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Analytical estimates of fractal and dynamical properties for one-dimensional expanding maps
Volume
128, number
6,7
PHYSICS
LETTERS
ANALYTICAL ESTIMATES OF FRACTAL AND DYNAMICAL FOR ONE-DIMENSIONAL EXPANDING MAPS G. TURCHETTI
11 April 1988
A
PROPERTIES
I
Dipartimento di Fisica della Universitcidi Bologna, INFN sezione di Bologna, Bologna, Ital_v
and S. VAIENTI Dipartimento di Fisica della Universitcidi Bologna, Bologna, Italy Received 12 August 1987; revised manuscript Communicated by A.P. Fordy
received
18 January
1988; accepted
for publication
11 February
1988
Given an expanding map of the interval whose repeller is a Cantor-like set, we give an analytic method to approximate its dynamic and fractal properties with the corresponding variables for linear expanding maps, which can be exactly coumputed. The procedure has been applied to the logistic map and very accurate results have been obtained.
In the last years many studies have been devoted to understanding the geometric and dynamic properties of strange repellers [ l-101. These are invariant sets under the iteration of a map characterized by the fact that each orbit, which starts in a neighborhood, moves away from them. Sometimes the repeller arises as a boundary separating the basins of two attractors: a trajectory with intial point close to the repeller will wander for a short time before falling into an attractor and this transient period is characterized by the expanding properties of the repeller itself. Iterating a rational map in the compactified complex plane, the complement of those points whose trajectory is Lyapunov stable is of special importance. This set universally known as Julia set [ 111 is probably the best understood repeller (when the map is hyperbolic) and it has inspired the definition of mixing repellers [ 11. To these mappings can be applied the powerful techniques of the thermodynamic formalism [ 12 1. In this Letter we show how to approximate these mappings with linear mappings whose dynamic and Work supported
0375-9601/8g/$ ( North-Holland
in part by a MPI 60% grant.
03.50 0 Elsevier Science Publishers Physics Publishing Division )
topological variables can be exactly computed. Given a real one-dimensional expanding map T with s inverses defined on an interval [a, 61 whose repeller is J c [a, 61, the basic idea is to approximate T’(x) with a linear map L(,) (x) with s” inverses such that the preimages of [a, b] for T” and L,,,, are the same namely T-n([u, b])=L,\([a,b]), see figs. 1 and 2. Denoting with C, the repeller of L(,,) (x), which is a linear Cantor set, the Hausdorff distance of C, from J vanishes as IZ+CO. The main result is that the Hausdorff dimension and the escape rate of C, have limits for n-+co, which are the dimension and the escape rate of J. If we endow both C, and J with the balanced measure, the Lyapunov exponent and the correlation dimension of C, have limits which are the Lyapunov exponent of J and an upper bound to the correlation dimension of J respectively. The numerical convergence is also quite good and a high accuracy can be reached. Compared to all the previous methods used to compute the dynamic and fractal properties of the repellers, the present one is rigorous, accurate and the required computing time so small that it can be used on a personal computer. We first recall the rigorous definitions of escape B.V.
343
Volume
128. number
6,7
PHYSICS
LETTERS
I I April 1988
A
defined [6] in terms of the distribution of the periodic fixed points of T" and it was shown to be equal to the pressure [ 121 of -log] T’ (x) 1 [ 131: P,-=-P(T,
-log
IT’(x)]).
(1)
The pressure of a continuous function @ J-tR is given by the variational principle [ 1 ] P(T,@)= max{h(~)+150(X)d~(X)} where the maximum is taken on the set of the T invariant probability measures ,u supported on J and h(p) is the Kolmogorov entropy for p. Another useful quantity to describe the geometric complexity of a repeller J is the energy integral defined by
-4
(2)
Fig. I. We show the quadratic approximation L,,) (x).
map T(x) =x2-
5/2 and its linear
rate and correlation dimensions. Given N,, points in a neigbourhood A of the repeller, the experimental escape rate pE measures their exponential decay under iterations of the map T,namely if N,, is the number of points in A after n iterations pEw -n-l log( N,,/N,). The theoretical escape ratePT was
A theorem by Frostman gence abscissa cu(p)=inf{cr,
@(cu,~)=+co}
J
i
1,
I I
1
:
9
Fig. 2. We show the first iteration of map T, namely r2(x) (x*-5/2)*-5/2 anditslinearapproximationL,,, (x).
