = 0 whence w = b n .As θ decreases, all collisions of orbit points involve bs turning into as, until we reach θ − with code a n .We omit the details.Above the line we have the critical words of the curves, and below the line the Farey fractions with corresponding boundary words [see theorems 1 ii) and 6].The bottom row displays the critical words at the Farey points, all of length smaller than 7, including the empty word ε of zero length at θ ± .The boundary word at a Farey point is the concatenation of a critical word and a periodic word, whose period is given by the denominator of the fraction.
The arithmetical and combinatorial aspects of a chain are illustrated in figure 1. Applying theorem 1 ii) recursively along a chain, from θ − to θ + , we deduce that every letter b of the initial word changes to an a without omissions or repetitions.This translates into the following arithmetical statement.
Corollary 2. For any integers n > m and m 0, let F be the subset of F n lying between m/n and (m + 1)/n, and for each p/q ∈ F consider the congruences x ≡ n (mod q), x ≡ 0 (mod q) (which coincide if q divides n).Then, as p/q ranges in F, the solutions of this family of congruences form a complete set of residues modulo n, and each non-zero residue is a solution of exactly one congruence.
In the above statement there is no restriction on the numerator m, because the restriction that appears in theorem 1 plays no role in the proof of part ii).
The number M = |F| − 1 of curves in a chain depends on its affine parameters (i, j).Let n = |i|.Such a number is independent of n in only three cases, namely j = 0, j = n − 1 (M = 1), and n 3 odd and 2j + 1 = n (M = 2), plus the corresponding values for negative i.In all other cases, for fixed j, we have M (i, j) → ∞ as |i| → ∞.In this case the average order of M (n) is 3n/π 2 , which may be deduced from that of the Farey series [8, theorem 331].
Our next result provides alternative characterisations of the Farey points of a chain.Lemma 3. Let L i,j be a chain and let ζ = (θ, ρ) ∈ L i,j .The following statements are equivalent: ζ is not a Farey point of L i ,j .
Proof.For i = 0 all statements above are false, hence equivalent.If i = 0 and ζ also belongs to L 0,0 or L 0,−1 , then all statements are true since the critical word of these chains is the empty word, which is both positive and negative by definition.
We now assume that i = 0, ρ = 0, 1, and we let w be the critical word at ζ.We shall prove that i) ⇒ iii) ⇒ ii) ⇒ i).i) ⇒ iii).If ζ is a Farey point of L i,j , then from theorem 1 i) and the fact that ρ = 0, 1 we have that θ = p/q with q < |i| and q |i|.We define c := |i|/q 1 and i and j by i = i − sign(i) c q and j = j − sign(i) c p. Hence 1 |i | < q < |i|, sign(i ) = sign(i) and one checks that i θ − j = i θ − j = ρ.So ζ lies at the intersection of L i,j and L i ,j .Furthermore, if w is the critical word at ζ with the same sign as i, it has the minimal length |i | by construction.Since q > |w| = |i |, theorem 1 i) shows that ζ cannot be a Farey point on L i ,j .
iii) ⇒ ii).If iii) holds, then θ = (j − j )/(i − i ), and hence at ζ the critical orbit is periodic with period |i| − |i |.Since the length of the critical word is necessarily smaller than the period, we have |w| We infer from part iii) of the lemma, which itself relies on theorem 1, that Farey points exist only in the context of a given chain.More precisely, at a Farey point B w on a chain there is always a direction in parameter space along another chain for which the critical word w does not change for sufficiently small displacements (and so B w is not a Farey point on the second chain).This fact is expressed concisely by the following statement: Corollary 4. A Farey point of a chain belongs to a critical curve transversal to the chain.

Curves through rational points
We now characterise all critical curves through a rational critical point ζ.As discussed in the previous section, equation (3) for rational ζ has a doubly-infinite family of solutions corresponding to as many chains passing through ζ.We shall identify the chains for which ζ belongs to a critical curve, and those for which ζ is a Farey point; in the latter case we also determine the adjacent curves on the chain.In what follows, by the Farey points of a critical curve we shall mean those belonging to the closure of the curve.
If θ = p/q, then the orbit of 0 consists of q equally spaced points on the unit interval, and therefore ζ must be of the form (p/q, r/s), with s dividing q.We call q the denominator of ζ.We develop p/q in continued fractions, and of the two possible continued fractions representations we choose the one whose last coefficient is unity (see [8, theorem 162]).Then the index n of the last convergent p n /q n = p/q is determined unambiguously.At various junctures we shall discuss the implications of this choice.We let (11) τ = q r s , τ ± = q r s ± 1 q where q = (−1) n−1 q n−1 .
