- Email: [email protected]
On the periodic orbits of a strongly chaotic system
Physica D 32 (1988) 451-460 North-Holland. Amsterdam
ON THE PERIODIC ORBITS OF A STRONGLY
CHAOTIC
R. AURICH* and F. STEINER II. Institut fir Theoretwhe Physik, Universitiit
Chauwee
Homburg.
Luruper
SYSTEM
149, 2000 Humhurg
50, Fed. Rep. Germut!r
Received 30 March 1988 Revised manuscript received 14 July 1988 Communicated by F. H. Busse
A point particle sliding freely on a two-dimensional surface of constant negative curvature (Hadamard-Gutzwiller model) exemplifies the simplest chaotic Hamiltonian system. Exploiting the close connection between hyperbolic geometry and the group SU (1, l)/{ k 11, we construct an algorithm (symbolic dynamics). which generates the periodic orbits of the system. For the simplest compact Riemann surface having as its fundamental group the “octagon group”, we present an enumeration of more than 206 million periodic orbits. For the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m + fin (m, n E N are governed by a particular Beatty sequence), which reveals a strange arithmetical structure of chaos, Knowledge of the length spectrum is crucial for quantization via the Selberg trace formula (periodic orbit theory), which in turn is expected to unravel the mystery of quantum chaos.
1. Introduction In this paper we are presenting for the first time a detailed description of the spectrum of periodic orbits (closed geodesics) for the special case of a compact Riemann surface of genus g = 2. This is a surface of constant negative Gaussian curvature, K = -1 in appropriate units, which can be considered topologically as a double doughnut. Such surfaces are currently under intensive investigation [l], because they yield the most chaotic models of Hamiltonian systems with a few degrees of freedom. In 1898 this point of view had already been stressed by Hadamard who investigated the free motion of a mass point on such a surface. But now physicists are interested in the connection between classical chaos and its counterpart in quantum mechanics, which remains a mystery until now. There are some hints that the eigenvalue spectrum
of the quantum
mechanical
system
*Supported by Deutsche Forschungsgemeinschaft Contract No. DFG-Ste 241/4-l.
dis-
under
0167-2789/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
plays some unique statistical properties [2], if the classical counterpart is chaotic. The most promising way - as emphasized by Gutzwiller [3] - is to employ the Selberg trace formula [4-61, which yields a kind of duality relation between the lengths of periodic orbits of the classical system and the eigenvalues of the quantum mechanical system. Until now a direct use of this remarkable formula was impossible, because of lack of knowledge of the classical periodic orbits. In this paper we describe the properties of the length spectrum for the special case of genus g = 2. A first application of the Selberg trace formula using this length spectrum as an input can be found in our recent paper [71. The classical dynamics on the pseudosphere, the whole surface of constant negative curvature, can be described conveniently by one of the models which map the infinitely extended pseudosphere on a special domain of the complex plane. We use the so-called Poincare disc, where the special domain is the interior of the unit circle, i.e. each point of the pseudosphere is mapped into the unit circle on the complex plane. There one has the
B.V.
452
R. Aurtch
metric
tensor
SL, =
with
Cl
_
of hyperbolic
geometry
Periodic
(i, j = 1,2)
and
motion
of a particle
surface
M of constant
xf +x2” < 1. The classical
of mass m sliding negative
freely on a
curvature
is gov-
by the Lagrangian
c,haotrc system
the closed periodic
orbits:
(2)
and takes place along euclidean circles (geodesics) intersecting perpendicularly the unit circle. The classical Hamiltonian is given by
1 H = 2mPlg”PJ>
where p, = mg,‘(dxJ/dt) and g’J is the inverse of g,]. (We call the dynamical system defined by (2) and (3) the Hadamard-Gutzwiller model.) The energy E = L = H is the only constant of and the dynamics
where b denotes T(b) its period. periodic we obtain
orbit
a primitive periodic orbit and With I(b) = hT( 6) = length of
b with energy
Huber’s
E and period
T(b),
law [12], see eq. (12) below.
Thus the length spectrum {f(b)} on M shows an exponential proliferation of long periodic orbits.
dx’ dx’ L = Tg,,-;i-idt
motion,
orhtts o/c1 mm,&
-;‘_ x;)2 6iJ7
z = x1 + ix,
erned
and F. Steiner/
is the geodesic
how on
M,
2. The length spectrum of periodic orbits
2.1. The fundamentul generating boosts
domain and the
Here we briefly describe how to construct a compact Riemann surface of constant negative curvature. The starting point is the whole surface of constant negative curvature called pseudosphere, on which one can define boosts in analogy to translations on the plane. These boosts together with the rotations span the three-dimensional Lorentz group 9 on the pseudosphere. Considering a discrete subgroup G ~9 containing only boosts, one can tessellate the pseudosphere by identifying the points z and z’, where Z’ = bz and b E G. Fig. 1 shows the tessellation of the pseudosphere in regular octagons. The distortion of the
There are no invariant tori in phase space, the system has very sensitive dependence on initial conditions, and almost all orbits are dense [8]. The system has the Anosov property [9]: neighbouring trajectories diverge with time at the rate e”‘, i.e. the trajectories are unstable, a typical property of chaos. From Jacobi’s equation for the geodesic deviation one obtains for the Liapunov exponent w = J2E/m. Pesin’s equality [lo], h = w, relates w to the Kolmogorov-Sinai entropy h [ll], which in turn determines the exponential proliferation of
octagons near the boundary of the Poincare disc is due to the metric (1). The tessellation can be viewed as cutting out a piece of the whole pseudosphere and gluing together the edges according to the connection rules of the group G. As already mentioned, we use the Poincare disc as the model for the pseudosphere. There the Lorentz group is the pseudo-unitary group SU (l,l)/{ k l}, because in the Poincart disc D a boost or a rotation can be represented by 2 x 2 matrices LY
P l(Y12- IpI’=
T= i p*
L-x* I’
1.
