- Email: [email protected]
Spectral decomposition of piecewise linear monotonic maps
Pergamun
Spectral Decomposition of PiecewiseLinear Monotonic Maps I. ANTOMOU ’ and BI QIAO InterTIdtlOnai
Solvay Institutes for Physics and Chcmistty, CP 231, Campus Plaine ULB, Bd du Triomphe, 1050 Brussels, Belgium. (Accepted 28 Febraury 1996)
Abstract-
We construct a generalized spectral decomposition of the Frobcnius-Pcrron operator for one class of piecewise linear monotonic maps by using a general, iterative, operator method which is applicable in principle for any mixing dynamical system. The Jordan b&k structure is specified and analyzed. The eigenvalues in the decomposition are related to the decay rates of the correlation functions. We explicitly define appropriate generalized function spaces which provide mathematical meaning to the spectral decomposition. Copyright @I996 Elsevier Science Ltd.
1. INTRODUCTION
The probabilistic non-local approach for the study of chaotic dynamics [1,2] aims in the study of the Frobenius-Perron and the Koopman operators [3] which describe the evolution of the probability densities and the observables of the system correspondingly. The probabilistic approach to dynamical systems has well-known advantages over the conventional topological approaches based on trajectories, namely: (1) the evolution law for the probability densities is linear even if the underlying dynamics is non-linear; (2) for unstable systems, trajectories, although existing as mathematical concepts, are operationally unattainable due to intrinsic computational limitations, while the computation of the evolution of probabilities is stable ; (3) the spectral decomposition of the linear evolution operator employing the methods of functional analysis provides a new computational tool for the study of the evolution ofdynamical systems 141. The FrobeniusPerron operator U is the dual of the Koopman operator Y defined [3,4] for any dynamical system S acting on the measurable space X as follows:
Yf Ix) = f (S (xl)
(1)
@PIf) = (PIw
(2)
where
dv(x)p (x)f(x) ‘Also at: Theoretische Naturkunde, Free University of Brussels, Belgium, 1895
(3)
I. ANTONIOU
1896
and BI QIAO
is the expectation value of the observable f in the density p with respect to the reference measure Y on X. If v is the invariant measure then U 1 = 1, where 1 (x) = 1 is the constant equilibrium density. The spectral decomposition of the Frobenius-Perron operator is therefore the dual of the spectral decomposition of the Koopman operator. The knowledge of the eigenvalues and the associated eigenfunctions of the Frobenius-Perron operator allows us to answer effectively all relevant questions concerning the decay properties of the correlation functions, as well as the analyticity properties of the power spectra of chaotic dynamical systems.Furthermore the spectral decomposition of the Frobenius-Perron operator amounts to an efficient solution of the prediction problem by means of probabilities [4] and can also provide a spectral condition of controllability for chaotic dynamical systems[5]. In the caseof non-invertible chaotic maps like the exact endomorphisms [3,6], i.e. non-invertible maps which have positive Kolmogrov-Sinai entropy [6], there is no spectral decomposition for the Koopman operator in the Hilbert spaceof square integrable functions becausethe Koopman operator is a unilateral shift [4,7]. As has been recently shown [4,7], spectral decompositions exist in extensions of the eigenvalue problem to suitable dual pairs corresponding to Gelfand triplets or rigged Hilbert spaces [4,8-lo]. When the extended operator results from meromorphic extensions of the resolvent, the eigenvalues correspond to the resonances of the power spectrum discussed by Pollicott and Ruelle [l l-141. Such generalized eigenvectors and eigenvalues for exact systems have also recently been obtained by several authors [7,15-261. A sufficient condition for resonancesto arise from a meromorphic continuation of the resolvent is that the Koopman operator is a Fredholm-Riesz operator [27]. In this paper we discuss the spectral decomposition for one class of the piecewise linear monotonic maps. The key idea of our approach is the construction of a more manageable intermediate operator which is intertwined with the initial operator. The eigenvalue problem for the intermediate operator is easier than the original problem. From the spectral decomposition of the intermediate operator we obtain the spectral decomposition of the original operator by suitable intertwining operators constructed iteratively. This method [4] is based on the previous work of the Brussels-Austin group led by Prigogine on subdynamics [28-321. We consider here piecewise monotonic Markov maps on the unit interval S : [O, 11 - [O, 11. These maps satisfy the following properties [6,33,34]: (i) The unit interval [O, 1I is partitioned into finite cells 1i = [?ji-i, &) , for i = 1, 2,. , N corresponding to the boundary points go,&,. . . ,& with 50 z O,~N z 1. Themap Sis linear over each cell Zi, i.e. S(X) = 2 (O(iX+ fli) 1~ (x) E $5’i i=l
(x)
, ~(i, fii E R.
(4)
i=l
where Sj (Xl = S (Xl II, (Xl = ((xix + Si) II, (Xl is the part of S on the cell 4 and 14 (x) is the indicator function of the cell Ii. (ii) The map is strictly monotonic over each cell 4, i.e.
(iii) The Markov property is expressed by the compatibility of the partition 11,. . , IN with the action of the map S. The map S should not destroy the partition Ii.. , IN, i.e. S maps every cell onto a union of cells. We shall restrict our discussion to the class of piecewise monotonic Markov maps which in addition satisfy the property:
Spectral decomposition of piecewise linear monotonic maps
1897
(iv) Each cell is mapped onto the unit interval, i.e. Slk = 10, l), k = 1,. , A? (see Figs 14). The piecewise linear maps satisfying (+0-(v) share the chaotic properties of the piecewise monotonic maps [6], namely they have strictly positive Kolmogorov-Sinai entropy because they are exact endomorphims. Moreover the invariant measure is absolutely continuous. The Renyi maps [4] and the family of the Tent maps [25] are special cases. The Frobenius-Perron operator for the maps satisfying (i)-(iv) acts on the densities p as follows [31:
UP(X)
=I
N d&T’ (x)
p (s;’
dx
i=I I
(x))
I
(6)
Here p (x) = 2 (x) is the Radon-Nykodym derivative of any measure p absolutely continuous with respect to the Lebesgue measure. The invariant measure is the Lebesgue measure. Indeed
I2- ( l[O,l] (s;‘x)
~lto.11 Lx) = : j=j
ai
N
=
=
CI I
i=l
5&
[O,l’ (x)
(7)
t’ l[O.ll (xl ( i=, 1 O(i 0
= l[O.l] (x).
2. CONSTRUCTION OF THE SPECTRAL DECOMPOSITION
In order to obtain the spectral decomposition of the Frobenius-Perron operator or the Koopman operator we choose the following initial biorthonormal system: Iqan) = (%I
(-1)“p
= (x4!,
(x)
> form,nlO,1,2
,...