Given a rational point ζ with ρ = 0, 1, we consider, among the positive chains containing ζ that with minimal value of i, and denote its affine parameters by (i + , j + ).In a similar manner, the negative chain with minimal value of −i will be denoted by (i − , j − ).These pairs will be called the dominant affine parameters of the point ζ.Likewise, we shall speak of the dominant chains (or lines), and the dominant curves of ζ.We will show that these objects exist and are unique.For uniformity of exposition, we treat the cases ρ = 0, 1 as follows.We shall regard the segment ρ = 0 as the positive dominant chain of any point ζ = (θ, 0), and the segment ρ = 1 as the negative dominant chain of any point ζ = (θ, 1).In both cases, the chain consists of a single curve with the empty word.Accordingly, if ρ = 0, we let (i + , j + ) = (0, 0) and (i − , j − ) = (−q, −p).If ρ = 1 we let (i + , j + ) = (q, p − 1) and (i − , j − ) = (0, −1).Finally, we define the upper and lower neighbours of a rational point ζ = (p/q, r/s) to be the points (12) ζ ↑= ζ + 0, respectively.If ζ is a critical point, then so are ζ ↑ and ζ ↓, with one of them missing if ρ = 0, 1.
We shall consider the four quadrants with origin at ζ and label all objects that are pertinent to such quadrants -points and curves-with the superscripts I, II, III, IV.The superscript ± will refer to the sign of curves, so that the positive sign refers to I, III and the negative sign to IV, II.Theorem 5. Let ζ = (θ, ρ) be a rational critical point.If ρ = 0, 1 then ζ is an interior point of the dominant curves of ζ, and the common Farey point of four infinite pencils of curves, arranged pairwise as adjacent curves in two pencils of chains of opposite sign.Furthermore, all Farey points distinct from ζ in a pencil belong to (the closure of ) a dominant curve of a neighbour of ζ, this association being bi-unique -see figure 2. The same applies if ρ = 0, 1, but in this case one neighbour and two pencils are missing (three pencils, if θ = 0, 1).
The sequences (13) are the affine parameters of all chains containing ζ. Assume first that ρ = 0, 1, that is, s = 1.If we had i t = 0 for some t, then s would also divide q n−1 , which in turn would yield s = 1, contrary to the assumption.So there are exactly two values t ± of t in (13) for which 0 < |i t | < q, namely (14) t + = −q r/s , t − = −q r/s = t + − 1 while all other values of t give |i t | > q.
With reference to (14) define and similarly for j ± ( ), so that the sign of i ± ( ) is ±1 for all .To obtain explicit formulae for i ± ( ) and j ± ( ) as functions of p/q and r/s, we use (13)(14)(15), keeping in mind that ru = qr/s and −x = − x .We obtain the dominant affine parameter of ζ: where {•} denotes the fractional part, and τ was defined in (11).Finally, we obtain, for = 0, 1, 2, . . .
The above expression give all affine parameters of all chains though ζ in the case ρ = 0, 1.From ( 6) the sign of i t is the sign of the chain containing ζ, and |i t | = |w|.Then, formulae (17) and lemma 3 give two infinite sequences of chains of opposite sign, such that ζ is an interior point of a curve in the chains [i ± (0), j ± (0)], and a Farey point in all other chains, as claimed.
We now assume further that ρ = 1/q, (q − 1)/q.As indicated above, we shall use the superscripts I, II, III, IV, which refer to the four quadrants with origin at ζ, to label the points in the four sequences and other relevant quantities.We denote the dominant affine parameters of the upper neighbour ζ ↑ -see (12)-by (i I , j I ) and (i II , j II ), and those of ζ ↓ by (i III , j III ) and (i IV , j IV ).(Thus I, III are positive and II, IV are negative.)Considering ( 11) and ( 16), we find (18) and j σ = i σ p q − ρ σ , where ρ σ = r/s + 1/q if σ = I, II and ρ σ = r/s − 1/q if σ = III, IV.
Next, for each pencil, we determine the external Farey point of each curve having ζ as the common Farey point.The corresponding parameters are given in (17) with 1. Then |i ± ( )| > q > |i σ | for every σ, and therefore, from lemma 3 iii) (with (i , j ) = (i σ , j σ ) and (i, j) = (i ± ( ), j ± ( ))), we see that the curves of the pencil in sector σ cannot extend further than the dominant line (i σ , j σ ).We will now show that all Farey points in fact belong to that line.
Let us consider the Farey sequence F, given in (10), for the affine parameters (i ± ( ), j ± ( )).From theorem 1 i) and the fact that p/q = θ ± , the rotation numbers of the Farey points of the curves adjacent to ζ are three successive terms of F: p l /q l < p/q < p r /q r .Then (see [8, section 3.1]), we have p r q − q r p = 1.Moreover p/q is the mediant of p r /q r and p l /q l , and we shall compute the latter from the former.