(4)
R. Aunch and F. Steiner/
Penodic orbits of a strongly chaotic system
The
2g
generators
453
differ
e * i(kn’2g) in the offdiagonal mines
the direction
as the geodesic the action
by the
factor
of (6) which
deter-
of the invariant
curve
line, defined
V which is mapped
of b, onto itself. The invariant
the b,‘s intersect at the origin domain.
only
each other at angles of 6 = 7r/2g
t = 0 yielding
Eq. (7) follows
bolic geometry:
under lines of
a regular
fundamental
from elementary
The dotted triangle
hyper-
in fig. 1 has an
area A
4qr(g - 1)
AA=8g=
gg
=Tr-(“+/I+&
and from (Y= S/2 =~/4g and tains p = “/4g = (Y.Furthermore
y =n/2 one ob(see fig. 1)
cos p 71 cash b = sin = cot a = cot 4g ) Fig. 1. The tagons.
if one
tesellation
defines
linear
fractional
of the pseudosphere
the action
in regular
oc-
of T on z E D as the
and realizing that b, translates a point z on the line b by the distance I, = 2b, one arrives at eq. (7).
transformation
2.2. The periodic orbits
The boosts are hyperbolic transformations, i.e. + 1Tr (T)l = 1%a) > 1, whereas rotations are elliptic ones, 1!J?(Y1 < 1. If the subgroup G is generated by the 2g generators
1
(g 2 2)
1 cash A 2
ei(k+g)
1 sinhA 2
’
Now we describe our method to find all periodic orbits, at least theoretically. In practice this method yields the lower part of the length spectrum which leads already to very interesting physical results as discussed in ref. [7]. Every group element b E G belongs to a periodic orbit. Hence one can find the length spectrum by calculating all group elements by forming the
b, =
product e-i(k42g)
sinh 210
’
10 cash z
\
out of the generating
b, and their
/ k=0,...,2g-
1,
(6) b = b,lb,z . . . b,,z =
(and their inverses,
boosts
inverses:
(8)
k = 2g,. . . ,4g - l), where
(7)
The length l(b) of the periodic orbit belonging to this b E G depends only on the real part of cx, I(b)=2arcosh]%a].
the resulting fundamental domain, which has 4g edges and an area of A = 4~( g - l), represents a regular surface of genus g.
(9)
To confirm eq. (9) consider the trace of the matrix b which is equal to the trace of the matrix b,
representing the same boost in the PoincarC upper half-plane* : e1/2
order 0
G is a subgroup
of the
quotient
group
one must take 1591a1 instead of SU(Ll)/{fl}, ‘%a, which then gives eq. (9). However, if one would generate all group elements successively one would count the orbits repeatedly because two group elements h and 6 have the same orbit if they belong to the same conjugacy class [b] = { b’lb’ = ibk’, 6 E G}. In other words, one must generate the conjugacy classes instead of the group elements itself. In addition, there is the identity b()b,‘b,b,‘b,‘b,b,‘b,
= 1.
and more
to the length
generating spectrum
12 or larger. The column contains
boosts
may
at length
of
“multiplicity”
a number
in
in parenthe-
sis which must be added to the multiplicity if one is interested in the multiplicity of all orbits includ-
e-//2 1
=2cosh;. Since
of twelve
table I sometimes
2%Ha=Trb=Trb,=Tr ( 0
ucts
contribute
(10)
which has to be used in its many equivalent forms in the process of enumerating the conjugacy classes. Therefore we developed a computer program which successively generated the conjugacy classes for the surface of genus g = 2 corresponding to the regular octagon shown in fig. 1. It calculated all conjugacy classes which had elements consisting of a product up to 11 generating boosts b,. This program provided more than 206 million conjugacy classes to which belong more than 204000 orbits of different lengths. The first 80 different lengths of primitive orbits are given in table I. The table shows that many orbits have the same length which is denoted as multiplicity**. This multiplicity is expected to be a bit too low at greater lengths because our computation takes only into account the conjugacy classes consisting of products of at most 11 generating boosts. Prod-
*The Poincxe upper half-plane has the advantage that one always can choose half-plane coordinates in which II, is diagonal. **The multiplicities given in table I take into account also the traversals backwards in time.
ing multiple
traversals,
The integers below.
m and n are explained
Some periodic
i.e. non-primitive
orbits
are shown
tion in their fundamental
domains
orbits.
in section
2.4
as an illustrain fig. 2. Figs.
2(a) to 2(f) display orbits in the regular octagon. whereas figs. 2(g) and 2(h) represent examples in the case of surfaces respectively.
of genus
g = 3 and
g = 8
Table I reveals unusual high multiplicities which are partly due to the symmetry of the regular octagon considered. Mostly, there are 16 orbits of the same type, because a generic orbit (e.g. fig. 2(c, d) can be rotated in steps of 71/4 yielding 8 different orbits due to the trivial symmetry of the octagon. Including backward traversals one arrives at 16 orbits. However, an orbit which is mapped under 4 (2 or 1) consecutive rotations by IT/~ onto itself belongs to a family of 8 (4 or 2) orbits (e.g. fig. 2(b, a, f), respectively. Further symmetries of the regular octagon (e.g. the shortest diameter I, is equal to the hyperbolic distance between two adjacent vertices) lead to additional degeneracies of orbit lengths. Despite of these symmetries the system loses, however, nothing of its chaotic properties. We would like to emphasize that the above mentioned symmetries are by far not enough to explain the chaotic behaviour of the multiplicities which will be discussed in the next section. 2.3. The asymptotic behaviour of the length spectrum Let us define the number N(I) orbits of length less than 1 by
N(l) :=
of all primitive
c g,, /,,< I
where the gn’s are the multiplicities
of the lengths
R. Aurich
and F. Steiner/
Periodic
orh~ts
of u stron&
455
chaotic system
Table I The first 80 lengths of the length spectrum of the regular octagon.
Length of periodic orbit
Multiplicity
3.0571418390 4.8969048954 5.8280707754 6.1142836779 6.6720057699 7.1073758741 7.2631634751 7.5956918304 7.8806922887 8.1300755289 8.2249036233 8.4368496405 8.6284636565 8.7027505564 8.8714798107 9.0270717197 9.1714255169 9.2282950896 9.3592716579 9.4821941493 9.5309770571 9.6440656486 9.7510997583 9.7938097907 9.8933106878 9.9880946367 10.0785887303 10.1149054114 10.1999558888 10.2815363765 10.3143770353 10.3915072941 10.4657729697 10.5373792543 10.5663046423 10.6344594911 10.7003680283 10.7270445783 10.7900178989 10.8510687003
I,
which
are
ordered
The
periodic
orbits
long
lengths
as shown
as
24 24 48 0 (24) 96 4X 48 8 96 48 192 48 96 48 288 12 48 (24) 96 192 48 192 96 336 0 (24) 192 192 192 96 384 96 192 32 3X4 96 288 96 288 96 576 4X
0 c I,,<
proliferate
I, ~1,
<
exponentially
by Huber
m
n
1 3 5 5 7 9 9 11 13 15 15 17 19 19 21 23 25 25 27 29 29 31 33 33 35 37 39 39 41 43 43 45 47 49 49 51 53 53 55 57
1 2 3 4 5 6 7 X 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
..’