344
=
(3)
is a lower bound to the Hausdorff dimension a(p) 6Du. Moreover if the inverse Mellin transform of a)((~, ,D) exists, then it is equal to the correlation integral C(l) = [ 1 dp(x)
I
[ 141 asserts that the diver-
dp(y)
tV/-
Ix--~1 )
(4)
J
and for some models the correlation dimension u = lim&ogC( 1) /log (I) can be proved to exist and to be equal to the divergence abscissa [ 8,15 1. The one-dimensional systems we consider are defined by the following properties. Let T be a smooth map (of class C’) from an open neighborhood of the closed interval [ 0,l ] into R (any closed interval [a, b] can be reduced to [0,1 ] with a linear transformation) such that (i) T-'([O,ll) = LO,11; (ii) TisexpandingonT-‘([0, 11); ]T’(x)I>y>l for xcT-'([0,11); (iii) T - I([0,1 ] ) is the disjoint union of s closed intervals. With these prescriptions the repeller J= nTEO T -"([0,1 ] ) is a completely invariant Cantor set. The above conditions require that T(x) is monotonic with modulus of the derivative larger than 1 in any interval of [0,1 ] where 0 < T(x)< 1 and that
Volume
128, number
PHYSICS
6,7
there are s disjoint subintervals I,, .... I, of [0, 1 ] which are mapped by T on [0, 11. These intervals are the preimages of [0, 1 ] by T and we write T - ’ ( [0, 1 ] ) = U i=, Ik. As a consequence there are s” disjoint subintervals I{“), . ... I$’ which are mapped on [0, 1 ] by T”; we shall write where Ih”)nIj’) =Q) for T-n( [0, l])=U;:,Ip) k# j and we shall denote with T;” (x) the inverse of the restriction of T”(x) to the interval I,(“) where it is single valued. To these systems an ergodic type theorem applies ]161>
LETTERS
The Hausdorff dimension sitive solution of [ 141 A”H+ ... SADH=l s I
h-m
z f(T;”
(~0
1)
=
~/(x1
@B
(xl
,
DH is the unique real po-
.
(10)
Given the balanced measure p(B the divergence abscissa a(~~) (which will be simply denoted cy in the following) is given by the real positive solution of [151 2;” +...+;1,,+
=s2
and the Lyapunov A=
lim r
1I April 1988
A
s
log1 T’ (x)
(11) exponent
I d@B
n (pLg) =/i is given by
(x>
C
s”,=, J
11 >
-=[O,
(5)
where ,uB is a T-invariant ergodic measure on J, f( x) is a continuous function on [0, 1 ] and the result holds uniformly for X~E [0, 11. We recall that a measure for which (5) holds, enjoys the property
= - ; [log@, ) +...+log(i,)]
dfiB(x)=
fkil
J
~flT~‘(x))
dp,(x)
(6)
J
and is called balanced measure [ 17 1. A very important subclass of these cantorian mappings is given by the linear cantorian mappings. These are defined by maps L(x) which are piecewise linear on s disjoint intervals I,, .... I, of [0, 1 ] and map each of them on [0, 1 ] so that we can write L-‘( [0, l])=U;,,Ik where I,fV,=(d for kzj. On each interval Ik we denote with Lr_’ (x) the inverse of L(x) and write
(12)
Given a nonlinear map T(x) we approximate it with a sequence L,,) (x) of linear cantorian maps in such a way that T-“(LO,
If(x)
.
l])=L,:([O,
11).
(13)
Eq. (13) defines the linear cantorian map L(,, (x) and its scales lk are given by the lengths of the intervals IA”) = T,” ( [0, 1 ] ) Ai”) =diameter{Iifl)},
k= 1, . ... S” .