(8)
Biorthonormality is verified directly:
(9)
The system is complete in the algebra of real analytic functions on [O, 11. Indeed for any analytic function f on [0, 1I we have:
1898
I. ANTONIOU
and BI QIAO
(f (xl 1 = +f:f/(n) (0)(2 I n=O +CO. = f(x) -6(“) (-l)” EC I n! n=O
(x) (x”(. >
Therefore:
1 P’n)(WI = 1. n=O
The matrix elements of the Frobenius-Perron operator U with respect to the biorthonormal system 1%) , (qm 1are:
(12)
I 01
m
zn= =
xl j=l
E j=l
-1 1 Lxj I 7 (Ej
-
Ej-1)
(13)
$ J
From property (4) the slope oci is:
where (T (nj) is the sign of the slope ‘Xi. Therefore the eigenvalues z, can also be expressedas:
j=l
Equilibrium corresponds to the eigenvalue zo = 1. Indeed:
(14)
1899
Spectral decomposition of piecewise linear monotonic maps
(15) From (14) we seethat degeneracy in the spectrum may occur if and only if:
5 s”bk) IlkIn+’ = iP (o(k) Izkr+l, for n f k=l
m.
(16)
k=l
In the case of Markov partitions into cells with equal length IZ, ] = IZz 1 = eigenvalues are given by the simple formula:
n odd
= jZN] = i the
(17)
Here N, and N- with N+ + N- = N denote the number of branches with a positive and negative slope, respectively. Therefore for this simple case condition (16) implies that degeneracy occurs if and only if n is even and m is odd and N+ = ; [N + N”‘-+‘I
.
Because of the possibility of degeneracy we shall consider the simple spectra in Section 2.1 and degenerate spectra in Section 2.2. 2.1. Simple spectrum
In order to obtain the eigenvectors of the Frobenius-Perron operator we use the algorithm presented in [4] based on the creation and destruction operators C, D. The right and left eigenvectors of U are given by the formulae:
(hi = ((pi+G)PiI Ifi) = 1(Pi + Di)+ @i) .
(18) (19)
The action of the creation and destruction operators C, D can be expressedin terms of their matrix elements with respect to the initial biorthonormal system ( Cpj), (vi I by
Ml = (PiI + 2 (GPiI a/r) (qkl k=O
Ifi) = I+[) + T (DiVkI Qi) I@k). k=O
The matrix elements of the operators C and D can be obtained recursively from the equations:
1900
I. ANTONIOU
and BI QIAO
(20)
From the above equations, by using the formula (12) we can obtain the equations for the matrix elements of C,,,D, as:
and
Therefore the eigenvectors of the Frobenius-Perron operator U are obtained by
(24) k=O
and
14) = I@‘“)+ 2 (DdPkI%>1%)
cm
k=n+l
where
(fol = bol = 1
(26)
which confirms that the invariant measure of this system is the Lebesgue measure. The spectral decomposition of the Frobenius-Perron operator and the Koopman operator is:
i=o
W’)
for any observable f and any density p in the domains which provide meaning to the spectral decomposition. 2.2. Degenerate spectrum
We shall consider the simple casewhen 111I = 112I = = 11~1 I = f, and N+ = N- + 1. More general casesof degeneracy can be treated in the same way. The matrix elements of U in this simple case can be expressedas:
1901
Spectral decomposition of piecewise linear monotonic maps
m! L i.P Pr!(m- n)! N”+’ i=l
W%?l %) = (Pm1J%) =
&,
(Cyi) (-fii)“-“,
m > fl
m = n, n even
(28)
1 -jjm+1* m = n, n odd I 0, m-en Then the Koopman operator V is represented as a lower-triangular matrix with respect to the biorthonormal system I%), (q,,, [ :
with diagonal part
(30) and off diagonal part
From (30) we see that the eigenprojectors of VOfor k = 1,2,. the parts 0, of intermediate operator 0 are given by [4]: 0, = P,UP, + P,UQ&P,
are 2-fold degenerate. Then
= P,UP,,
(32)
here P,UQ,C,P, = 0 because C, and P,JQ,, are the upper-triangular operators without diagonal parts. The upper-triangular property of PJJQ,, is obvious from (28). The upper-triangular property of C, can be proved by the upper-triangular property of P,,UQ,, [4]. Therefore the intermediate operator 0 is:
0 = fy@. = FPJIP, n=O n=O
(33)
From (33) we can obtain
0’
=
I@O) (PO1 + F & k=l
[ 1*2k)
(p2k
+
1@2k-l)
(Q32k-1
11
(34)
k=l
Equation (34) shows that the non-vanishing of the lower-triangular matrix elements VI@ UC - I ) IS a necessary and sufficient condition for the appearance of Jordan blocks in O+ and therefore for U becausethe intertwining operators Szand R-l transfer the Jordan-block structure of 0 to U through U = SZOSZ-’[4] : b2kI
1902
I. ANTONIOU
v =
j W’)‘@o)
and 31 QlAO
OmI
+~$[l(=')+Fik)(@,k/ k=l
+
i(n-')+FZk-l)(R~Zk-iI]
(351
This enabIes us to arrive at: (f2k
I&k-l)
(fip2ki
=
1 = @;ZQ72k-I
(&k-I
i&k)
j =
((pk
1=
=
(a-‘)+
=
[(~k+D*ck)-‘(~+~k)lt~2k).
=
w)+
=
[@k
+ ck)
( h-
+ ck)
Q)Zk-1
(35)
k = 0, 1,.
Q)Zk 1 ,
j ,
k =
(37)
1,2,.
@2k)
k=O,
l,...
(38)
h-1) + Dkck)-’
(pk
+ Dk)]+
+2k-1)
,
k=
1,2,...
(39)
\vhere
(fol = (Pot = (11 (fi
I =
(PA
+
(GPlI
w
(40) (4
=
(f2l
=
(A
+
(
bml
(41)
%) (4701
1
(42) x2 + 7 _ l b2l If Wo) - + (cm I @o) (472t FfPII -1 $ 9 and the matrix elements of the operator C or D can be calculated from the following C, D operator equations: =
PjCi =
- Zi
lj
DiPj = &
,
I
(PjCiU] Pi - PjC’lPi - PjU]CiPi) , j < i
(43)
(fiU1 Pj - fiDiU1 Pj - Pib’l Dip,) , i < j.
WI
1903
Spectral decomposition of piecewise linear monotonic maps
0.6 0.4 0.2
The relations (43) and (44) can be obtained from general C, D operator equations in [4] by means of the upper-triangular property of P,,UQn. Therefore, when (~lgk] VI&-r) f 0, k = 1,2,... , the Frobenius-Perron operator U and the Koopman operator V have the Jordan decomposition as
=
b
I&)
(fol
f)
+g
&
[b
l&k)
(f2kl
f)
+
b
lfik-I)
&k-1
1 f)]
(45) + +f k=l
b2kI
v
Ifik-I)
(p
I&k)
(hk-l
1 f)
where the real numbers & and (Q)2k] V ]@k-1) are the eigenvalues and weights of each irreducible block k, respectively. When (Q)2k]V ]@2k-1) = 0, k = 1,2, . . , the Frobenius-Perron operator U and the Koopman operator V have the diagonal spectral decomposition with 2-fold degeneracy as:
WPI f) = (PI w =
(Pim)(/oi~)+~~[(PIrik)(/2*/~)+(Pl~k-l)(f.k-l~i)].
3. EXAMPLES
We shall discuss four examples of piecewise monotonic Markov maps. (1) S1 : ro, 11 - 10, 11 (Fig. 1) defined by:
(46)
I. ANTONIOU
1904
and BI QIAO
I _ 3X, x E
1-
: 0, f 1 3x - 1, x E ;, ; 1 _
Sl (x) =
2, x E
3x -
(47)
1 ;, >
From (6) the formula of the Frobenius-Perron operator is expressedby
(48)
m!