With the notation as above, we have the candidate values for p r and q r : As |k| increases, p r k /q r k approaches p/q, and the approach is from the right if q r k is positive.The fraction p r k /q r k belongs to the |i ± ( )|th Farey sequence if 0 < q r k |i ± ( )|, and so we let k ± be the largest value of k for which this property holds.We find where τ ± was defined in (11).Using the quadrant superscripts, we let p I ( ) = p r k + , q I ( ) = q r k + , p IV ( ) = p r k − , q IV ( ) = q r k − .To harmonise the notation, we shall also use the symbols where i ± ( ) was defined in (17).
To compute p II,III ( ) and q II,III ( ) using the mediant property we let p II,III ( ) = a ± p − p IV,I ( ) and q II,III ( ) = a ± q − q IV,I ( ), where a ± is a positive integer.The required value of a ± is the largest such that q III,II |i ± ( )|, which is Performing the calculation explicitly gives: We have constructed four infinite sequences of Farey points, and we must now show that the Farey points of each sequence belong to (the closure of) a dominant curve of a neighbour of ζ.We begin to show that these points are collinear, and to this end, we let where all quantities refer to the same superscript.Using the above and (21), we find where ζ↑↓ was defined in (12).
We have proved that the Farey point distinct from ζ in the σ-pencil, namely ζ σ ( ), = 1, 2 . . .belong to the dominant line with parameters (i σ , j σ ).It remains to show that all these Farey points belong to the closure of the dominant curve (infinitely many of them do, since ζ ↑↓ is an interior point of the dominant curve).By theorem 1 i), we must show that the element of the |i σ |th Farey sequence which lies to the right (for σ = I, IV) or to the left (for σ = II, III) of p/q lies at least as far out as p σ (1)/q σ (1).From ( 17) and (18) we have |i σ | q |i σ (1)|, and hence the corresponding Farey sequences satisfy | which proves our assertion.
The final item in the proof are the boundary cases ρ = 0, 1.By definition, a rational critical point ζ with ρ = 0 has the following affine parameters for positive and negative chains: for = 0, 1, 2, . . .
We first consider the case where ρ = 0 and θ = 0, 1, i.e., ζ is of the form ζ = (θ, ρ) = (p/q, 0) with q 2. In this case, the III-and IV-pencils are missing.Using ( 18) and (25), we see that the θ-coordinate, denoted by θ I ( ), of the intersection point of the line 1 and the positive dominant line of ζ ↑= (p/q, 1/q) is given by Similarly, the θ-coordinate, denoted by θ II ( ), of the intersection point of the line ρ = i − ( )θ − j − ( ), 1 and the negative dominant line of ζ ↑ is given by θ II ( ) = p +p q +q q > 0 p +p+p q +q+q q < 0.
Using the condition for neighbouring terms in a Farey sequence, we see that p/q and θ I ( ) (resp.θ II ( ) and p/q) are neighbours in ).Thus, all the intersection points are the Farey points distinct from ζ in the I-and II-pencils.Since | , all these Farey points belong to the closure of the dominant curves of ζ ↑.In the case where ρ = 1 and θ = 0, 1, i.e., ζ is of the form ζ = (θ, ρ) = (p/q, 1) with q 2, the missing pencils are I and II.In this case, we can do the same to show that the Farey points distinct from ζ in the III-and IV-pencils, respectively, belong to the positive and negative dominant curves of ζ ↓.Lastly, we consider ζ with q = 1, i.e., the four corners (0, 0), (1, 0), (0, 1), (1, 1) of the parameter space, each of which has only one pencil.It is easy to see that the Farey points of the I-pencil of (0, 0) and those of the II-pencil of (1, 0) belong to ρ = 1 (0 θ 1), i.e., the negative dominant curve of (0, 1) and (1,1).We can also see that the Farey points of the IV-pencil of (0, 1) and those of the III-pencil of (1, 1) belong to ρ = 0 (0 θ 1), i.e., the positive dominant curve of (0, 0) and (1, 0).We remark that the particular choice of continued fraction representation for p/q has little effect of the above argument.It merely causes a shift by one unity of the quantities t ± in (14).
