Remark on continued fractions, Mobius transformations and cycles Текст научной статьи по специальности «Математика»

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REMARK ON CONTINUED FRACTIONS, MÖBIUS TRANSFORMATIONS AND CYCLES

V.V. KISIL

School of Mathematics, University of Leeds, Leeds, UK kisilv@maths.leeds.ac.uk

We review interrelations between continued fractions, Möbius transformations and representations of cycles by 2 x 2 matrices. This leads us to several descriptions of continued fractions through chains of orthogonal or touching horocycles. One of these descriptions was proposed in a recent paper by A. Beardon and I. Short. The approach is extended to several dimensions in a way which is compatible to the early propositions of A. Beardon based on Clifford algebras.

Keywords: continued fractions, Möbius transformations, cycles, Clifford algebra

B.B. КИСИЛЬ. ЗАМЕЧАНИЕ О НЕПРЕРЫВНЫХ ДРОБЯХ, ПРЕОБРАЗОВАНИЯХ МЕБИУСА И ЦИКЛАХ

Рассмотрена взаимосвязь непрерывных дробей, преобразований Мебиуса и представлений циклов матрицами 2 x 2. B результате получено описание непрерывных дробей посредством цепей ортогональных или касательных орициклов. Одно из этих описаний было недавно предложено в статье A. Beardon и I. Short. Изложенный подход обобщается на высшие размерности способом, который согласуется с более ранним методом A. Beardon, основанным на алгебрах Клиффорда.

Ключевые слова: непрерывные дроби, преобразования Мебиуса, циклы, алгебра Клиффорда

Introduction

Continued fractions remain an important and attractive topic of current research [1—3], [4], §E.3. A fruitful and geometrically appealing method considers a continued fraction as an (infinite) product of linear-fractional transformations from the Möbius group. see Sec. 1. of this paper for an overview, papers [5—8], [9], Ex. 10.2 and in particular [10] contain further references and historical notes. Partial products of linear-fractional maps form a sequence in the Möbius group, the corresponding sequence of transformations can be viewed as a discrete dynamical system [10,11]. Many important questions on continued fractions, e.g. their convergence, can be stated in terms of asymptotic behaviour of the associated dynamical system. Geometrical generalisations of continued fractions to many dimensions were introduced recently as well [3,12].

Any consideration of the Möbius group introduces cycles — the Möbius invariant set of all circles and straight lines. Furthermore, an efficient treatment cycles and Möbius transformations are realised through certain 2 x 2 matrices, which we will review in Sec. 2., see also [9], [13], § 4.1, [14], [4], § 4.2, [15,16]. Linking the above relations we may propose the main thesis of the present note:

Claim 1 (Continued fractions and cycles). Properties of continued fractions may be illustrated and demonstrated using related cycles, in particular, in the form of respec-

tive 2 x 2 matrices.

One may expect that such an observation has been made a while ago, e.g. in the book [9], where both topics were studied. However, this seems did not happen for some reasons. It is only the recent paper [17], which pioneered a connection between continued fractions and cycles. We hope that the explicit statement of the claim will stimulate its further fruitful implementations. In particular, Sec. 3. reveals all three possible cycle arrangements similar to one used in [17]. Secs. 4.-5. show that relations between continued fractions and cycles can be used in the multidimensional case as well.

As an illustration, we draw on Fig. 1 chains of tangent horocycles (circles tangent to the real line, see [17] and Sec. 3.) for two classical simple continued fractions:

e = 2 +

1

1 +

2 +

1 +

1 + ...

n = 3 +

7 +

15 +

1+

1

1

1

1

1

1

1

One can immediately see, that the convergence for n is much faster: already the third horocycle is too small to be visible even if it is drawn. This is related to larger coefficients in the continued fraction for n.

Paper [17] also presents examples of proofs based on chains of horocycles. This intriguing construction was introduced in [17] ad hoc. Guided by the above claim we reveal sources of this and other similar arrangements of horocycles. Also, we can produce multidimensional variant of the framework.

1. Continued Fractions

We use the following compact notation for a continued fraction:

K (an\bn)

ai

bi +

a2

ai a2 аз bi + b2 + Ьз +.

b2 +

a3

Ьз + ••

(1)

Without loss of generality we can assume aj = 0 for all j. The important particular case of simple continued fractions, with an = 1 for all n, is denoted by K(bn) = K(1|bn). Any continued fraction can be transformed to an equivalent simple one.