. for
Length of periodic orbit 10.8758208811 10.9343449461 10.9912049969 11.0464930504 11.0689538604 11.1221615882 11.1739903506 11.2450675676 11.2938444648 11.3414600077 11.3608557221 11.4069202001 11.4519475300 11.4703055581 11.5139434106 11.5566494092 11.5984625221 11.6155292590 11.6561415509 11.6959455360 11.7122036393 11.7509179318 11.7888970234 11.8044196553 11.8414054395 11.8777196407 11.913386213X 11.9279754971 11.9627646508 11.9969589861 12.0443403136 12.0771791386 12.1094874621 12.1227186314 12.1543053200 12.1854008904 12.1981408207 12.2285673558 12.2585379275 12.2708217905 ____________
eration
-
This
behaviour
terns,
whereas
+ @q-“-l)
)
(12)
i
is characteristic a power
I-+ co.
law
for
determines
chaotic
sys-
the prolif-
II
96 288 432 96 288 192 384 576 288 288 96 672 192 480 4X 672 48 576 192 (4x) 4x0 288 528 96 192 3x4 576 40 560 3x4 3x4 576 288 3x4 96 864 3x4 3x4 48 (24) 76x 96
57 59 61 63 63 65 67 69 71 73 73 75 77 77 79 Xl 83 83 X5 x7 X7 x9 91 91 93 95 97 97 99 101 103 105 107 107 109 111 111 113 115 115
41 42 43 44 45 46 47 49 50 51 52 53 54 55 56 57 5x 59 60 61 62 63 64 65 66 67 6x 69 70 71 73 74 75 76 77 78 79 x0 81 X2
The
with
lengths ;
f $ ! n=O
N(1)
01
of periodic
systems. well
[12]:
Multiplicity
our
orbits
asymptotic computation
as depicted
account
the correction
I = 14.5
the calculated
the prediction which
is
in the case expression
(12).
a reflection
down
in fig.
of
the
coincides
the we
shortest take
into
up to N = 3. Above
staircase see
to
3, where
terms
of integrable (12)
function
dotted
missing
line lengths
is below in fig. as
3, ex-
(b)
,p
I 5
3
9
7
11
IS
IS
I
Fig. 3. The staircase function N(I) as defined in eq. (11) in comparison with Huber’s law (12) (dotted line).
(4
plained in section 2.2. With the aid of our computation we have empirically found the law which determines the proliferation of the different lengths themselves:
without
regard
to their multiplicities.
The law
(fl
with p = 11.3 is in excellent agreement with our numerical results. In order to check eq. (14) consider the length differences between two adjacent lengths
(9) Fig. 2. (a) The to h,,h>h;‘h;‘. riodic orb;t h,,h, ‘h,h, ‘h,h, riodic orbit
(h)
periodic orbit in the regular octagon belonging (m, n) = (23.16). I,, = 9.02707.. (b) The pein the regular octagon belonging to I. (m, n) = (5,3); I, = 5.82807.. (c) The pein the regular octagon belonging to b,,b,h~lb,b,b~‘. (m,n)=(39,27), /,,=10.07858 (d) The periodic orbit in the regular octagon belonging to b,b,b,b, ‘b,bim’. (m,n)=(lS,ll): I,,=8.22490... (e) The periodic orbit in the regular octagon belonging to b I b 3b I b 3b z ‘. (m, n) = (15,ll); I, = 8.22490. _. (0 The periodic orbit in the regular octagon belonging to b0,230 b h b b -lb, ‘b,‘b, ‘. (m, n) = (25343,17920); I,, = 23.05309.. (g) The periodic orbit in the case of genus g = 3 belonging to b 0 b 0 b 5h0 h 3 (h) The periodic orbit in the case of genus g = X belonging to h, b,b,b,.
(“length
density”)
1 A6 I ntl - 1, =x--T-Fe
d*(l)
1
,,2 (15)
and define s(n)
:= (In+1 - I,@
(16)
which should fluctuate around 1 if eq. (14) is correct. Indeed this behaviour is confirmed in fig. 4, where we use p = 8fi as will be explained later. The exceptionally high spikes seen in fig. 4 are due to missing orbits which are generated by more than 11 boosts and to non-primitive orbits which
6(n) 20
15
10
OS
00
Fig. 4. The length differences shown where p = 80.
weighted
by exp (1,,/2)/p
are
Fig. 5. The multiplicity g, of the orbit length I,, plotted as squares is compared with the mean multiplicity (17) (the smooth curve).
are not taken into account in fig. 4 (see for example n = 4 or n = 24 in table I). The law (14) uniquely determines the mean multiplicity g(l): To be consistent with Huber’s law, the following asymptotic law should hold: 1500
g^(l) - py.
-
(17)
Fig. 5 reveals large fluctuations around this mean multiplicity g(l). If one compares, however, the mean value S, over the interval [[, - 4, I, + 41, s,=;
(18)
gk,
c I, - : < I, i I, + i
with the mean multiplicity
S(l), one observes
good
agreement until lengths of order 14.5 (fig. 6). Above these lengths the mean multiplicity S, lies somewhat below &I) for reasons explained above.
2.4.
Arithmetical
properties
has matrix
line
of periodic orbits
In the case of the regular b=
Fig. 6. The mean multiplicity (18) is plotted as dashed and is compared with g(1) eq. (17) (the smooth curve).
octagon
each boost
EG
elements
a=a+\/2b,
consisting
-
of algebraic -
p=d1+\/2
numbers -
(c+\/2d),
(19)
R. A “rich and F. Sterner/
45x
where
a, h, c and d are of the form n, + in,,
a+fib
Penodic
orbits of u stronglv
n,, n2 E Z. Indeed,
chuotlc gxttw
any boost b E G has the representation
il+fi(c+fid)
b=
(20) I v’il_t(
c* + fid*)
a* + fib*
because on the one hand the generating g = 2 combined with
I>
boosts b, have this form as follows from eqs. (6) and (7) for genus
cosh$=cot;=I+&,
(21)
and on the other hand every product
of matrices
of the form (20) is again a matrix
of this form:
(22) where
cx= (a + fib)(
a^+ 06)
+ (1-t
a)(
c + fid)(
c^*+ ad’*)
= a’ + fib’,
(23)
(24) and a’, b’, c’, d’ are again of the form n; + in;. Thus each orbit length is simply determined by 1Sal = m + fin, m, n E N, and because of this important fact these two integers are given in table I too. There one makes a surprising discovery: The n’s are running through all natural numbers with two exceptions, because there are no orbits at n = 48 and tr = 72. These voids are probably filled up if one takes into account conjugacy classes with elements consisting of twelve and more generating boosts. The rule for the integers m is not so simple. They run through all odd integers but there are sometimes “pairs” of odd integers without any systematics, as it appears at first sight. Nevertheless one can reproduce the sequence (m, n) by the following “fi-rule”: _ n = 1,2,3, . . . ~ m is that odd natural
number
which minimizes
1m/n - fi I.