(14)
and the positive numbers I 1, .... A, are the scales of the linear Cantor, the repeller of L( x), which we denote with C. For these linear mappings the pressure of the function /3loglL’(x)J for PER, L(x) and L’(x) restricted to the Cantor C is given by
In figs. 1 and 2 we describe the construction of the piecewise linear maps L, , ) (x) and LC2)(x) approximating the quadratic map T(x) =x2-: and its first iterate T”(x). Denoting with C,, the Cantor which is the repeller of L(,, (x) and with p:“‘, OF), a(“), A’“’ the theoretical escape rate, Hausdorff dimension, divergence abscissa and Lyapunov exponent for L(,, (x), endowed with the balanced measure, the following results have been proved. ( 1) The Hausdorff distance 6( C,, J ) converges to zero as n-+co. This result follows from theorem 3.8 of ref. [ 141 and we recall that the Hausdorff distance 6(A, B) between two sets A, B in the metric space with distance d( , ) is defined as [ 141
P(L,Plog
(8)
6(A, B) =sup inf d(x, y) +sup
(9)
(2 ) The pressure for L(,, times 1/n converges the pressure for T
L;’ (x)=ah+tkjlkx, QZ‘tl,
XE[O,l],
Lk’([O,
l])EIk,
JL’(x)~)=log(~;~+...+&~).
In particular
the theoretical
p-r=-log(A,
+...+A,).
(7)
escape is given by
aeA
.ytB
atB
inf d(x, y)
.
(15)
VEA
to
345
Volume
128, number 6,7
=P(T.Plog
Ir(x)o
PHYSICS
1
(16)
where ~IE~Rand the convergence is uniform with respect to j3 on any compact subset of [R.The proof will be given in a forthcoming paper [ 18 1. Since the escape rate is related to the pressure by eq. ( 1) it follows that 1/n times the escape rate from C, converges to the escape rate from J, lim -I,,) n .Z,n
=p7 .
(17)
From the Bowen-Ruelle formula P(T, - DH log I T’ (x) I ) = 0 it follows that the Hausdorff dimension of C,? converges to the Hausdorff dimension of J, limD2)
,1-x2
=DH .
(18)
Inspecting eqs. (8 ) and ( 11) for piecewise linear maps L(x) with s scales, we see that the abscissa of divergence cy with respect to the balanced measure is the unique real positive root of the equation
LETTERS
IL’(x)I)=10g(s2)=2P(L,0).
(19)
This result gives a dynamical meaning to the abscissa of divergence and is formally similar to the Bowen-Ruelle formula quoted above. ( 3 ) The abscissa of divergence cy, of the map L,,) converges to a limit iir, limo(“)-fi n-ax
9
(23) where h(,uB) is the Kolmogorov entropy. The convergence of the information dimension is then an obvious consequence of ( 23 ). We have applied the above results to the quadratic map x’ = T(x; p) =x2-p,
(21)
Before analyzing the numerical results we quote the last convergence result. (4) For the balanced measure the Lyapunov exponent of L,,, times 1/n converges to the Lyapunov exponent of T, 346
p>2,
(24)
for which numerical non-rigorous results are available and rigorous theoretical bounds have been derived for the Hausdorffdimension. For the map (24) the interval [ 0, 1 ] is replaced by the interval [ -4, q] where q= (1 +dG)/2. The bounds to the Hausdorff dimension read [ 2 1,221
bJz=J
(
2 log 2 d
log 2 Y’
-’ (25)
where q’ = ,/p - q and the upper bound is nontrival for p>2.72. Another rigorous upper bound which applies to the above systems [22] is
(20)
where the pressure for the map T(x) takes also the value 2P( T, 0). However a can only be proved to be an upper bound to the divergence absissa a!(~~) of T and a lower bound to the information dimension D, (,uB) with respect to the balanced measure [ 191, a(fi,)~ol~D,(~~g).
1988
The information dimension [ 191 with respect to the balanced measure can be computed using the relation [20]
1+
P(L,alog
I I Apr11
.4
DH<
log 2 pT.+log 2
(26)
and can be used to check the numerical results. The observed convergence is rather fast and its rate increases as we increase p. Also for p close to 2 we can remarkably improve the convergence if we use the Thiele continued fraction as extrapolation algorithm, see appendix 3 of ref. [ 23 1. For p> 10 the extrapolated results exhibit a five digit accuracy using only the first three iterations of T, a computation that can be made with a pocket calculator. In table 1 we quote for p= 1 the sequences n -‘pp), Dg), a(“) and n -‘A(“) and the corresponding extrapo-
Volume
128, number
6,7
PHYSICS
Table 1 For p= 3 and 1 G n < 12 we quote the linear cantorian abscissa of divergence and their Thiele extrapolations.