It is easy to see, that continued fractions are related to the following linear-fractional (Möbius) transformation, cf. [5-8]:

Sn = Si О S2 о ... О Sn,

where

(z)

bJ +z

(2)

These Möbius transformations are considered as bijec-tive maps of the Riemann sphere C = C U {to} onto

ab , c d,

tional transformation z

itself. If we associate the matrix

to a liner-frac-

aZ+d, then the composition of two such transformations corresponds to multiplication of the respective matrices. Thus, relation (2) has the matrix form:

(P— Pn) _ (0 ai) (0 02) (0 a„)

\Qn-i Qn) _ V1 bi) V1 b2J V1 '

(3)

The last identity can be fold into the recursive formula:

Pn-i Qn-i

P Q

n \ = / Pn-2

=

Pn-2

Qn-2

Pn-i

Qn-i

\ (0 an\

) [} bn)

(4)

This is equivalent to the main recurrence relation:

Pn bnPn— i + anPn-2 Qn — bnQn-i + anQn-2

Pi = ai, Po

Qi — bi, Qo

0,

■■ 1, (5)

for n = 1, 2,3,....

The meaning of entries Pn and Qn from the matrix (3) is revealed as follows. Möbius transformation (2)-(3) maps 0 and to to

Pn-1

Qn — Sn(0),

Qn

Qn-i

— Sn(<x>).

(6)

It is easy to see that Sn(0) is the partial quotient of (1):

Pn _ ai a2 an

Qn bi + b2 + ... + bn

(7)

Properties of the sequence of partial quotients jQ—^ in

terms of sequences {an} and {bn} are the core of the continued fraction theory. Equation (6) links partial quotients with the Möbius map (2). Circles form an invariant family under Möbius transformations, thus their appearance for continued fractions is natural. Surprisingly, this happened only recently in [17].

2. Möbius Transformations and Cycles

If M

ab c d

is a matrix with real entries then

for the purpose of the associated Möbius transformations M : z ^ CZ+b we may assume that det M = ±1. The collection of all such matrices form a group. Möbius maps commute with the complex conjugation z ^ z. If det M > 0 then both the upper and the lower halfplanes are preserved; if det M < 0 then the two halfplanes are swapped. Thus, we can treat M as the map of equivalence classes z ~ z, which are labelled by respective points of the closed upper half-plane. Under this identification we consider any map produced by M with det M = ±1 as the map of the closed upper-half plane to itself.

The characteristic property of Möbius maps is that circles and lines are transformed to circles and lines. We use the word cycles for elements of this Möbius-invariant family [15,16,18]. We abbreviate a cycle given by the equation

k(u2 + v2) - 2lv - 2nu + m — 0

(8)

a

j

to the point (k, l, n, m) of the three dimensional projective space PR3. The equivalence relation z ~ z is lifted to the equivalence relation

(k,l,n,m) ~ (k,l, —n,m)

(9)

in the space of cycles, which again is compatible with the Möbius transformations acting on cycles.

The most efficient connection between cycles and Möbius transformations is realised through the construction, which may be traced back to [9] and was subsequently rediscovered by various authors [13], § 4.1, [14], [4], § 4.2, see also [15, 16]. The construction associates a cycle (k,l,n,m) with the 2 x 2 matrix C =

l +j}n —— mn), see discussion in [16], § 4.4 for a

justification. This identification is Möbius covariant: the

Möbius transformation defined by M = ^ ^ maps

a cycle with matrix C to the cycle with matrix MCM-1. Therefore, any Möbius-invariant relation between cycles can be expressed in terms of corresponding matrices. The central role is played by the Möbius-invariant inner product [16], § 5.3:

(c , C^ = R tr(CC),

which is a cousin of the product used in GNS construction of C*-algebras. Notably, the relation:

= 0 or km + mk - 2nn - 2Ù = 0, (11)

describes two cycles C = (k,l,m,n) and C = (k, l, m, n) orthogonal in Euclidean geometry. Also, the inner product (10) expresses the Descartes-Kirillov condition [4], Lem. 4.1(c), [16], Ex. 5.26 of C and C to be externally tangent:

(c + C,C + C^ = 0 or

(l + k)2 + (n + n)2 - (m + m)(k + k) = 0, (12)

where the representing vectors C = (k,l,n,m) and C = (k,l,m,n) from PR3 need to be normalised by the conditions {C, C) = 1 and (C, C\ = 1.