From (28) the matrix elements of V with respect to the biorthonormal system I@,) , (91,~I are:
-L ((-I)"+ n!(m - n)! 3m+' w?M
S)
= (%A
ml)
I
=
iJ
m = n, n
1
-3n+l'
0,
1 +2"-"),
m>n
even
m = n, n
(491
odd
m-en
From (49) we seethat the Koopman operator has Jordan blocks decomposition: +m @+
=
I@O)
(PO1
+
1
. &
[l@Zk)
(Q)ZkI
I@2k-1)
+
(Q)Zk-I
I]
k=l + +f k=l
(Q)ZkI
v
i@2k-1)
I+‘Zk)
(Q)Zk-I
I
v = (ROQ-I)+ =
16)
($3 1 +
ff k=l
j&
(Q)2kI
v
I@2k-I)
+ E
[i&k)
(f2k
I&k)
1 + Ifik-1)
(hk-1
(hk-I
11
(51)
1
k=l
The principal vectors of V are: (f2kl
(f2k-11
I&)
&k-I)
=
=
1 [ (pk
=
((Pk+Ck)Q)2kj,
=
((Pk+Ck)p2k-,I,
+ Dkck)-’
I[(~k+~kck)-‘(P*+Dk)]+@2k-l)~
(pk
(52)
k=o,l,...
k= 1,2,...
+ Dk)]+
+2k)
J
k = 0, l,... k= 1,2,...
(53) (54) (55)
The first, second and third right principal vectors can be easily calculated by using the above formula:
Spectral decomposition of piecewise linear monotonic maps
1905
Fig. 2. A piecewise linear monotonic map S2 (.u).
(fol = (11
(56)
VII = (ml; Gmr@o) bol = X-- 2 ( I
(57)
(fil
=
kP21 +p(?4
@o) (PO1
(58)
9 - -
=
3 I
(
where we can check:
Uf,(x, = u x2-; ( > = $(2-;)+$(x-;)
(5%
(60)
This example with Jordan block structure is like the example mentioned by Driebe [26]. (2) S2 : [O, 1I - [O. 11defined by
$2(x)=
1-4x,
x E
0,;
4x-1,
XE
12 - -
I 11 -2x+2,
i4 XE -,I "r 4
>
1
(61)
I. ANTONIOU
1906
and BI QIAO
I .o
0.8
0.6
0.4
0.2
Fig. 3. A piecewise linear monotonic map & (x).
where IZl I = IZzl f 1131,and N+ = 1, N- = 2. (Fig. 2). The Frobenius-Perron operator is: (62)
From (13), (24) and (25) we can obtain the eigenvalues and the eigenvectors. The eigenvalues are:
(63) The first and second right eigenvectors of U are: fo (xl = 1
(64)
and 1 j-i (x) = x - -. (65) 2 This shows that the eigenvaluesjump from positive to negative when n even changes to n odd, and vice versa. (3) Ss : [0, 1I - LO,1I defined by
s3
(xl = -
where IZrI = 1121= 1131= 114I = $, and N+ = N- = 2 (Fig. 3).
(66)
Spectral decomposition of piecewise linear monotonic maps
1907
0.8 0.6 0.4 0.2
0.4
0.2
0.6
0.8
I .o
Fig. 4. A piecewise linear monotonic map S4 L’c)
The formula of the Frobenius-Perron operator is: UP(x)=~[p(-~x+~)+p(~x+~)+P(-~x+~)+P(~x+~)].
(67)
From (13) the eigenvalues are:
z,=
0, n odd 1 n even ’
1&I’
Therefore the eigenvalues are zero when n is odd. Comparing S3 with S2 we see that S3 is symmetric with respect to the midpoint of the unit interval. This geometric symmetry is the reason that all odd eigenvalues of U are zero. (4) S4 : [O, 11 - [0, 11 defined by
s4 (x)
(69)
=
where 111I = IZ2I = 113I = i and iV+ = N- + 1 = 2 (Fig. 4) is a Tent map [25]. The Frobenius-Perron operator is: (70) From (13) (36) and (37) the eigenvalues are: -
n odd n even
and the first, second and third right eigenvectors of U are:
(71)
I. ANTONIOU
1908
and BI QIAO
fo(xl = 1
(72)
1 fi (xl = x - 2
(73)
2 (74) 3 There exists a two-fold degeneracy,but there is no Jordan block structure. The matrix elements of the Frobenius-Perron operators of S4 and Si are:
fi Lx) = 2 - -.
bml ti 1%)= & b?bl VI IW = &
{(-1Y
{(-I)”
(-2Y+
(-2)“-”
+ 2,-‘},
m>n
+ (-1)” (-3)“-“},
(75)
m > n.
Therefore:
(cpmI Vi I%) = 0, for n odd, m > n
(76)
m > n.
(77)
(tp,l I4 IQ,,) f 0, for n odd,
Formulae (76) and (77) show why there is no Jordan decomposition for S4 but there is a Jordan decomposition for Si. Similarly we seethat there is no Jordan decomposition for the FrobeniusPerron operator of the Tent maps [25]
T, =
(78)
because (QmlVl@2k-1) =O, wherem = 2,3 ,... , n = 0, l,... ,
[qp
fork=
1,2,...
and [y 1denotes the integer part of real number y.
4. THE MEANING OF SPECTRAL DECOMPOSITION
The spectral decompositions (27), (45) and (46) are meaningful for any observable f and any density p in suitably chosen domains. Since the eigenvectors or the principal vectors (8) of the Koopman operator contain Dirac’s delta functions and its derivatives, the spectral decomposition (27), (45) and (46) make no sensein the Hilbert space L* of square integrable functions on the unit interval. A natural way to give meaning to the spectral decomposition (27) (45) and (46) is to extend the Koopman operator V beyond the Hilbert spaceL* to a larger space x@ of continuous anti-linear functionals on a properly chosen dense subspace Cpof the Hilbert space L*. In other words, relations (27), (45) and (46) should hold for any density function p in the test spaceQ,and any anti-linear functional f in the anti-dual space ‘a. In this way the spectral decompositions (27), (45) and (46) can be understood as a natural extensions of the Gelfand-Maurin theory of
Spectral decomposition of piecewise linear monotonic maps
I909
the generalized spectral decomposition of norma operators with continuous spectrum [g-lo]. This generalization [4,7] is necessarybecausethe original Gelfand-Maurin theory holds onIy for operators with a spectral theorem in Hilbert spaceIike the normal operators, while the Koopman operator, being a unilateral shift, has no spectral decomposition [4,7] in Hilbert space. A simple choice for the test function space is the space P of polynomials which is the union of the spaces3” of polynomiaIs with the degree smaller or equal to n :
n=l
We can define a L-F topology [9] on ?’ in the folIowing way: Let u’ = ct &xk f P, the norm on P is defined by
then (80) provides the strict inductive limit topology llwll,, = I[wll,+l on P [93. Therefore we can conclude: (1) The space 2’ is a nuclear LF-space, thus it is complete and barreled, hence it is a dense subspace in the Hilbert space L2. (2) The space P is stable with respect to the Frobenius-Perron operator U and U is continuous with respect to the LF- topology of P, because U preserves the degree of the polynomials. In fact, from (6) we have
= T+
Iti I ($$j” I
which shows that UIP c 2’. The extended Koopman operator V can be written as follows: In the case of the simple spectrum:
or in the case of degeneracy and Jordan blocks:
' = 16) (fat + :$[&k, k=l
(hk[
+
i&k-i)(fZk-,I]
(83)
Both can acquire meaning in the rigged Hilbert space P C L& C xP or to the dual pair PI x2? The eigenvalues zj of U coincide with the Pollicott-RuelIe resonances. In fact, the PollicottRuelle resonances are the inverses of the zeros of the Fredholm determinant [I l-141:
1910
I. ANTONIOU
and Bl QIAO
d(z) z det(1 -zU) (84)
The triangular property of the matrix elements (12) gives: TrU” = 1 (U”piI @i) i=O +m
(85)
= CG i=O
which allows us to arrive at:
= gne-“;qG)” i=O neln(
=f
1-x;)
(86)
i=O +a =
Cll(l
-ZZi>
i=O +oO
= Crr(l i=o
-ze-“).