LETTERS
approximations
I 1 April 1988
A
to the Lyapunov
exponent,
n
,,t<“Jn-’
A’“, &,,n-’
p’“‘n-I
plZ’!,,n-’
DP;’
DE%,
1 2 3 4 5 6 I 8 9 10 11 12
1.1435 1.1431 1.1395 1.1372 1.1358 1.1348 1.1341 1.1336 1.1332 1.1329 1.1326 1.1324
1.1435 1.1426 1.1441 1.1286 1.1306 1.1301 1.1299 1.1300 1.1300 1.1300 1.1300 1.1300
0.45037 0.42672 0.42012 0.4 1680 0.41481 0.41348 0.41252 0.41181 0.41126 0.41081 0.41045 0.41015
0.45037 0.40308 0.40840 0.40682 0.40676 0.4068 1 0.40682 0.40683 0.40682 0.40682 0.40682 0.40682
0.60616 0.61412 0.61751 0.61925 0.6203 1 0.62101 0.62152 0.62190 0.66219 0.62243 0.62262 0.62279
0.60616 0.62209 0.62595 0.62458 0.62459 0.62458 0.62456 0.62457 0.62457 0.62458 0.62457 0.62456
Table 2 For three values of p, we give the Hausdorff dimension after n = 11and n = 12 backward the lower bound (D; ) and upper bounds (Dg ) given by eq. (25) and (26).
3 10 30
lated sequences DAi” Dj,l”
0.62262 0.38198 0.29069
0.62457 0.38223 0.29074
0.62279 0.3820 0.29069
for 1~ n < 12. In table 2 we compare and the extrapolated values Dk’l’iext), DA”‘” (ext) with the bounds given by (25 ) and (26 ). Finally for a comparison with refs. [7,24] we quote @t-‘2) (ext) for cr=2.4725, 3.75, 20, 48.75, 90, which reads kp$“) (ext) =0.263796, 0.552765, 1.47904, 1.93565, 2.24574. In fig. 3 we show DA12’ (ext), &pf”) (ext) and &4(12)(ext) for 2qp~4.5 and we see that the limits agree with 1, 0 and log 2 respectively which are the values for the Julia set with p= 2. This set is no-longer a Cantor but the closed interval [ - 2, 2 1. In the same limit a behavior like (p- 2) “’ of pT and 1 -DH is observed. For p> 2 the curves are smooth and suggest a regular dependence on p of the relevant dynamical and fractal variables. The divergence abscissa has a poorer convergence to 1 when p-t2 which can be explained as follows. The pressure we denote P(p; p) =P( T, /3 log 1T’ (x; p) ) ) has forp-t2 a limit which can be analytically computed [ 25 ] and reads P(/3;2)=2/?log2for/3>1,P(~;2)=(1+/Qlog2for p< 1. The divergence abscissa cr is a solution of the
0.62456 0.38223 0.29074
0.60744 0.38125 0.29059
iterations,
escape rate, Hausdorff
aO’l 0.60616 0.59904 0.59847 0.59867 0.59888 0.59904 0.59915 0.59923 0.59930 0.59935 0.59939 0.59943
dimension,
ffr:xL 0.60616 0.59193 0.59779 0.60196 0.59996 0.59964 0.59978 0.59986 0.59982 0.59982 0.59982 0.59982
with their Thiele extrapolations
0.8707 0.41841 0.30224
and
0.63015 0.38370 0.29109
equation P( cu; 2) = 2 log 2 exactly at the point where the derivative of the pressure is discontinuous. Our sequence of approximations provides a limit curve for the pressure P(P; 2) where the edge at j?= 1 is rounded off and precisely at that point the convergence will be the worst.
2
IP 4.5
Fig. 3. We show the dependence on p of the escape rate (line a), Hausdorff dimension (line b) and Lyapunov exponent (line c) for 2-cpS4.5.