3. Continued Fractions and Cycles

Let M

be a matrix with real entries

and the determinant det M equal to ±1, we denote this by 6 = det M. As mentioned in the previous section, to calculate the image of a cycle C under Möbius transformations M we can use matrix similarity MCM-1.

If M

Pu—i Pn

is the matrix (3) associated to

^Qn— i Qr

a continued fraction and we are interested in the partial fractions , it is natural to ask:

Which cycles C have transformations MCM-1 depending on the first (or on the second) columns of M only? It is a straightforward calculation with matrices1 to check the following statements:

Lemma 1. The cycles (0, 0,1,m) (the horizontal lines v = m) are the only cycles, such that: their images un-

bJ are independent

der the Möbius transformation

from the column

(Ü>

The image associated to the col-

umn ( cj is the horocycle (c m, acm, 6, a2m), which touches the real line at a and has the radius .

c mc2

In particular, for the matrix (4) the horocycle is touching the real line at the point Qn-1 = Sn(œ) (6).

Qn - 1

Lemma 2. The cycles (k, 0,1,0) (with the equation k(u2 + v2) — 2v = 0) are the only cycles, such that their

images under the Möbius transformation ^ ^ arein-

a

dependent from the column . The image associated

b

a.

(10) to the column J is the horocycle (d2k, bdk, ó, b2k),

which touches the real line at d and has the radius jd¡?.

In particular, for the matrix (4) the horocycle is touching the real line at the point QQ" = Sn(0) (6). In view of these partial quotients the following cycles joining them are of interest.

Lemma 3. A cycle (0,1,n, 0) (any non-horizontal line passing 0) is transformed by (2)-(3) to the cycle

(2QnQn-i, PnQn-i + QnPn-i, 5n, 2PnPn-i), which passes points QQ" = Sn(0) and Q"-1 = S(œ) on the

Qn Qn-1

real line.

The above three families contain cycles with specific relations to partial quotients through Möbius transformations. There is one degree of freedom in each family: m, k and n, respectively. We can use the parameters to create an ensemble of three cycles (one from each family) with some Möbius-invariant interconnections. Three most natural arrangements are illustrated by Fig. 2. The first row presents the initial selection of cycles, the second row — their images after a Möbius transformation (colours are preserved). The arrangements are as follows:

1. The left column shows the arrangement used in the paper [17]: two horocycles are tangent, the third cycle, which we call connecting, passes three points of pair-wise contact between horocycles and the real line. The connecting cycle is also orthogonal to horocycles and the real line. The arrangement corresponds to the following values m = 2, k = 2, n = 0. These parameters are uniquely defined by the above tangent and orthogonality conditions together with the requirement that the horocycles' radii agreeably depend from the consecutive partial quotients' denomina-

This calculation can be done with the help of the tailored Computer Algebra System (CAS) as described in [16], App. B, [19].

a

c

tors:

i

and

i

Q

respectively. This follows

2QîLi

from the explicit formulae of image cycles calculated in Lemmas 1 and 2.

The central column of Fig. 2 presents two orthogonal horocycles and the connecting cycle orthogonal to them. The initial cycles have parameters m _ \/2, k _ \/2, n _ 0. Again, these values follow from the geometric conditions and the alike dependence of radii from the partial quotients' de--¿L- and Q

nominators:

2Q—

Finally, the right column has the same two orthog-

onal horocycles, but the connecting cycle passes one of two horocycles' intersection points. Its mirror reflection in the real axis satisfying (9) (drawn in the dashed style) passes the second intersection point. This corresponds to the values m = \/2, k = \/2, n = ±1. The connecting cycle makes the angle 45° at the points of intersection with the real axis. It also has the radius

V2 2

Pn

Qn

— the geomet-

2 \QnQn-i\

ric mean of radii of two other cycles. This again repeats the relation between cycles' radii in the first case.

Fig. 2. Various arrangements for three cycles. The first row shows the initial position, the second row — after a Möbius transformation (colours are preserved). The left column shows the arrangement used in the paper [17]: two horocycles touching, the connecting cycle is passing their common point and is orthogonal to the real line. The central column presents two orthogonal horocycles and the connecting cycle orthogonal to them. The horocycles in the right column are again orthogonal but the connecting cycle passes one of their intersection points and makes 45° with the real axis.

i

Three configurations have fixed ratio \f2 between respective horocycles' radii. Thus, they are equally suitable for the proofs based on the size of horocycles, e.g. [17], Thm. 4.1.