So the sequence (m, n) is simply a sequence which approximates $2 supplemented by the subsidiary condition that m is an odd natural number. Therefore the length spectrum is completely determined by its arithmetical properties, see eq. (26) below. Unfortunately we are unable to formulate a likewise simple rule for the multiplicity.
R. Aurich
and F. Steiner/
Penodlc
450
orbits of a strong!&, chuotzc .~~.~trnl
We would like to mention a connection between the “pairs” in the m sequence sequence. The “pairs” of m occur at the following values of n:
If l/x
+ l/y
appears
Beatty
.
3,6,10,13,17,20,23,27,30;.. The same sequence
and the so-called
(25)
in a special case of Beatty’s theorem
= 1, where x and y are positive
irrational
[13] which states:
numbers,
then the sequences
[~],[2~],[3~],...,[~1,[2~1,[3~l,... together
include
every positive
By the special
integer just once, where [x] means
choice x = 2 + fi
and
y = fi
the sequence
(25). Let us now show that the law (14) follows from the “fi-rule”. I n-l 2
cosh
= m+fin,
the integral
[x],[~x],[~x], Writing
part of x. . . . is exactly
the sequence
eq. (19) in the form
m,nEN
(26)
we could use m = [fin]. But to get a more manageable expression we instead use the asymptotically correct value m = fin despite of the fact that this m is not an integer of course. Then we can asymptotically 1
2e
express eq. (26) as
in/2 _
2&l
(27)
or n
_
1
-
eL/2
Realizing
that
Z?(l)-
(28)
.
4&C
1, is the nth length we have recovered
-&e1/2,
whencep=8&.
The theoretical value p = 8fi determined from fig. 4.
3. Summary
eq. (14):
= 11.31 . . .
(29) is in excellent
and discussion
In this paper we have presented for the first time an explicit enumeration of the shortest periodic orbits (classical paths) for a prototype example of a chaotic Hamiltonian system: a point particle sliding freely on a two-dimensional surface of constant negative curvature. Our computations crucially relied on the intimate relation
between
agreement
with
the periodic
our
orbits
empirical
value
and the topology,
11.3
ge-
ometry and group theory of compact Riemann surfaces of genus g 2 2. To keep things as simple as possible, we have chosen g = 2 and considered the most regular fundamental region, i.e. the hyperbolic octagon shown in fig. 1. The corresponding Riemann surface M can be identified with D/G, the action of a particular Fuchsian group, the “octagon group” G on the PoincarC disc D.
460
R. Aurich
The octagon generators boosts,
group
and F. Sterner/
G can be constructed
b, (k = 0,1, . . . ,7) playing
whose
explicit
matrix
Periodtc orhrts o/u strong!\,
from 8
the role of
representation
has
been given in eq. (6). By forming all possible “words” of arbitrary length from the alphabet consisting of the 8 “letters” b, by matrix multiplication, one can construct in principle all group elements of G. Of great importance is the classitication of the group elements in terms of conjugacy classes, since such a classification is equivalent to an enumeration of periodic orbits on M. The (hyperbolic) length f(b) of a periodic orbit corresponding to a given group element b E G is uniquely determined by eq. (9). Although one is led along these lines to a well defined algorithm for calculating the periodic orbits together with their lengths, the actual computation could only be carried out on a large computer. Within the computer time available to us, we were able to reconstruct all words with a maximal length of 11 letters. As a result we obtained 206796242 primitive periodic orbits. The shortest 80 lengths and their multiplicities have been given in table I. A few typical examples of periodic orbits have been displayed in figs. 2(a-h). Surprisingly enough, the “length staircase” N(I), defined in eq. (11) is nicely described by the asymptotic behaviour (12) down to the shortest lengths, as can be seen from fig. 3. Only at the upper end of the staircase one observes a deviation, which is caused by missing words formed out of 12 or more letters. In section 2.4 we derived a curious representation of the group elements b E G, eq. (20) in terms of algebraic numbers of the form n, + fin,, n, E Z. Thus the length spectrum {1(b)} is completely determined by its arithmetical properties as given in eq. (26). We have thus discovered a close connection between the length spectrum of the octagon group G and number theory. It is hoped, that this connection can be deepened in order to shed some light on the multiplicity distribution and to reveal the arithmetical structure of chaos. Work in this direction is being carried out. Using the arithmetic properties of the length spectrum we could derive the asymptotic law (14) and
thus
chaotic system
predict
the value
which, originally,
8fi
for the parameter
was introduced
eter in eq. (14). The predicted
value
excellent
numerical
agreement
(see figs. 4-6) The
length
constitutes
with
our
p
as a free paramfor p is in results
which gave p = 11.3. spectrum
an important
obtained
in
this
paper
input in the Selberg trace
formula [3-61, which is an exact substitute for the Bohr-Sommerfeld-Einstein quantization rules for our chaotic system, and which can be interpreted as an (exact) instanton-gas evaluation of the corresponding Feynman path integral. It establishes a striking duality relation between the quantum mechanical energy spectrum and the lengths of the classical trajectories. On the basis of this exact “periodic orbit theory” we hope to unravel the mystery of quantum chaos. Investigations along these lines are under study; a preliminary announcement is given in ref. [7].