Therefore we have: e-y! = zi,
(87) Relation (87) confirms that the complex poles yi of the Fredholm determinant d (z) are the logarithms of eigenvalues Zi. The eigenvalues zi < 1, i = 1,2,. . . , can determine the rates of the correlation functions and the rate of approach to the equilibrium. This shows that the piecewise linear monotonic maps with 11 = IZ = = IN may have different spectra and different eigenvectors or principal vectors in the same rigged Hilbert space ‘p, although they are isomorphic systems because they have the same KS-entropy [6]. This observation shows that the rates of approach to equilibrium may be different for isomorphic dynamical systems. 5. CONCLUDING
REMARKS
From the previous discussion we see that the geometry of piecewise linear Markov maps is reflected in the spectral decomposition. The Frobenius-Perron operators of piecewise linear Markov maps can be diagonal with simple or degenerate spectrum or can have Jordan block structure depending on the geometry. The admissible test function spaces 2’ exclude the Dirac delta functions. This means that the trajectories S (v’ - S”y) , n = 0, 1,2,. . . are excluded from the domain of the FrobeniusPerron operator. Formulae (27), (45) and (46) can be used for probabilistic predictions using
Spectral decomposition of piecewise linear monotonic maps
1911
initial densities expandable in terms of admissible test functions only. This remark, however, goes beyond the prediction problem, as it also reflects the intrinsically probabilistic character of unstable dynamical systems.The spectral decompositions (27) (45) and (46) have, moreover, the property that the dynamical properties are reflected in the spectrum becausethe eigenvalues are the powers of the Lyapounov time. The extension of the dynamical formulation through the spectral decomposition to suitable rigged Hilbert spacesleads to a natural classification of chaotic maps in terms of the resonance spectrum [25]. Acknowledgements-We thank Professor Ilya Prigogine for his encouragement, support and comments which, to a large extent, motivated this work. We also profited a great deal from several discussions with Professors Z. Suchanecki and E Bosco. We also acknowledge the financial support of the Belgian Government under the Interuniversity Attraction Poles and the European Commission, D G III ESPRIT project ACTCS 9282.
REFERENCES I. I. Prigogine, From Being to Becoming. Freeman, New York (1980). 2. G Nicolis and I. Prigogine, Exploring Complexity. Freeman, New York (1989). 3. A. Lasota and M. Mackey, Chaos, Fractals, and Noise, 2nd edn. Springer, New York (1994). 4. I. Antoniou and S. Tasaki, Int. J Quantum Chem. 46, 425474 (1993). 5. I. Antoniou, V. Basios and F. Bosco, Int. 1 Brjiircafions Chaos. In press. 6. I. Kornfeld, V. Fomin and Ya. Sinai, Ergodic Theory. Springer, Berlin (1982). 7. I. Antoniou and S. Tasaki, 1 Phys. A: Math. Gen. 26, 73-94 (1993). 8. I. Gelfand and N. Vilenkin, Generalized Functions, Vol. 4. Academic Press, New York (1964). 9. F. Treves, Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967). IO. A. Bohm, Quantum Mechanics: Foundations and Applications, 3rd edn. Springer, Berlin (1993). 11. M. Pollicott, Invent. Math. 81, 413 (1985); ibid. 85, 147 (1958). 12. D. Ruelle, Phys. Rev. Left. 56, 405 (1989). 13. J. P Eckmann, Resonances in dynamical systems. In Proc. IXth Int. Congress on Mathematical Physics, edited by B. Simon et al. Hilger, Bristol (1989). 14. D. Ruelle, Puble, Math. ZHES 72, 175 (1990). 15. H. Mori, B. So and T. Ose, Prog. Theor. Phys. 66, 1266-1283 (1981). 16. M. Dorfle, 1 Star. Phys. 40, 93-132 (1985). 17. G. RoeprstortI, On the Exponential Decay of Correlations in Exact Dynamical Systems. Princeton (unpublished) (1987). 18. P. Gaspard, 1 Phys. A: Math. Gen. 25, LA83 (1992); Phys Lett. A 168, 13-17 (1992). 19. H. Hasegawa and D. Driebe, Phys. Lett. A 168, 18-24 (1992); ibid. 176, 193-201 (1993). 20. S. Tasaki, Z. Suchanecki and I. Antoniou, Phys. Left. A 179, 97-102; ibid. 103-110 (1993); Chaos, Solirons & Fractals 4, 227-254 (1994). 21. D. Mackernan and G. Nicolis, Phys. Rev. E 50, 988-999 (1994). 22. S. Tasaki, I. Antoniou and Z. Suchanecki, Phys. Left. A 179,97-102 (1993). 23. S. Tasaki, Z. Suchanecki and I. Antoniou, Phys. Let?. A 179, 103-110 (1993). 24. S. Tasaki, I. Antoniou and Z. Suchanecki, Chaos Solitons & Fractals 4, 227-254 (1994). 25. I. Antoniou and Bi Qiao, Phys. Left. A. 215, 280-290 (1996). 26. D. Driebe, Jordan blocks in a one dimentional Markov map. Computers Math. Appl. In press. 27. 0. Bandtlow, I. Antoniou and Z. Suchanecki, Resonancesof dynamical systems and Fredholm-Riesz operator in rigged Hilbert spaces. Computers Math. Appl. In press. 28. C. George, Physica 65, 277 (1973). 29. I. Prigogine, C. George, F. Henin and L. Rosenfeld, Chem. Scr 4, 5 (1973). 30. T. Petrosky and Hasegawa, Physica A 160 351 (H 1989). 31. T. Petrosky and I. Prigogine, Phys. A 175, 146 (1991). 32. T. Petrosky, I. Prigogine and S. Tasaki, Phys. A 173, 175 (1991). 33. C. Presten, Iterates of piecewise monotonic mappings on an interval. In Lecture Notes in Math, Vol. 1374. Springer, New York (1988). 34. R. Adler and D. Mayer, Ergodic Theory Symbolic Dynamicas and Hyperbolic Spaces, edited by T. Bedford, M. Keane and C. Senes. Oxford University Press, Oxford (1991).