On the other hand, there is a tiny computational

advantage in the case of orthogonal horocycles. Let we

p.

have the sequence pj of partial fractions pj = Q and want to rebuild the corresponding chain of horocycles. A horocycle with the point of contact pj has components (1,Pj,nj,p2), thus only the value of nj need to be calculated at every step. If we use the condition "to be tangent to the previous horocycle", then the quadratic relation (2.) shall be solved. Meanwhile, the orthogonality relation (11) is linear in nj.

4. Multi-dimensional Möbius maps and cycles

It is natural to look for multidimensional generalisations of continued fractions. A geometric approach based on Möbius transformation and Clifford algebras was proposed in [12]. The Clifford algebra Cl(n) is the

associative unital algebra over R generated by the elements e1,...,en satisfying the following relation:

ei ej + ej ei

—Щ,

where Sj is the Kronecker delta. An element of Cl(n) having the form x = x1e1 + ... + xnen can be associated with vectors (x1,..., xn) e Rn. The reversion a ^ a* in Cl(n) [13], (1.19(ii)) is defined on vectors by x* = x and extended to other elements by the relation (ab)* = b*a*. Similarly the conjugation is defined on vectors by x = —x and the relation ab = ba. We also use the notation |a| = aa > 0 for any product a of vectors. An important observation is that any non-zero vectors x have a multiplicative inverse: x-1 = Xz■

By Ahlfors [20] (see also [12], § 5, [13], Thm. 4.10)

ab

ear-fractional transformation of Rn if the following conditions are satisfied:

1. a, b, c and d are products of vectors in Rn;

a matrix M

with Clifford entries defines a lin-

2. ab*, cd*, c*a and d*b are vectors in Rn;

3. the pseudodeterminant 6 := ad* — bc* is a nonzero real number.

Clearly we can scale the matrix to have the pseudodeterminant 6 = ±1 without an effect on the related linear-fractional transformation. Define, cf. [13], (4.7)

5. Continued fractions from Clifford algebras and horocycles

There is an association between the triangular matrices and the elementary Möbius maps of Rn, cf. (2):

M =

f d* —b*\

c* a* J

and M*

id b\ I с aj'

(13)

(!b) :

(x + b)

-1

(15)

. ' bnen.

Then MM = 6I and M = kM*, where k =1 or —1 depending either d is a product of even or odd number of vectors.

To adopt the discussion from Section 2. to several dimensions we use vector rather than paravector formalism, see [13], (1.42) for a discussion. Namely, we consider vectors x £ Rn+1 as elements x =

xiei + ... + xnen + xn+ien+i in Cl(n + 1). Therefore we can extend the Möbius transformation defined

a b

by M = d) with a, b,c,d e Cl(n) to act on Rn+1.

Again, such transformations commute with the reflection R in the hyperplane xn+1 = 0:

R : xiei + ... + Xnen + Xn+ien+i ^

where x = xiei + ... + xnen, b = biei +

Similar to the real line case in Section 1., Bear-don proposed [12] to consider the composition of a series of such transformations as multidimensional continued fraction, cf. (2). It can be again represented as the product (3) of the respective 2x2 matrices. Another construction of multidimensional continued fractions based on horocycles was hinted in [17]. We wish to clarify the connection between them. The bridge is provided by the respective modifications of Lem. 1—3. Lemma 4. The cycles (0, en+i, m) (the "horizontal" hyperplane xn+i = m) are the only cycles, such that their

b

images under the Möbius transformation

dependent from the column

d

are in-

The image associated

X\C\ + ... + Xnen xn+1en+1.

Thus we can consider the Möbius maps acting on the equivalence classes x ~ R(x).

Spheres and hyperplanes in Rn+1 — which we 2 2

continue to call cycles — can be associated to 2 x 2 ma- k(u + v ) — 2v

to the column y^j is the horocycle (—m |c|2 , —mac + öen+i,m |a|2), which touches the hyperplane xn+1 = 0

trices [14], [13], (4.12):

l ^fö '

kxx — Ix — xl + m = 0 ^ C = ( k с)' (14)

at ai and has the radius —a?.