347
Volume 128, number
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PHYSICS
The present method can be extended to conformal expanding maps of the plane with a disconnected repeller. For example if the invariant set J is included in a closed disc D with the same diameter (this is the case for the Julia sets), then the scales of the approximate linear Cantors are again given by eq. ( 14) where I; denote the preimages by the map T” of the disk D. To conclude we notice that a procedure to determine the Hausdorff dimension of an attractor based on the generation of resealed Cantor sets was also recently developed in ref. [ 26 1.
References [ 1 ] D. Ruelle, Ergod. Theory Dynam. Syst. 2 ( 1982) 99. [ 21 H. Kantr and P. Grassberger, Physica D 17 ( 1985) 75. [ 3) S. Takesue, Prog. Theor. Phys. 76 ( 1986) 11. [4] P. Szepfalusy and T. Tel, Phys. Rev. A 34 (1986) 2520. [ 5 ] S. McDonald, C. Grebogi, E. Ott and J.A. Yorke, Physica D 17 (1985) 125. [ 61 L.P. Kadanoff and C. Tang, Proc. Natl. Acad. Sci. USA 8 1 (1984) 1276. [ 71 M. Widom, D. Bensimon, L.P. Kadanoff and S.J. Shenker, J. Stat. Phys. 32, (1983) 443. [ 81 D. Bessis, G. Servizi, G. Turchetti and S. Vaienti, Phys. Lett. A 119 (1987) 345.
348
LETTERS
A
I 1April 1988
[ 91 S. Pelikan. Trans. Am. Math. Sot. 292 ( 1985 ) 695. [ lo] P. Grassberger and I. Procaccia, Physica D 9 ( 1983) 189. [ 111 H. Brolin, Ark. Math. 6 (1965) 103. [ 121 D. Ruelle, Thermodynamic formalism (Addison-Wesley, Reading, I978 ). [ 131 J.P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57 ( I985 ) 617. [ 141 K.J. Falconer, The geometry of fractal sets (Cambrrdge Univ. Press, Cambridge, 1985 ) [ 151 D. Bessis. J.D. Fournier. G. Servizr, Ci. Turchetti and S. Vaienti. Phys. Rev. A 36 (1987) 920. [ 161 M.Ju. Ljubich, Ergod. Theory Dynam. Syst. 3 ( 1983) 35 1. [ 171 M.F. Barnsley and S. Demko. Proc. R. Sot. A 399 ( 1985) 243. [ 181 S. Vaienti, Some properties of mixing repellers, to be published in J. Phys. A. [ 191 L.S. Young, Ergod. Theory Dynam. Syst. 2 ( I982 ) 109. [ 201 F. Ledrappier and M. Misiurewiez, Ergod. Theory Dynam. Syst. 5 (1985) 595. [ 211 T.S. Pitcher and J.R. Kinney, Ark. Mat. 8 ( 1968) 25. [22] S. Vaienti. Nuovo Cimento B 99 (1987) 77. [ 231 W. Van Assche, G. Turchetti and D. Bessis, J. Stat. Phys. 43 (1986) 85. [ 24 ] T. Tel, preprint ICTP, Trieste ( 1986). [25] D. Bessis, G. Paladin. G. Turchetti and S. Vaientr, Gcneralized dimensions, entropies and Lyapunov exponents from the pressure for strange sets, to be published m J. Stat. Phys. [26] K.J. Tsang, Phys. Rev. Lett. 57 (1986) 1390.
128, number
6,7
PHYSICS
LETTERS
ANALYTICAL ESTIMATES OF FRACTAL AND DYNAMICAL FOR ONE-DIMENSIONAL EXPANDING MAPS G. TURCHETTI
11 April 1988
A
PROPERTIES
I
Dipartimento di Fisica della Universitcidi Bologna, INFN sezione di Bologna, Bologna, Ital_v
and S. VAIENTI Dipartimento di Fisica della Universitcidi Bologna, Bologna, Italy Received 12 August 1987; revised manuscript Communicated by A.P. Fordy
received
18 January
1988; accepted
for publication
11 February
1988
Given an expanding map of the interval whose repeller is a Cantor-like set, we give an analytic method to approximate its dynamic and fractal properties with the corresponding variables for linear expanding maps, which can be exactly coumputed. The procedure has been applied to the logistic map and very accurate results have been obtained.