Acknowledgements We would like to thank G. Miinster our attention to ref. [13].
for drawing
References [l] N.L. Balazs and A. Voros, Phys. Rep. 143 (1986) 109. [2] See e.g. M.V. Berry, Proc. R. Sot. London A400 (1985) 229; A413 (1987) 183. [3] M.C. Gutzwiller, Phys. Rev. Lett. 45 (1980) 150; Phys. Scripta T9 (1985) 184. [4] A. Selberg, J. Ind. Math. Sot. 20 (1956) 47. [5] F. Steiner, Phys. Lett. B 188 (1987) 447. [6] F. Steiner, in: Recent Developments in Mathematical Physics, and Proc. Schladming Conf. 1987 (Springer. Berlin, 1987). p. 305. [7] R. Aurich, M. Sieber and F. Steiner, Phys. Rev. Lett. 61 (1988) 483. [S] E. Artin, Abh. Math. Sem. UniversitZt Hamburg 3 (1924) 170. [9] D.V. Anosov, Proc. Steklov Inst. Math. 90 (1967). Am. Math. Sot. Trans. (1969). [lo] Ya.V. Pesin, Sov. Math. DokI. 17 (1976) 196. [ll] See e.g. Ya.G. Sinai, Introduction to Ergodic Theory (Princeton University Press, 1976). [12] H. Huber, Math. Ann. 138 (1959) 1. [13] H.S.M. Coxeter, Scripta Math. 19 (1953) 135; LG. Connell, Can. Math. Bulletin 2 (1959) 181, 190; 3 (1960) 17.
ON THE PERIODIC ORBITS OF A STRONGLY
CHAOTIC
R. AURICH* and F. STEINER II. Institut fir Theoretwhe Physik, Universitiit
Chauwee
Homburg.
Luruper
SYSTEM
149, 2000 Humhurg
50, Fed. Rep. Germut!r
Received 30 March 1988 Revised manuscript received 14 July 1988 Communicated by F. H. Busse
A point particle sliding freely on a two-dimensional surface of constant negative curvature (Hadamard-Gutzwiller model) exemplifies the simplest chaotic Hamiltonian system. Exploiting the close connection between hyperbolic geometry and the group SU (1, l)/{ k 11, we construct an algorithm (symbolic dynamics). which generates the periodic orbits of the system. For the simplest compact Riemann surface having as its fundamental group the “octagon group”, we present an enumeration of more than 206 million periodic orbits. For the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m + fin (m, n E N are governed by a particular Beatty sequence), which reveals a strange arithmetical structure of chaos, Knowledge of the length spectrum is crucial for quantization via the Selberg trace formula (periodic orbit theory), which in turn is expected to unravel the mystery of quantum chaos.
1. Introduction In this paper we are presenting for the first time a detailed description of the spectrum of periodic orbits (closed geodesics) for the special case of a compact Riemann surface of genus g = 2. This is a surface of constant negative Gaussian curvature, K = -1 in appropriate units, which can be considered topologically as a double doughnut. Such surfaces are currently under intensive investigation [l], because they yield the most chaotic models of Hamiltonian systems with a few degrees of freedom. In 1898 this point of view had already been stressed by Hadamard who investigated the free motion of a mass point on such a surface. But now physicists are interested in the connection between classical chaos and its counterpart in quantum mechanics, which remains a mystery until now. There are some hints that the eigenvalue spectrum
of the quantum
mechanical
system
*Supported by Deutsche Forschungsgemeinschaft Contract No. DFG-Ste 241/4-l.
dis-
under
0167-2789/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
plays some unique statistical properties [2], if the classical counterpart is chaotic. The most promising way - as emphasized by Gutzwiller [3] - is to employ the Selberg trace formula [4-61, which yields a kind of duality relation between the lengths of periodic orbits of the classical system and the eigenvalues of the quantum mechanical system. Until now a direct use of this remarkable formula was impossible, because of lack of knowledge of the classical periodic orbits. In this paper we describe the properties of the length spectrum for the special case of genus g = 2. A first application of the Selberg trace formula using this length spectrum as an input can be found in our recent paper [71. The classical dynamics on the pseudosphere, the whole surface of constant negative curvature, can be described conveniently by one of the models which map the infinitely extended pseudosphere on a special domain of the complex plane. We use the so-called Poincare disc, where the special domain is the interior of the unit circle, i.e. each point of the pseudosphere is mapped into the unit circle on the complex plane. There one has the
B.V.
452
R. Aurtch
metric
tensor
SL, =
with
Cl
_
of hyperbolic
geometry
Periodic
(i, j = 1,2)
and
motion
of a particle
surface
M of constant
xf +x2” < 1. The classical
of mass m sliding negative
freely on a
curvature
is gov-
by the Lagrangian
c,haotrc system
the closed periodic
orbits:
(2)
and takes place along euclidean circles (geodesics) intersecting perpendicularly the unit circle. The classical Hamiltonian is given by
1 H = 2mPlg”PJ>
where p, = mg,‘(dxJ/dt) and g’J is the inverse of g,]. (We call the dynamical system defined by (2) and (3) the Hadamard-Gutzwiller model.) The energy E = L = H is the only constant of and the dynamics
where b denotes T(b) its period. periodic we obtain
orbit
a primitive periodic orbit and With I(b) = hT( 6) = length of
b with energy
Huber’s
E and period
T(b),
law [12], see eq. (12) below.
Thus the length spectrum {f(b)} on M shows an exponential proliferation of long periodic orbits.
dx’ dx’ L = Tg,,-;i-idt
motion,
orhtts o/c1 mm,&
-;‘_ x;)2 6iJ7
z = x1 + ix,
erned
and F. Steiner/
is the geodesic
how on
M,
2. The length spectrum of periodic orbits
2.1. The fundamentul generating boosts
domain and the
Here we briefly describe how to construct a compact Riemann surface of constant negative curvature. The starting point is the whole surface of constant negative curvature called pseudosphere, on which one can define boosts in analogy to translations on the plane. These boosts together with the rotations span the three-dimensional Lorentz group 9 on the pseudosphere. Considering a discrete subgroup G ~9 containing only boosts, one can tessellate the pseudosphere by identifying the points z and z’, where Z’ = bz and b E G. Fig. 1 shows the tessellation of the pseudosphere in regular octagons. The distortion of the
There are no invariant tori in phase space, the system has very sensitive dependence on initial conditions, and almost all orbits are dense [8]. The system has the Anosov property [9]: neighbouring trajectories diverge with time at the rate e”‘, i.e. the trajectories are unstable, a typical property of chaos. From Jacobi’s equation for the geodesic deviation one obtains for the Liapunov exponent w = J2E/m. Pesin’s equality [lo], h = w, relates w to the Kolmogorov-Sinai entropy h [ll], which in turn determines the exponential proliferation of
octagons near the boundary of the Poincare disc is due to the metric (1). The tessellation can be viewed as cutting out a piece of the whole pseudosphere and gluing together the edges according to the connection rules of the group G. As already mentioned, we use the Poincare disc as the model for the pseudosphere. There the Lorentz group is the pseudo-unitary group SU (l,l)/{ k l}, because in the Poincart disc D a boost or a rotation can be represented by 2 x 2 matrices LY
P l(Y12- IpI’=
T= i p*
L-x* I’
1.