Spectral Decomposition of PiecewiseLinear Monotonic Maps I. ANTOMOU ’ and BI QIAO InterTIdtlOnai
Solvay Institutes for Physics and Chcmistty, CP 231, Campus Plaine ULB, Bd du Triomphe, 1050 Brussels, Belgium. (Accepted 28 Febraury 1996)
Abstract-
We construct a generalized spectral decomposition of the Frobcnius-Pcrron operator for one class of piecewise linear monotonic maps by using a general, iterative, operator method which is applicable in principle for any mixing dynamical system. The Jordan b&k structure is specified and analyzed. The eigenvalues in the decomposition are related to the decay rates of the correlation functions. We explicitly define appropriate generalized function spaces which provide mathematical meaning to the spectral decomposition. Copyright @I996 Elsevier Science Ltd.
1. INTRODUCTION
The probabilistic non-local approach for the study of chaotic dynamics [1,2] aims in the study of the Frobenius-Perron and the Koopman operators [3] which describe the evolution of the probability densities and the observables of the system correspondingly. The probabilistic approach to dynamical systems has well-known advantages over the conventional topological approaches based on trajectories, namely: (1) the evolution law for the probability densities is linear even if the underlying dynamics is non-linear; (2) for unstable systems, trajectories, although existing as mathematical concepts, are operationally unattainable due to intrinsic computational limitations, while the computation of the evolution of probabilities is stable ; (3) the spectral decomposition of the linear evolution operator employing the methods of functional analysis provides a new computational tool for the study of the evolution ofdynamical systems 141. The FrobeniusPerron operator U is the dual of the Koopman operator Y defined [3,4] for any dynamical system S acting on the measurable space X as follows:
Yf Ix) = f (S (xl)
(1)
@PIf) = (PIw
(2)
where
dv(x)p (x)f(x) ‘Also at: Theoretische Naturkunde, Free University of Brussels, Belgium, 1895
(3)
I. ANTONIOU
1896
and BI QIAO
is the expectation value of the observable f in the density p with respect to the reference measure Y on X. If v is the invariant measure then U 1 = 1, where 1 (x) = 1 is the constant equilibrium density. The spectral decomposition of the Frobenius-Perron operator is therefore the dual of the spectral decomposition of the Koopman operator. The knowledge of the eigenvalues and the associated eigenfunctions of the Frobenius-Perron operator allows us to answer effectively all relevant questions concerning the decay properties of the correlation functions, as well as the analyticity properties of the power spectra of chaotic dynamical systems.Furthermore the spectral decomposition of the Frobenius-Perron operator amounts to an efficient solution of the prediction problem by means of probabilities [4] and can also provide a spectral condition of controllability for chaotic dynamical systems[5]. In the caseof non-invertible chaotic maps like the exact endomorphisms [3,6], i.e. non-invertible maps which have positive Kolmogrov-Sinai entropy [6], there is no spectral decomposition for the Koopman operator in the Hilbert spaceof square integrable functions becausethe Koopman operator is a unilateral shift [4,7]. As has been recently shown [4,7], spectral decompositions exist in extensions of the eigenvalue problem to suitable dual pairs corresponding to Gelfand triplets or rigged Hilbert spaces [4,8-lo]. When the extended operator results from meromorphic extensions of the resolvent, the eigenvalues correspond to the resonances of the power spectrum discussed by Pollicott and Ruelle [l l-141. Such generalized eigenvectors and eigenvalues for exact systems have also recently been obtained by several authors [7,15-261. A sufficient condition for resonancesto arise from a meromorphic continuation of the resolvent is that the Koopman operator is a Fredholm-Riesz operator [27]. In this paper we discuss the spectral decomposition for one class of the piecewise linear monotonic maps. The key idea of our approach is the construction of a more manageable intermediate operator which is intertwined with the initial operator. The eigenvalue problem for the intermediate operator is easier than the original problem. From the spectral decomposition of the intermediate operator we obtain the spectral decomposition of the original operator by suitable intertwining operators constructed iteratively. This method [4] is based on the previous work of the Brussels-Austin group led by Prigogine on subdynamics [28-321. We consider here piecewise monotonic Markov maps on the unit interval S : [O, 11 - [O, 11. These maps satisfy the following properties [6,33,34]: (i) The unit interval [O, 1I is partitioned into finite cells 1i = [?ji-i, &) , for i = 1, 2,. , N corresponding to the boundary points go,&,. . . ,& with 50 z O,~N z 1. Themap Sis linear over each cell Zi, i.e. S(X) = 2 (O(iX+ fli) 1~ (x) E $5’i i=l
(x)
, ~(i, fii E R.
(4)
i=l
where Sj (Xl = S (Xl II, (Xl = ((xix + Si) II, (Xl is the part of S on the cell 4 and 14 (x) is the indicator function of the cell Ii. (ii) The map is strictly monotonic over each cell 4, i.e.
(iii) The Markov property is expressed by the compatibility of the partition 11,. . , IN with the action of the map S. The map S should not destroy the partition Ii.. , IN, i.e. S maps every cell onto a union of cells. We shall restrict our discussion to the class of piecewise monotonic Markov maps which in addition satisfy the property:
Spectral decomposition of piecewise linear monotonic maps
1897
(iv) Each cell is mapped onto the unit interval, i.e. Slk = 10, l), k = 1,. , A? (see Figs 14). The piecewise linear maps satisfying (+0-(v) share the chaotic properties of the piecewise monotonic maps [6], namely they have strictly positive Kolmogorov-Sinai entropy because they are exact endomorphims. Moreover the invariant measure is absolutely continuous. The Renyi maps [4] and the family of the Tent maps [25] are special cases. The Frobenius-Perron operator for the maps satisfying (i)-(iv) acts on the densities p as follows [31:
UP(X)
=I
N d&T’ (x)
p (s;’
dx
i=I I
(x))
I
(6)
Here p (x) = 2 (x) is the Radon-Nykodym derivative of any measure p absolutely continuous with respect to the Lebesgue measure. The invariant measure is the Lebesgue measure. Indeed
I2- ( l[O,l] (s;‘x)
~lto.11 Lx) = : j=j
ai
N
=
=
CI I
i=l
5&
[O,l’ (x)
(7)
t’ l[O.ll (xl ( i=, 1 O(i 0
= l[O.l] (x).
2. CONSTRUCTION OF THE SPECTRAL DECOMPOSITION
In order to obtain the spectral decomposition of the Frobenius-Perron operator or the Koopman operator we choose the following initial biorthonormal system: Iqan) = (%I
(-1)“p
= (x4!,
(x)
> form,nlO,1,2
,...