In the last years many studies have been devoted to understanding the geometric and dynamic properties of strange repellers [ l-101. These are invariant sets under the iteration of a map characterized by the fact that each orbit, which starts in a neighborhood, moves away from them. Sometimes the repeller arises as a boundary separating the basins of two attractors: a trajectory with intial point close to the repeller will wander for a short time before falling into an attractor and this transient period is characterized by the expanding properties of the repeller itself. Iterating a rational map in the compactified complex plane, the complement of those points whose trajectory is Lyapunov stable is of special importance. This set universally known as Julia set [ 111 is probably the best understood repeller (when the map is hyperbolic) and it has inspired the definition of mixing repellers [ 11. To these mappings can be applied the powerful techniques of the thermodynamic formalism [ 12 1. In this Letter we show how to approximate these mappings with linear mappings whose dynamic and Work supported
0375-9601/8g/$ ( North-Holland
in part by a MPI 60% grant.
03.50 0 Elsevier Science Publishers Physics Publishing Division )
topological variables can be exactly computed. Given a real one-dimensional expanding map T with s inverses defined on an interval [a, 61 whose repeller is J c [a, 61, the basic idea is to approximate T’(x) with a linear map L(,) (x) with s” inverses such that the preimages of [a, b] for T” and L,,,, are the same namely T-n([u, b])=L,\([a,b]), see figs. 1 and 2. Denoting with C, the repeller of L(,,) (x), which is a linear Cantor set, the Hausdorff distance of C, from J vanishes as IZ+CO. The main result is that the Hausdorff dimension and the escape rate of C, have limits for n-+co, which are the dimension and the escape rate of J. If we endow both C, and J with the balanced measure, the Lyapunov exponent and the correlation dimension of C, have limits which are the Lyapunov exponent of J and an upper bound to the correlation dimension of J respectively. The numerical convergence is also quite good and a high accuracy can be reached. Compared to all the previous methods used to compute the dynamic and fractal properties of the repellers, the present one is rigorous, accurate and the required computing time so small that it can be used on a personal computer. We first recall the rigorous definitions of escape B.V.
343
Volume
128. number
6,7
PHYSICS
LETTERS
I I April 1988
A
defined [6] in terms of the distribution of the periodic fixed points of T" and it was shown to be equal to the pressure [ 121 of -log] T’ (x) 1 [ 131: P,-=-P(T,
-log
IT’(x)]).
(1)
The pressure of a continuous function @ J-tR is given by the variational principle [ 1 ] P(T,@)= max{h(~)+150(X)d~(X)} where the maximum is taken on the set of the T invariant probability measures ,u supported on J and h(p) is the Kolmogorov entropy for p. Another useful quantity to describe the geometric complexity of a repeller J is the energy integral defined by
-4
(2)
Fig. I. We show the quadratic approximation L,,) (x).
map T(x) =x2-
5/2 and its linear
rate and correlation dimensions. Given N,, points in a neigbourhood A of the repeller, the experimental escape rate pE measures their exponential decay under iterations of the map T,namely if N,, is the number of points in A after n iterations pEw -n-l log( N,,/N,). The theoretical escape ratePT was
A theorem by Frostman gence abscissa cu(p)=inf{cr,
@(cu,~)=+co}
J
i
1,
I I
1
:
9
Fig. 2. We show the first iteration of map T, namely r2(x) (x*-5/2)*-5/2 anditslinearapproximationL,,, (x).