(4)
R. Aunch and F. Steiner/
Penodic orbits of a strongly chaotic system
The
2g
generators
453
differ
e * i(kn’2g) in the offdiagonal mines
the direction
as the geodesic the action
by the
factor
of (6) which
deter-
of the invariant
curve
line, defined
V which is mapped
of b, onto itself. The invariant
the b,‘s intersect at the origin domain.
only
each other at angles of 6 = 7r/2g
t = 0 yielding
Eq. (7) follows
bolic geometry:
under lines of
a regular
fundamental
from elementary
The dotted triangle
hyper-
in fig. 1 has an
area A
4qr(g - 1)
AA=8g=
gg
=Tr-(“+/I+&
and from (Y= S/2 =~/4g and tains p = “/4g = (Y.Furthermore
y =n/2 one ob(see fig. 1)
cos p 71 cash b = sin = cot a = cot 4g ) Fig. 1. The tagons.
if one
tesellation
defines
linear
fractional
of the pseudosphere
the action
in regular
oc-
of T on z E D as the
and realizing that b, translates a point z on the line b by the distance I, = 2b, one arrives at eq. (7).
transformation
2.2. The periodic orbits
The boosts are hyperbolic transformations, i.e. + 1Tr (T)l = 1%a) > 1, whereas rotations are elliptic ones, 1!J?(Y1 < 1. If the subgroup G is generated by the 2g generators
1
(g 2 2)
1 cash A 2
ei(k+g)
1 sinhA 2
’
Now we describe our method to find all periodic orbits, at least theoretically. In practice this method yields the lower part of the length spectrum which leads already to very interesting physical results as discussed in ref. [7]. Every group element b E G belongs to a periodic orbit. Hence one can find the length spectrum by calculating all group elements by forming the
b, =
product e-i(k42g)
sinh 210
’
10 cash z
\
out of the generating
b, and their
/ k=0,...,2g-
1,
(6) b = b,lb,z . . . b,,z =
(and their inverses,
boosts
inverses:
(8)
k = 2g,. . . ,4g - l), where
(7)
The length l(b) of the periodic orbit belonging to this b E G depends only on the real part of cx, I(b)=2arcosh]%a].
the resulting fundamental domain, which has 4g edges and an area of A = 4~( g - l), represents a regular surface of genus g.
(9)
To confirm eq. (9) consider the trace of the matrix b which is equal to the trace of the matrix b,
representing the same boost in the PoincarC upper half-plane* : e1/2
order 0
G is a subgroup
of the
quotient
group
one must take 1591a1 instead of SU(Ll)/{fl}, ‘%a, which then gives eq. (9). However, if one would generate all group elements successively one would count the orbits repeatedly because two group elements h and 6 have the same orbit if they belong to the same conjugacy class [b] = { b’lb’ = ibk’, 6 E G}. In other words, one must generate the conjugacy classes instead of the group elements itself. In addition, there is the identity b()b,‘b,b,‘b,‘b,b,‘b,
= 1.
and more
to the length
generating spectrum
12 or larger. The column contains
boosts
may
at length
of
“multiplicity”
a number
in
in parenthe-
sis which must be added to the multiplicity if one is interested in the multiplicity of all orbits includ-
e-//2 1
=2cosh;. Since
of twelve
table I sometimes
2%Ha=Trb=Trb,=Tr ( 0
ucts
contribute
(10)
which has to be used in its many equivalent forms in the process of enumerating the conjugacy classes. Therefore we developed a computer program which successively generated the conjugacy classes for the surface of genus g = 2 corresponding to the regular octagon shown in fig. 1. It calculated all conjugacy classes which had elements consisting of a product up to 11 generating boosts b,. This program provided more than 206 million conjugacy classes to which belong more than 204000 orbits of different lengths. The first 80 different lengths of primitive orbits are given in table I. The table shows that many orbits have the same length which is denoted as multiplicity**. This multiplicity is expected to be a bit too low at greater lengths because our computation takes only into account the conjugacy classes consisting of products of at most 11 generating boosts. Prod-
*The Poincxe upper half-plane has the advantage that one always can choose half-plane coordinates in which II, is diagonal. **The multiplicities given in table I take into account also the traversals backwards in time.
ing multiple
traversals,
The integers below.
m and n are explained
Some periodic
i.e. non-primitive
orbits
are shown
tion in their fundamental
domains
orbits.
in section
2.4
as an illustrain fig. 2. Figs.
2(a) to 2(f) display orbits in the regular octagon. whereas figs. 2(g) and 2(h) represent examples in the case of surfaces respectively.
of genus
g = 3 and
g = 8
Table I reveals unusual high multiplicities which are partly due to the symmetry of the regular octagon considered. Mostly, there are 16 orbits of the same type, because a generic orbit (e.g. fig. 2(c, d) can be rotated in steps of 71/4 yielding 8 different orbits due to the trivial symmetry of the octagon. Including backward traversals one arrives at 16 orbits. However, an orbit which is mapped under 4 (2 or 1) consecutive rotations by IT/~ onto itself belongs to a family of 8 (4 or 2) orbits (e.g. fig. 2(b, a, f), respectively. Further symmetries of the regular octagon (e.g. the shortest diameter I, is equal to the hyperbolic distance between two adjacent vertices) lead to additional degeneracies of orbit lengths. Despite of these symmetries the system loses, however, nothing of its chaotic properties. We would like to emphasize that the above mentioned symmetries are by far not enough to explain the chaotic behaviour of the multiplicities which will be discussed in the next section. 2.3. The asymptotic behaviour of the length spectrum Let us define the number N(I) orbits of length less than 1 by
N(l) :=
of all primitive
c g,, /,,< I
where the gn’s are the multiplicities
of the lengths
R. Aurich
and F. Steiner/
Periodic
orh~ts
of u stron&
455
chaotic system
Table I The first 80 lengths of the length spectrum of the regular octagon.