To complete our analysis of the critical curves of the map g, we now characterise the words of all curves incident to a rational point ζ, in terms of the word at ζ. Theorem 6.Let ζ be a rational critical point with denominator q, let w σ ( ), 0 be the word of the th curve adjacent to ζ in the quadrant σ.Let u + (resp.u − ) be the word of the positive (resp.negative) dominant curve of ζ.Let v + (resp.v − ) be the word obtained by switching the first letter of u + (resp.u − ).Then |u Proof.Assume first that ρ = 0, 1.Let Then u σ is a periodic boundary word at ζ, for the initial conditions 0 or ρ.The periodic orbit has no other boundary point because ζ lies in the interior of the curve of both u + and u − .Thus |u σ | = q.Let w σ ( ) be the word of the curve adjacent to ζ in the quadrant σ, and let n σ ( ) be the length of this word.From (17) we find with the usual convention on sign, and the quotient of division of n σ ( ) by q is given by n σ ( )/q = {±τ } + = .
The word w σ ( ) is now computed from theorem 1, part ii).Then w σ ( ) will consist of repetitions of a modification of u σ , followed by a modification of u ± , where the modifications are performed on the symbols congruent to |i σ (0)| modulo q if the curve is on the right of ζ (σ = I, IV), or those congruent to 0 modulo q, except the first symbol, if the curve is on the left of ζ (σ = II, III).This gives w σ ( ) in the statement.
We now consider the case ρ = 0.By definition, u + = ε and u − = b q , where ε denotes the empty word [see discussion preceding (12)].Obviously, |u + u − | = |u − u + | = q.For σ = I, II, we denote by w σ ( ) the word of the curve adjacent to ζ in the quadrant σ and put n σ ( ) = |w σ ( )|, as above.From (25) we have The word w of length n I ( ) = q at ζ is given by w = b q .Thus, theorem 1 ii) gives Likewise, the word w of length n II ( ) = q( + 1) at ζ is w = b q( +1) , and the same theorem gives This completes the proof of the case ρ = 0. We can do the same for the proof of the case ρ = 1.

Triple points and convergents
In the previous section we considered the solutions of (3) for fixed rational (θ, ρ), resulting in infinitely many chains passing through that point.A global view on symbolic dynamics may be gained by considering a different set of chains, namely those corresponding to the set N n of solutions of (3) with |i| bounded by n, and j subject to the bounds J(i) given in ( 9): By construction, the boundary words associated to the chains in N n contain all possible critical words of length not exceeding n and either sign.For brevity, we shall not develop the analysis of N n here, but merely prove a geometric theorem, illustrated in figure 3, which deals with a subset of N n consisting of six chains near a rational critical point.This result displays a connection between intersections of chains and convergents of continued fractions.
Let ζ = (p/q, r/s) be a rational critical point with two neighbours ζ ↑ and ζ ↓, that is, r/s = 0, 1.We recall (see beginning of section 4) that p k /q k denote the convergents of the continued fraction expansion of p/q = p n /q n , chosen so to have the last coefficient equal to one.Then p n−1 /q n−1 is the rational closest to p/q among those with denominator less than q.(This is not the case if the last coefficient is greater than one.)We define where τ, τ − , τ + are given in (11).for some integer k, where the sign is that of q .Since r/s = 1/q, (q − 1)/q, none of the τ s is an integer, and so we have With this in mind, we match each sequence in (29) with two µ-sequences so as to transform it into an arithmetic progression. (30) The above argument shows that in general a rational critical point has two µ-sequences corresponding to two triple points, which we now compute.We recall that p k /q k are the convergents of p/q = p n /q n .We let p = (−1) n−1 p n−1 and q = (−1) n−1 q n−1 [cf.(11)], whence pq − p q = 1.

7 εFigure 1 .
Figure1.The partition of the positive chain L 7,5 with affine parameters (i, j) =(7,5), into four critical curves and five Farey points (solid circles), determined according to theorem 1 i).Along the chain, all boundary words have length i = 7. Above the line we have the critical words of the curves, and below the line the Farey fractions with corresponding boundary words [see theorems 1 ii) and 6].The bottom row displays the critical words at the Farey points, all of length smaller than 7, including the empty word ε of zero length at θ ± .The boundary word at a Farey point is the concatenation of a critical word and a periodic word, whose period is given by the denominator of the fraction.
holds, then ζ is necessarily a rational point, and the denominator of θ is less than |i|, being a divisor of |i| − |w|.Thus θ ∈ F |i| , and ζ is a Farey point from theorem 1 i).

Figure 2 .
Figure 2. The critical curves through (or adjacent to) the rational point ζ = (3/5, 2/5) (see theorem 5), with its two dominant lines (black) and four pencils of curves concurrent at ζ (grey), the latter being their common Farey point.The other Farey point of the curves in a pencil lie on a dominant lines of one of ζ's neighbours (blue), each pencil paired with a different line.The six dominant lines feature two concurrent triples, as detailed in theorem 7.

A
triple point of ζ is a common point of three concurrent dominant lines, which comprise one line from each of ζ, ζ ↑, ζ ↓.The next result characterises all triple points.

− 1
as the triple points for case 7.