\c\2 m\c\2

Lemma 5. The cycles (k,en+1,0) (with the equation 0) are the only cycles, such that

a b

lm

\k 4'

their images under the Möbius transformation

where k,m e R and l e Rn+i. For brevity we also encode a cycle by its coefficients (k,l,m). A justification of (14) is provided by the identity:

are independent from the column

ac.

c

d

The image asso-

ciated to the column is the horocycle (k |d| , kbd+ Sen+i, —kbb), which touches the hyperplane xn+i = 0

(1 x)

l m x

Vk Uw

at

bd

and has the radius

kxx — Ix — xl + m,

since x = —x for x e Rn. The identification is also Mobius-covariant in the sense that the transformation associated with the Ahlfors matrix M send a cycle C to the cycle MCM * [13], (4.16). The equivalence x - R(x) is extended to spheres:

lm

\k 4

(T

Я(Г),

The proof of the above lemmas are reduced to multiplications of respective matrices with Clifford entries.

Lemma 6. A cycle C = (0, l, 0), where l = x + ren+i and 0 = x e Rn, r e R, that is any non-horizontal hyperplane passing the origin, is transformed into MCM * = (cxd+dxc, axd+bx c+6ren+i,axb+bxa). This cycle passes points -^j? and .

c d, then the centre of

since it is preserved by the Möbius transformations with coefficients from Cl(n).

Similarly to (10) we define the Möbius-invariant inner product of cycles by the identity (c,C^ = Ktr(CC), where K denotes the scalar part of a Clifford number. The orthogonality condition (C,C) = 0

If x

(2 |c|2 |d|2

MCM*

ас |d|2 + bd |c|2 , (ac)(db) + (bd)(ca))

is

_ ( ac, !( ЙТ

+

bd ) + ôr d2 ) + 2d2d |

2 en+1, that is, the centre be-

means that the respective cycle is geometrically orthogonal in Rn+1.

longs to the two-dimensional plane passing the points a? and d? and orthogonal to the hyperplane xn+i =

0.

Proof. We note that en+ix = —xen+i for all x e Rn. Thus, for a product of vectors d e Cl(n) we have

x

c

1

m

en+id — d*en+i. Then

References

cen+id + de n+ic — (cd* — dc*)en+i — (cd* — (cd*)*)en+i —0,

1.

due to the Ahlfors condition 2. Similarly, aen+ib + 2 ben+\d = 0 and aen+\d + ben+\c = (ad* — bc*)en+i = Sen+i.

The image MCM* of the cycle C = (0, l, 0) is

(cld + dlc, ald + blc, alb + bla). From the above calculations for l = x + ren+i it becomes (cxd + dxc, axd + 3-bxc + Sren+i, axb + bxa). The rest of statement is verified by the substitution. □

4.

Thus, we have exactly the same freedom to 5. choose representing horocycles as in Section 3.: make two consecutive horocycles either tangent or orthogonal. To visualise this, we may use the two-dimensional g. plane V passing the points of contacts of two consecutive horocycles and orthogonal to xn+i = 0. It is natural to choose the connecting cycle (drawn in blue in Fig. 2) 7. with the centre belonging to V, this eliminates excessive degrees of freedom. The corresponding parameters are described in the second part of Lem. 6. Then, the inter- g. section of horocycles with V are the same as in Fig. 2.

Thus, the continued fraction with the partial quo-

tients pQQ

G Rn can be represented by the chain of

tangent/orthogonal horocycles. The observation made at the end of Section 3. on computational advantage of orthogonal horocycles remains valid in multidimensional situation as well.

As a further alternative we may shift the focus from horocycles to the connecting cycle (drawn in blue in Fig. 2). The part of the space Rn encloses inside the connecting cycle is the image under the corresponding Möbius transformation of the half-space of Rn cut by the hyperplane (0,l, 0) from Lem. 6. Assume a sequence of connecting cycles Cj satisfies the following two conditions, e.g. in Seidel-Stern-type theorem [17], Thm 4.1: 12

1. for any j, the cycle C ■ is enclosed within the cycle

9.

10.

11.

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14.

Cj-i;

2. the sequence of radii of Cj tends to zero. Under the above assumption the sequence of partial fractions converges. Furthermore, if we use the connecting cycles in the third arrangement, that is generated by the cycle (0,x + en+1, 0), where ||x|| = 1, x e Rn, then the above second condition can be replaced by following

(2') the sequence of x^j^ of (n + 1)-th coordinates of the centres of the connecting cycles Cj tends to zero.

Thus, the sequence of connecting cycles is a useful tool 15. to describe a continued fraction even without a relation to horocycles.

Summing up, we started from multidimensional continued fractions defined through the composition of 16. Möbius transformations in Clifford algebras and associated to it the respective chain of horocycles. This establishes the equivalence of two approaches proposed 17. in [12] and [17] respectively.

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Статья поступила в редакцию 05.09.2015.