344
=
(3)
is a lower bound to the Hausdorff dimension a(p) 6Du. Moreover if the inverse Mellin transform of a)((~, ,D) exists, then it is equal to the correlation integral C(l) = [ 1 dp(x)
I
[ 141 asserts that the diver-
dp(y)
tV/-
Ix--~1 )
(4)
J
and for some models the correlation dimension u = lim&ogC( 1) /log (I) can be proved to exist and to be equal to the divergence abscissa [ 8,15 1. The one-dimensional systems we consider are defined by the following properties. Let T be a smooth map (of class C’) from an open neighborhood of the closed interval [ 0,l ] into R (any closed interval [a, b] can be reduced to [0,1 ] with a linear transformation) such that (i) T-'([O,ll) = LO,11; (ii) TisexpandingonT-‘([0, 11); ]T’(x)I>y>l for xcT-'([0,11); (iii) T - I([0,1 ] ) is the disjoint union of s closed intervals. With these prescriptions the repeller J= nTEO T -"([0,1 ] ) is a completely invariant Cantor set. The above conditions require that T(x) is monotonic with modulus of the derivative larger than 1 in any interval of [0,1 ] where 0 < T(x)< 1 and that
Volume
128, number
PHYSICS
6,7
there are s disjoint subintervals I,, .... I, of [0, 1 ] which are mapped by T on [0, 11. These intervals are the preimages of [0, 1 ] by T and we write T - ’ ( [0, 1 ] ) = U i=, Ik. As a consequence there are s” disjoint subintervals I{“), . ... I$’ which are mapped on [0, 1 ] by T”; we shall write where Ih”)nIj’) =Q) for T-n( [0, l])=U;:,Ip) k# j and we shall denote with T;” (x) the inverse of the restriction of T”(x) to the interval I,(“) where it is single valued. To these systems an ergodic type theorem applies ]161>
LETTERS
The Hausdorff dimension sitive solution of [ 141 A”H+ ... SADH=l s I
h-m
z f(T;”
(~0
1)
=
~/(x1
@B
(xl
,
DH is the unique real po-
.
(10)
Given the balanced measure p(B the divergence abscissa a(~~) (which will be simply denoted cy in the following) is given by the real positive solution of [151 2;” +...+;1,,+
=s2
and the Lyapunov A=
lim r
1I April 1988
A
s
log1 T’ (x)
(11) exponent
I d@B
n (pLg) =/i is given by
(x>
C
s”,=, J
11 >
-=[O,
(5)
where ,uB is a T-invariant ergodic measure on J, f( x) is a continuous function on [0, 1 ] and the result holds uniformly for X~E [0, 11. We recall that a measure for which (5) holds, enjoys the property
= - ; [log@, ) +...+log(i,)]
dfiB(x)=
fkil
J
~flT~‘(x))
dp,(x)
(6)
J
and is called balanced measure [ 17 1. A very important subclass of these cantorian mappings is given by the linear cantorian mappings. These are defined by maps L(x) which are piecewise linear on s disjoint intervals I,, .... I, of [0, 1 ] and map each of them on [0, 1 ] so that we can write L-‘( [0, l])=U;,,Ik where I,fV,=(d for kzj. On each interval Ik we denote with Lr_’ (x) the inverse of L(x) and write
(12)
Given a nonlinear map T(x) we approximate it with a sequence L,,) (x) of linear cantorian maps in such a way that T-“(LO,
If(x)
.
l])=L,:([O,
11).
(13)
Eq. (13) defines the linear cantorian map L(,, (x) and its scales lk are given by the lengths of the intervals IA”) = T,” ( [0, 1 ] ) Ai”) =diameter{Iifl)},
k= 1, . ... S” .
(14)
and the positive numbers I 1, .... A, are the scales of the linear Cantor, the repeller of L( x), which we denote with C. For these linear mappings the pressure of the function /3loglL’(x)J for PER, L(x) and L’(x) restricted to the Cantor C is given by
In figs. 1 and 2 we describe the construction of the piecewise linear maps L, , ) (x) and LC2)(x) approximating the quadratic map T(x) =x2-: and its first iterate T”(x). Denoting with C,, the Cantor which is the repeller of L(,, (x) and with p:“‘, OF), a(“), A’“’ the theoretical escape rate, Hausdorff dimension, divergence abscissa and Lyapunov exponent for L(,, (x), endowed with the balanced measure, the following results have been proved. ( 1) The Hausdorff distance 6( C,, J ) converges to zero as n-+co. This result follows from theorem 3.8 of ref. [ 141 and we recall that the Hausdorff distance 6(A, B) between two sets A, B in the metric space with distance d( , ) is defined as [ 141
P(L,Plog
(8)
6(A, B) =sup inf d(x, y) +sup
(9)
(2 ) The pressure for L(,, times 1/n converges the pressure for T
L;’ (x)=ah+tkjlkx, QZ‘tl,
XE[O,l],
Lk’([O,
l])EIk,
JL’(x)~)=log(~;~+...+&~).