Length of periodic orbit
Multiplicity
3.0571418390 4.8969048954 5.8280707754 6.1142836779 6.6720057699 7.1073758741 7.2631634751 7.5956918304 7.8806922887 8.1300755289 8.2249036233 8.4368496405 8.6284636565 8.7027505564 8.8714798107 9.0270717197 9.1714255169 9.2282950896 9.3592716579 9.4821941493 9.5309770571 9.6440656486 9.7510997583 9.7938097907 9.8933106878 9.9880946367 10.0785887303 10.1149054114 10.1999558888 10.2815363765 10.3143770353 10.3915072941 10.4657729697 10.5373792543 10.5663046423 10.6344594911 10.7003680283 10.7270445783 10.7900178989 10.8510687003
I,
which
are
ordered
The
periodic
orbits
long
lengths
as shown
as
24 24 48 0 (24) 96 4X 48 8 96 48 192 48 96 48 288 12 48 (24) 96 192 48 192 96 336 0 (24) 192 192 192 96 384 96 192 32 3X4 96 288 96 288 96 576 4X
0 c I,,<
proliferate
I, ~1,
<
exponentially
by Huber
m
n
1 3 5 5 7 9 9 11 13 15 15 17 19 19 21 23 25 25 27 29 29 31 33 33 35 37 39 39 41 43 43 45 47 49 49 51 53 53 55 57
1 2 3 4 5 6 7 X 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
..’
. for
Length of periodic orbit 10.8758208811 10.9343449461 10.9912049969 11.0464930504 11.0689538604 11.1221615882 11.1739903506 11.2450675676 11.2938444648 11.3414600077 11.3608557221 11.4069202001 11.4519475300 11.4703055581 11.5139434106 11.5566494092 11.5984625221 11.6155292590 11.6561415509 11.6959455360 11.7122036393 11.7509179318 11.7888970234 11.8044196553 11.8414054395 11.8777196407 11.913386213X 11.9279754971 11.9627646508 11.9969589861 12.0443403136 12.0771791386 12.1094874621 12.1227186314 12.1543053200 12.1854008904 12.1981408207 12.2285673558 12.2585379275 12.2708217905 ____________
eration
-
This
behaviour
terns,
whereas
+ @q-“-l)
)
(12)
i
is characteristic a power
I-+ co.
law
for
determines
chaotic
sys-
the prolif-
II
96 288 432 96 288 192 384 576 288 288 96 672 192 480 4X 672 48 576 192 (4x) 4x0 288 528 96 192 3x4 576 40 560 3x4 3x4 576 288 3x4 96 864 3x4 3x4 48 (24) 76x 96
57 59 61 63 63 65 67 69 71 73 73 75 77 77 79 Xl 83 83 X5 x7 X7 x9 91 91 93 95 97 97 99 101 103 105 107 107 109 111 111 113 115 115
41 42 43 44 45 46 47 49 50 51 52 53 54 55 56 57 5x 59 60 61 62 63 64 65 66 67 6x 69 70 71 73 74 75 76 77 78 79 x0 81 X2
The
with
lengths ;
f $ ! n=O
N(1)
01
of periodic
systems. well
[12]:
Multiplicity
our
orbits
asymptotic computation
as depicted
account
the correction
I = 14.5
the calculated
the prediction which
is
in the case expression
(12).
a reflection
down
in fig.
of
the
coincides
the we
shortest take
into
up to N = 3. Above
staircase see
to
3, where
terms
of integrable (12)
function
dotted
missing
line lengths
is below in fig. as
3, ex-
(b)
,p
I 5
3
9
7
11
IS
IS
I
Fig. 3. The staircase function N(I) as defined in eq. (11) in comparison with Huber’s law (12) (dotted line).
(4
plained in section 2.2. With the aid of our computation we have empirically found the law which determines the proliferation of the different lengths themselves:
without
regard
to their multiplicities.
The law
(fl
with p = 11.3 is in excellent agreement with our numerical results. In order to check eq. (14) consider the length differences between two adjacent lengths
(9) Fig. 2. (a) The to h,,h>h;‘h;‘. riodic orb;t h,,h, ‘h,h, ‘h,h, riodic orbit
(h)
periodic orbit in the regular octagon belonging (m, n) = (23.16). I,, = 9.02707.. (b) The pein the regular octagon belonging to I. (m, n) = (5,3); I, = 5.82807.. (c) The pein the regular octagon belonging to b,,b,h~lb,b,b~‘. (m,n)=(39,27), /,,=10.07858 (d) The periodic orbit in the regular octagon belonging to b,b,b,b, ‘b,bim’. (m,n)=(lS,ll): I,,=8.22490... (e) The periodic orbit in the regular octagon belonging to b I b 3b I b 3b z ‘. (m, n) = (15,ll); I, = 8.22490. _. (0 The periodic orbit in the regular octagon belonging to b0,230 b h b b -lb, ‘b,‘b, ‘. (m, n) = (25343,17920); I,, = 23.05309.. (g) The periodic orbit in the case of genus g = 3 belonging to b 0 b 0 b 5h0 h 3 (h) The periodic orbit in the case of genus g = X belonging to h, b,b,b,.
(“length
density”)
1 A6 I ntl - 1, =x--T-Fe
d*(l)
1
,,2 (15)
and define s(n)
:= (In+1 - I,@
(16)
which should fluctuate around 1 if eq. (14) is correct. Indeed this behaviour is confirmed in fig. 4, where we use p = 8fi as will be explained later. The exceptionally high spikes seen in fig. 4 are due to missing orbits which are generated by more than 11 boosts and to non-primitive orbits which
6(n) 20
15
10
OS
00
Fig. 4. The length differences shown where p = 80.
weighted
by exp (1,,/2)/p
are
Fig. 5. The multiplicity g, of the orbit length I,, plotted as squares is compared with the mean multiplicity (17) (the smooth curve).
are not taken into account in fig. 4 (see for example n = 4 or n = 24 in table I). The law (14) uniquely determines the mean multiplicity g(l): To be consistent with Huber’s law, the following asymptotic law should hold: 1500
g^(l) - py.
-
(17)
Fig. 5 reveals large fluctuations around this mean multiplicity g(l). If one compares, however, the mean value S, over the interval [[, - 4, I, + 41, s,=;
(18)
gk,
c I, - : < I, i I, + i
with the mean multiplicity
S(l), one observes
good
agreement until lengths of order 14.5 (fig. 6). Above these lengths the mean multiplicity S, lies somewhat below &I) for reasons explained above.
2.4.