(8)
Biorthonormality is verified directly:
(9)
The system is complete in the algebra of real analytic functions on [O, 11. Indeed for any analytic function f on [0, 1I we have:
1898
I. ANTONIOU
and BI QIAO
(f (xl 1 = +f:f/(n) (0)(2 I n=O +CO. = f(x) -6(“) (-l)” EC I n! n=O
(x) (x”(. >
Therefore:
1 P’n)(WI = 1. n=O
The matrix elements of the Frobenius-Perron operator U with respect to the biorthonormal system 1%) , (qm 1are:
(12)
I 01
m
zn= =
xl j=l
E j=l
-1 1 Lxj I 7 (Ej
-
Ej-1)
(13)
$ J
From property (4) the slope oci is:
where (T (nj) is the sign of the slope ‘Xi. Therefore the eigenvalues z, can also be expressedas:
j=l
Equilibrium corresponds to the eigenvalue zo = 1. Indeed:
(14)
1899
Spectral decomposition of piecewise linear monotonic maps
(15) From (14) we seethat degeneracy in the spectrum may occur if and only if:
5 s”bk) IlkIn+’ = iP (o(k) Izkr+l, for n f k=l
m.
(16)
k=l
In the case of Markov partitions into cells with equal length IZ, ] = IZz 1 = eigenvalues are given by the simple formula:
n odd
= jZN] = i the
(17)
Here N, and N- with N+ + N- = N denote the number of branches with a positive and negative slope, respectively. Therefore for this simple case condition (16) implies that degeneracy occurs if and only if n is even and m is odd and N+ = ; [N + N”‘-+‘I
.
Because of the possibility of degeneracy we shall consider the simple spectra in Section 2.1 and degenerate spectra in Section 2.2. 2.1. Simple spectrum
In order to obtain the eigenvectors of the Frobenius-Perron operator we use the algorithm presented in [4] based on the creation and destruction operators C, D. The right and left eigenvectors of U are given by the formulae:
(hi = ((pi+G)PiI Ifi) = 1(Pi + Di)+ @i) .
(18) (19)
The action of the creation and destruction operators C, D can be expressedin terms of their matrix elements with respect to the initial biorthonormal system ( Cpj), (vi I by
Ml = (PiI + 2 (GPiI a/r) (qkl k=O
Ifi) = I+[) + T (DiVkI Qi) I@k). k=O
The matrix elements of the operators C and D can be obtained recursively from the equations:
1900
I. ANTONIOU
and BI QIAO
(20)
From the above equations, by using the formula (12) we can obtain the equations for the matrix elements of C,,,D, as:
and
Therefore the eigenvectors of the Frobenius-Perron operator U are obtained by
(24) k=O
and
14) = I@‘“)+ 2 (DdPkI%>1%)
cm
k=n+l
where
(fol = bol = 1
(26)
which confirms that the invariant measure of this system is the Lebesgue measure. The spectral decomposition of the Frobenius-Perron operator and the Koopman operator is:
i=o
W’)
for any observable f and any density p in the domains which provide meaning to the spectral decomposition. 2.2. Degenerate spectrum
We shall consider the simple casewhen 111I = 112I = = 11~1 I = f, and N+ = N- + 1. More general casesof degeneracy can be treated in the same way. The matrix elements of U in this simple case can be expressedas:
1901
Spectral decomposition of piecewise linear monotonic maps
m! L i.P Pr!(m- n)! N”+’ i=l
W%?l %) = (Pm1J%) =
&,
(Cyi) (-fii)“-“,
m > fl
m = n, n even
(28)
1 -jjm+1* m = n, n odd I 0, m-en Then the Koopman operator V is represented as a lower-triangular matrix with respect to the biorthonormal system I%), (q,,, [ :
with diagonal part
(30) and off diagonal part
From (30) we see that the eigenprojectors of VOfor k = 1,2,. the parts 0, of intermediate operator 0 are given by [4]: 0, = P,UP, + P,UQ&P,
are 2-fold degenerate. Then
= P,UP,,
(32)
here P,UQ,C,P, = 0 because C, and P,JQ,, are the upper-triangular operators without diagonal parts. The upper-triangular property of PJJQ,, is obvious from (28). The upper-triangular property of C, can be proved by the upper-triangular property of P,,UQ,, [4]. Therefore the intermediate operator 0 is:
0 = fy@. = FPJIP, n=O n=O
(33)
From (33) we can obtain
0’
=
I@O) (PO1 + F & k=l
[ 1*2k)
(p2k
+
1@2k-l)
(Q32k-1
11
(34)
k=l
Equation (34) shows that the non-vanishing of the lower-triangular matrix elements VI@ UC - I ) IS a necessary and sufficient condition for the appearance of Jordan blocks in O+ and therefore for U becausethe intertwining operators Szand R-l transfer the Jordan-block structure of 0 to U through U = SZOSZ-’[4] : b2kI
1902
I. ANTONIOU
v =
j W’)‘@o)
and 31 QlAO
OmI
+~$[l(=')+Fik)(@,k/ k=l
+
i(n-')+FZk-l)(R~Zk-iI]
(351
This enabIes us to arrive at: (f2k
I&k-l)
(fip2ki
=
1 = @;ZQ72k-I
(&k-I
i&k)
j =
((pk
1=
=
(a-‘)+
=
[(~k+D*ck)-‘(~+~k)lt~2k).
=
w)+
=
[@k
+ ck)
( h-
+ ck)
Q)Zk-1
(35)
k = 0, 1,.
Q)Zk 1 ,
j ,
k =
(37)
1,2,.
@2k)
k=O,
l,...
(38)
h-1) + Dkck)-’
(pk
+ Dk)]+
+2k-1)
,
k=
1,2,...
(39)
\vhere
(fol = (Pot = (11 (fi
I =
(PA
+
(GPlI
w
(40) (4
=
(f2l
=
(A
+
(
bml
(41)
%) (4701
1
(42) x2 + 7 _ l b2l If Wo) - + (cm I @o) (472t FfPII -1 $ 9 and the matrix elements of the operator C or D can be calculated from the following C, D operator equations: =
PjCi =
- Zi
lj
DiPj = &
,
I
(PjCiU] Pi - PjC’lPi - PjU]CiPi) , j < i
(43)
(fiU1 Pj - fiDiU1 Pj - Pib’l Dip,) , i < j.
WI
1903
Spectral decomposition of piecewise linear monotonic maps
0.6 0.4 0.2
The relations (43) and (44) can be obtained from general C, D operator equations in [4] by means of the upper-triangular property of P,,UQn. Therefore, when (~lgk] VI&-r) f 0, k = 1,2,... , the Frobenius-Perron operator U and the Koopman operator V have the Jordan decomposition as
=
b
I&)
(fol
f)
+g
&
[b
l&k)
(f2kl
f)
+
b
lfik-I)
&k-1
1 f)]
(45) + +f k=l
b2kI
v
Ifik-I)
(p
I&k)
(hk-l
1 f)
where the real numbers & and (Q)2k] V ]@k-1) are the eigenvalues and weights of each irreducible block k, respectively. When (Q)2k]V ]@2k-1) = 0, k = 1,2, . . , the Frobenius-Perron operator U and the Koopman operator V have the diagonal spectral decomposition with 2-fold degeneracy as:
WPI f) = (PI w =
(Pim)(/oi~)+~~[(PIrik)(/2*/~)+(Pl~k-l)(f.k-l~i)].
3. EXAMPLES
We shall discuss four examples of piecewise monotonic Markov maps. (1) S1 : ro, 11 - 10, 11 (Fig. 1) defined by:
(46)
I. ANTONIOU
1904
and BI QIAO
I _ 3X, x E
1-
: 0, f 1 3x - 1, x E ;, ; 1 _
Sl (x) =
2, x E
3x -
(47)
1 ;, >
From (6) the formula of the Frobenius-Perron operator is expressedby
(48)
m!