Arithmetical
properties
has matrix
line
of periodic orbits
In the case of the regular b=
Fig. 6. The mean multiplicity (18) is plotted as dashed and is compared with g(1) eq. (17) (the smooth curve).
octagon
each boost
EG
elements
a=a+\/2b,
consisting
-
of algebraic -
p=d1+\/2
numbers -
(c+\/2d),
(19)
R. A “rich and F. Sterner/
45x
where
a, h, c and d are of the form n, + in,,
a+fib
Penodic
orbits of u stronglv
n,, n2 E Z. Indeed,
chuotlc gxttw
any boost b E G has the representation
il+fi(c+fid)
b=
(20) I v’il_t(
c* + fid*)
a* + fib*
because on the one hand the generating g = 2 combined with
I>
boosts b, have this form as follows from eqs. (6) and (7) for genus
cosh$=cot;=I+&,
(21)
and on the other hand every product
of matrices
of the form (20) is again a matrix
of this form:
(22) where
cx= (a + fib)(
a^+ 06)
+ (1-t
a)(
c + fid)(
c^*+ ad’*)
= a’ + fib’,
(23)
(24) and a’, b’, c’, d’ are again of the form n; + in;. Thus each orbit length is simply determined by 1Sal = m + fin, m, n E N, and because of this important fact these two integers are given in table I too. There one makes a surprising discovery: The n’s are running through all natural numbers with two exceptions, because there are no orbits at n = 48 and tr = 72. These voids are probably filled up if one takes into account conjugacy classes with elements consisting of twelve and more generating boosts. The rule for the integers m is not so simple. They run through all odd integers but there are sometimes “pairs” of odd integers without any systematics, as it appears at first sight. Nevertheless one can reproduce the sequence (m, n) by the following “fi-rule”: _ n = 1,2,3, . . . ~ m is that odd natural
number
which minimizes
1m/n - fi I.
So the sequence (m, n) is simply a sequence which approximates $2 supplemented by the subsidiary condition that m is an odd natural number. Therefore the length spectrum is completely determined by its arithmetical properties, see eq. (26) below. Unfortunately we are unable to formulate a likewise simple rule for the multiplicity.
R. Aurich
and F. Steiner/
Penodlc
450
orbits of a strong!&, chuotzc .~~.~trnl
We would like to mention a connection between the “pairs” in the m sequence sequence. The “pairs” of m occur at the following values of n:
If l/x
+ l/y
appears
Beatty
.
3,6,10,13,17,20,23,27,30;.. The same sequence
and the so-called
(25)
in a special case of Beatty’s theorem
= 1, where x and y are positive
irrational
[13] which states:
numbers,
then the sequences
[~],[2~],[3~],...,[~1,[2~1,[3~l,... together
include
every positive
By the special
integer just once, where [x] means
choice x = 2 + fi
and
y = fi
the sequence
(25). Let us now show that the law (14) follows from the “fi-rule”. I n-l 2
cosh
= m+fin,
the integral
[x],[~x],[~x], Writing
part of x. . . . is exactly
the sequence
eq. (19) in the form
m,nEN
(26)
we could use m = [fin]. But to get a more manageable expression we instead use the asymptotically correct value m = fin despite of the fact that this m is not an integer of course. Then we can asymptotically 1
2e
express eq. (26) as
in/2 _
2&l
(27)
or n
_
1
-
eL/2
Realizing
that
Z?(l)-
(28)
.
4&C
1, is the nth length we have recovered
-&e1/2,
whencep=8&.
The theoretical value p = 8fi determined from fig. 4.
3. Summary
eq. (14):
= 11.31 . . .
(29) is in excellent
and discussion
In this paper we have presented for the first time an explicit enumeration of the shortest periodic orbits (classical paths) for a prototype example of a chaotic Hamiltonian system: a point particle sliding freely on a two-dimensional surface of constant negative curvature. Our computations crucially relied on the intimate relation
between
agreement
with
the periodic
our
orbits
empirical
value
and the topology,
11.3
ge-
ometry and group theory of compact Riemann surfaces of genus g 2 2. To keep things as simple as possible, we have chosen g = 2 and considered the most regular fundamental region, i.e. the hyperbolic octagon shown in fig. 1. The corresponding Riemann surface M can be identified with D/G, the action of a particular Fuchsian group, the “octagon group” G on the PoincarC disc D.
460
R. Aurich
The octagon generators boosts,
group
and F. Sterner/
G can be constructed
b, (k = 0,1, . . . ,7) playing
whose
explicit
matrix
Periodtc orhrts o/u strong!\,
from 8
the role of
representation
has
been given in eq. (6). By forming all possible “words” of arbitrary length from the alphabet consisting of the 8 “letters” b, by matrix multiplication, one can construct in principle all group elements of G. Of great importance is the classitication of the group elements in terms of conjugacy classes, since such a classification is equivalent to an enumeration of periodic orbits on M. The (hyperbolic) length f(b) of a periodic orbit corresponding to a given group element b E G is uniquely determined by eq. (9). Although one is led along these lines to a well defined algorithm for calculating the periodic orbits together with their lengths, the actual computation could only be carried out on a large computer. Within the computer time available to us, we were able to reconstruct all words with a maximal length of 11 letters. As a result we obtained 206796242 primitive periodic orbits. The shortest 80 lengths and their multiplicities have been given in table I. A few typical examples of periodic orbits have been displayed in figs. 2(a-h). Surprisingly enough, the “length staircase” N(I), defined in eq. (11) is nicely described by the asymptotic behaviour (12) down to the shortest lengths, as can be seen from fig. 3. Only at the upper end of the staircase one observes a deviation, which is caused by missing words formed out of 12 or more letters. In section 2.4 we derived a curious representation of the group elements b E G, eq. (20) in terms of algebraic numbers of the form n, + fin,, n, E Z. Thus the length spectrum {1(b)} is completely determined by its arithmetical properties as given in eq. (26). We have thus discovered a close connection between the length spectrum of the octagon group G and number theory. It is hoped, that this connection can be deepened in order to shed some light on the multiplicity distribution and to reveal the arithmetical structure of chaos. Work in this direction is being carried out. Using the arithmetic properties of the length spectrum we could derive the asymptotic law (14) and
thus
chaotic system
predict
the value
which, originally,
8fi
for the parameter
was introduced
eter in eq. (14). The predicted
value
excellent
numerical
agreement
(see figs. 4-6) The
length
constitutes
with
our
p
as a free paramfor p is in results
which gave p = 11.3. spectrum
an important
obtained
in
this
paper
input in the Selberg trace
formula [3-61, which is an exact substitute for the Bohr-Sommerfeld-Einstein quantization rules for our chaotic system, and which can be interpreted as an (exact) instanton-gas evaluation of the corresponding Feynman path integral. It establishes a striking duality relation between the quantum mechanical energy spectrum and the lengths of the classical trajectories. On the basis of this exact “periodic orbit theory” we hope to unravel the mystery of quantum chaos. Investigations along these lines are under study; a preliminary announcement is given in ref. [7].
Acknowledgements We would like to thank G. Miinster our attention to ref. [13].
for drawing
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