From (28) the matrix elements of V with respect to the biorthonormal system I@,) , (91,~I are:
-L ((-I)"+ n!(m - n)! 3m+' w?M
S)
= (%A
ml)
I
=
iJ
m = n, n
1
-3n+l'
0,
1 +2"-"),
m>n
even
m = n, n
(491
odd
m-en
From (49) we seethat the Koopman operator has Jordan blocks decomposition: +m @+
=
I@O)
(PO1
+
1
. &
[l@Zk)
(Q)ZkI
I@2k-1)
+
(Q)Zk-I
I]
k=l + +f k=l
(Q)ZkI
v
i@2k-1)
I+‘Zk)
(Q)Zk-I
I
v = (ROQ-I)+ =
16)
($3 1 +
ff k=l
j&
(Q)2kI
v
I@2k-I)
+ E
[i&k)
(f2k
I&k)
1 + Ifik-1)
(hk-1
(hk-I
11
(51)
1
k=l
The principal vectors of V are: (f2kl
(f2k-11
I&)
&k-I)
=
=
1 [ (pk
=
((Pk+Ck)Q)2kj,
=
((Pk+Ck)p2k-,I,
+ Dkck)-’
I[(~k+~kck)-‘(P*+Dk)]+@2k-l)~
(pk
(52)
k=o,l,...
k= 1,2,...
+ Dk)]+
+2k)
J
k = 0, l,... k= 1,2,...
(53) (54) (55)
The first, second and third right principal vectors can be easily calculated by using the above formula:
Spectral decomposition of piecewise linear monotonic maps
1905
Fig. 2. A piecewise linear monotonic map S2 (.u).
(fol = (11
(56)
VII = (ml; Gmr@o) bol = X-- 2 ( I
(57)
(fil
=
kP21 +p(?4
@o) (PO1
(58)
9 - -
=
3 I
(
where we can check:
Uf,(x, = u x2-; ( > = $(2-;)+$(x-;)
(5%
(60)
This example with Jordan block structure is like the example mentioned by Driebe [26]. (2) S2 : [O, 1I - [O. 11defined by
$2(x)=
1-4x,
x E
0,;
4x-1,
XE
12 - -
I 11 -2x+2,
i4 XE -,I "r 4
>
1
(61)
I. ANTONIOU
1906
and BI QIAO
I .o
0.8
0.6
0.4
0.2
Fig. 3. A piecewise linear monotonic map & (x).
where IZl I = IZzl f 1131,and N+ = 1, N- = 2. (Fig. 2). The Frobenius-Perron operator is: (62)
From (13), (24) and (25) we can obtain the eigenvalues and the eigenvectors. The eigenvalues are:
(63) The first and second right eigenvectors of U are: fo (xl = 1
(64)
and 1 j-i (x) = x - -. (65) 2 This shows that the eigenvaluesjump from positive to negative when n even changes to n odd, and vice versa. (3) Ss : [0, 1I - LO,1I defined by
s3
(xl = -
where IZrI = 1121= 1131= 114I = $, and N+ = N- = 2 (Fig. 3).
(66)
Spectral decomposition of piecewise linear monotonic maps
1907
0.8 0.6 0.4 0.2
0.4
0.2
0.6
0.8
I .o
Fig. 4. A piecewise linear monotonic map S4 L’c)
The formula of the Frobenius-Perron operator is: UP(x)=~[p(-~x+~)+p(~x+~)+P(-~x+~)+P(~x+~)].
(67)
From (13) the eigenvalues are:
z,=
0, n odd 1 n even ’
1&I’
Therefore the eigenvalues are zero when n is odd. Comparing S3 with S2 we see that S3 is symmetric with respect to the midpoint of the unit interval. This geometric symmetry is the reason that all odd eigenvalues of U are zero. (4) S4 : [O, 11 - [0, 11 defined by
s4 (x)
(69)
=
where 111I = IZ2I = 113I = i and iV+ = N- + 1 = 2 (Fig. 4) is a Tent map [25]. The Frobenius-Perron operator is: (70) From (13) (36) and (37) the eigenvalues are: -
n odd n even
and the first, second and third right eigenvectors of U are:
(71)
I. ANTONIOU
1908
and BI QIAO
fo(xl = 1
(72)
1 fi (xl = x - 2
(73)
2 (74) 3 There exists a two-fold degeneracy,but there is no Jordan block structure. The matrix elements of the Frobenius-Perron operators of S4 and Si are:
fi Lx) = 2 - -.
bml ti 1%)= & b?bl VI IW = &
{(-1Y
{(-I)”
(-2Y+
(-2)“-”
+ 2,-‘},
m>n
+ (-1)” (-3)“-“},
(75)
m > n.
Therefore:
(cpmI Vi I%) = 0, for n odd, m > n
(76)
m > n.
(77)
(tp,l I4 IQ,,) f 0, for n odd,
Formulae (76) and (77) show why there is no Jordan decomposition for S4 but there is a Jordan decomposition for Si. Similarly we seethat there is no Jordan decomposition for the FrobeniusPerron operator of the Tent maps [25]
T, =
(78)
because (QmlVl@2k-1) =O, wherem = 2,3 ,... , n = 0, l,... ,
[qp
fork=
1,2,...
and [y 1denotes the integer part of real number y.
4. THE MEANING OF SPECTRAL DECOMPOSITION
The spectral decompositions (27), (45) and (46) are meaningful for any observable f and any density p in suitably chosen domains. Since the eigenvectors or the principal vectors (8) of the Koopman operator contain Dirac’s delta functions and its derivatives, the spectral decomposition (27), (45) and (46) make no sensein the Hilbert space L* of square integrable functions on the unit interval. A natural way to give meaning to the spectral decomposition (27) (45) and (46) is to extend the Koopman operator V beyond the Hilbert spaceL* to a larger space x@ of continuous anti-linear functionals on a properly chosen dense subspace Cpof the Hilbert space L*. In other words, relations (27), (45) and (46) should hold for any density function p in the test spaceQ,and any anti-linear functional f in the anti-dual space ‘a. In this way the spectral decompositions (27), (45) and (46) can be understood as a natural extensions of the Gelfand-Maurin theory of
Spectral decomposition of piecewise linear monotonic maps
I909
the generalized spectral decomposition of norma operators with continuous spectrum [g-lo]. This generalization [4,7] is necessarybecausethe original Gelfand-Maurin theory holds onIy for operators with a spectral theorem in Hilbert spaceIike the normal operators, while the Koopman operator, being a unilateral shift, has no spectral decomposition [4,7] in Hilbert space. A simple choice for the test function space is the space P of polynomials which is the union of the spaces3” of polynomiaIs with the degree smaller or equal to n :
n=l
We can define a L-F topology [9] on ?’ in the folIowing way: Let u’ = ct &xk f P, the norm on P is defined by
then (80) provides the strict inductive limit topology llwll,, = I[wll,+l on P [93. Therefore we can conclude: (1) The space 2’ is a nuclear LF-space, thus it is complete and barreled, hence it is a dense subspace in the Hilbert space L2. (2) The space P is stable with respect to the Frobenius-Perron operator U and U is continuous with respect to the LF- topology of P, because U preserves the degree of the polynomials. In fact, from (6) we have
= T+
Iti I ($$j” I
which shows that UIP c 2’. The extended Koopman operator V can be written as follows: In the case of the simple spectrum:
or in the case of degeneracy and Jordan blocks:
' = 16) (fat + :$[&k, k=l
(hk[
+
i&k-i)(fZk-,I]
(83)
Both can acquire meaning in the rigged Hilbert space P C L& C xP or to the dual pair PI x2? The eigenvalues zj of U coincide with the Pollicott-RuelIe resonances. In fact, the PollicottRuelle resonances are the inverses of the zeros of the Fredholm determinant [I l-141:
1910
I. ANTONIOU
and Bl QIAO
d(z) z det(1 -zU) (84)
The triangular property of the matrix elements (12) gives: TrU” = 1 (U”piI @i) i=O +m
(85)
= CG i=O
which allows us to arrive at:
= gne-“;qG)” i=O neln(
=f
1-x;)
(86)
i=O +a =
Cll(l
-ZZi>
i=O +oO
= Crr(l i=o
-ze-“).
Therefore we have: e-y! = zi,
(87) Relation (87) confirms that the complex poles yi of the Fredholm determinant d (z) are the logarithms of eigenvalues Zi. The eigenvalues zi < 1, i = 1,2,. . . , can determine the rates of the correlation functions and the rate of approach to the equilibrium. This shows that the piecewise linear monotonic maps with 11 = IZ = = IN may have different spectra and different eigenvectors or principal vectors in the same rigged Hilbert space ‘p, although they are isomorphic systems because they have the same KS-entropy [6]. This observation shows that the rates of approach to equilibrium may be different for isomorphic dynamical systems. 5. CONCLUDING
REMARKS
From the previous discussion we see that the geometry of piecewise linear Markov maps is reflected in the spectral decomposition. The Frobenius-Perron operators of piecewise linear Markov maps can be diagonal with simple or degenerate spectrum or can have Jordan block structure depending on the geometry. The admissible test function spaces 2’ exclude the Dirac delta functions. This means that the trajectories S (v’ - S”y) , n = 0, 1,2,. . . are excluded from the domain of the FrobeniusPerron operator. Formulae (27), (45) and (46) can be used for probabilistic predictions using
Spectral decomposition of piecewise linear monotonic maps
1911
initial densities expandable in terms of admissible test functions only. This remark, however, goes beyond the prediction problem, as it also reflects the intrinsically probabilistic character of unstable dynamical systems.The spectral decompositions (27) (45) and (46) have, moreover, the property that the dynamical properties are reflected in the spectrum becausethe eigenvalues are the powers of the Lyapounov time. The extension of the dynamical formulation through the spectral decomposition to suitable rigged Hilbert spacesleads to a natural classification of chaotic maps in terms of the resonance spectrum [25]. Acknowledgements-We thank Professor Ilya Prigogine for his encouragement, support and comments which, to a large extent, motivated this work. We also profited a great deal from several discussions with Professors Z. Suchanecki and E Bosco. We also acknowledge the financial support of the Belgian Government under the Interuniversity Attraction Poles and the European Commission, D G III ESPRIT project ACTCS 9282.
REFERENCES I. I. Prigogine, From Being to Becoming. Freeman, New York (1980). 2. G Nicolis and I. Prigogine, Exploring Complexity. Freeman, New York (1989). 3. A. Lasota and M. Mackey, Chaos, Fractals, and Noise, 2nd edn. Springer, New York (1994). 4. I. Antoniou and S. Tasaki, Int. J Quantum Chem. 46, 425474 (1993). 5. I. Antoniou, V. Basios and F. Bosco, Int. 1 Brjiircafions Chaos. In press. 6. I. Kornfeld, V. Fomin and Ya. Sinai, Ergodic Theory. Springer, Berlin (1982). 7. I. Antoniou and S. Tasaki, 1 Phys. A: Math. Gen. 26, 73-94 (1993). 8. I. Gelfand and N. Vilenkin, Generalized Functions, Vol. 4. Academic Press, New York (1964). 9. F. Treves, Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967). IO. A. Bohm, Quantum Mechanics: Foundations and Applications, 3rd edn. Springer, Berlin (1993). 11. M. Pollicott, Invent. Math. 81, 413 (1985); ibid. 85, 147 (1958). 12. D. Ruelle, Phys. Rev. Left. 56, 405 (1989). 13. J. P Eckmann, Resonances in dynamical systems. In Proc. IXth Int. Congress on Mathematical Physics, edited by B. Simon et al. Hilger, Bristol (1989). 14. D. Ruelle, Puble, Math. ZHES 72, 175 (1990). 15. H. Mori, B. So and T. Ose, Prog. Theor. Phys. 66, 1266-1283 (1981). 16. M. Dorfle, 1 Star. Phys. 40, 93-132 (1985). 17. G. RoeprstortI, On the Exponential Decay of Correlations in Exact Dynamical Systems. Princeton (unpublished) (1987). 18. P. Gaspard, 1 Phys. A: Math. Gen. 25, LA83 (1992); Phys Lett. A 168, 13-17 (1992). 19. H. Hasegawa and D. Driebe, Phys. Lett. A 168, 18-24 (1992); ibid. 176, 193-201 (1993). 20. S. Tasaki, Z. Suchanecki and I. Antoniou, Phys. Left. A 179, 97-102; ibid. 103-110 (1993); Chaos, Solirons & Fractals 4, 227-254 (1994). 21. D. Mackernan and G. Nicolis, Phys. Rev. E 50, 988-999 (1994). 22. S. Tasaki, I. Antoniou and Z. Suchanecki, Phys. Left. A 179,97-102 (1993). 23. S. Tasaki, Z. Suchanecki and I. Antoniou, Phys. Let?. A 179, 103-110 (1993). 24. S. Tasaki, I. Antoniou and Z. Suchanecki, Chaos Solitons & Fractals 4, 227-254 (1994). 25. I. Antoniou and Bi Qiao, Phys. Left. A. 215, 280-290 (1996). 26. D. Driebe, Jordan blocks in a one dimentional Markov map. Computers Math. Appl. In press. 27. 0. Bandtlow, I. Antoniou and Z. Suchanecki, Resonancesof dynamical systems and Fredholm-Riesz operator in rigged Hilbert spaces. Computers Math. Appl. In press. 28. C. George, Physica 65, 277 (1973). 29. I. Prigogine, C. George, F. Henin and L. Rosenfeld, Chem. Scr 4, 5 (1973). 30. T. Petrosky and Hasegawa, Physica A 160 351 (H 1989). 31. T. Petrosky and I. Prigogine, Phys. A 175, 146 (1991). 32. T. Petrosky, I. Prigogine and S. Tasaki, Phys. A 173, 175 (1991). 33. C. Presten, Iterates of piecewise monotonic mappings on an interval. In Lecture Notes in Math, Vol. 1374. Springer, New York (1988). 34. R. Adler and D. Mayer, Ergodic Theory Symbolic Dynamicas and Hyperbolic Spaces, edited by T. Bedford, M. Keane and C. Senes. Oxford University Press, Oxford (1991).