In particular
the theoretical
p-r=-log(A,
+...+A,).
(7)
escape is given by
aeA
.ytB
atB
inf d(x, y)
.
(15)
VEA
to
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128, number 6,7
=P(T.Plog
Ir(x)o
PHYSICS
1
(16)
where ~IE~Rand the convergence is uniform with respect to j3 on any compact subset of [R.The proof will be given in a forthcoming paper [ 18 1. Since the escape rate is related to the pressure by eq. ( 1) it follows that 1/n times the escape rate from C, converges to the escape rate from J, lim -I,,) n .Z,n
=p7 .
(17)
From the Bowen-Ruelle formula P(T, - DH log I T’ (x) I ) = 0 it follows that the Hausdorff dimension of C,? converges to the Hausdorff dimension of J, limD2)
,1-x2
=DH .
(18)
Inspecting eqs. (8 ) and ( 11) for piecewise linear maps L(x) with s scales, we see that the abscissa of divergence cy with respect to the balanced measure is the unique real positive root of the equation
LETTERS
IL’(x)I)=10g(s2)=2P(L,0).
(19)
This result gives a dynamical meaning to the abscissa of divergence and is formally similar to the Bowen-Ruelle formula quoted above. ( 3 ) The abscissa of divergence cy, of the map L,,) converges to a limit iir, limo(“)-fi n-ax
9
(23) where h(,uB) is the Kolmogorov entropy. The convergence of the information dimension is then an obvious consequence of ( 23 ). We have applied the above results to the quadratic map x’ = T(x; p) =x2-p,
(21)
Before analyzing the numerical results we quote the last convergence result. (4) For the balanced measure the Lyapunov exponent of L,,, times 1/n converges to the Lyapunov exponent of T, 346
p>2,
(24)
for which numerical non-rigorous results are available and rigorous theoretical bounds have been derived for the Hausdorffdimension. For the map (24) the interval [ 0, 1 ] is replaced by the interval [ -4, q] where q= (1 +dG)/2. The bounds to the Hausdorff dimension read [ 2 1,221
bJz=J
(
2 log 2 d
log 2 Y’
-’
where q’ = ,/p - q and the upper bound is nontrival for p>2.72. Another rigorous upper bound which applies to the above systems [22] is
(20)
where the pressure for the map T(x) takes also the value 2P( T, 0). However a can only be proved to be an upper bound to the divergence absissa a!(~~) of T and a lower bound to the information dimension D, (,uB) with respect to the balanced measure [ 191, a(fi,)~ol~D,(~~g).
1988
The information dimension [ 191 with respect to the balanced measure can be computed using the relation [20]
1+
P(L,alog
I I Apr11
.4
DH<
log 2 pT.+log 2
(26)
and can be used to check the numerical results. The observed convergence is rather fast and its rate increases as we increase p. Also for p close to 2 we can remarkably improve the convergence if we use the Thiele continued fraction as extrapolation algorithm, see appendix 3 of ref. [ 23 1. For p> 10 the extrapolated results exhibit a five digit accuracy using only the first three iterations of T, a computation that can be made with a pocket calculator. In table 1 we quote for p= 1 the sequences n -‘pp), Dg), a(“) and n -‘A(“) and the corresponding extrapo-
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128, number
6,7
PHYSICS
Table 1 For p= 3 and 1 G n < 12 we quote the linear cantorian abscissa of divergence and their Thiele extrapolations.
LETTERS
approximations
I 1 April 1988
A
to the Lyapunov
exponent,
n
,,t<“Jn-’
A’“, &,,n-’
p’“‘n-I
plZ’!,,n-’
DP;’
DE%,
1 2 3 4 5 6 I 8 9 10 11 12
1.1435 1.1431 1.1395 1.1372 1.1358 1.1348 1.1341 1.1336 1.1332 1.1329 1.1326 1.1324
1.1435 1.1426 1.1441 1.1286 1.1306 1.1301 1.1299 1.1300 1.1300 1.1300 1.1300 1.1300
0.45037 0.42672 0.42012 0.4 1680 0.41481 0.41348 0.41252 0.41181 0.41126 0.41081 0.41045 0.41015
0.45037 0.40308 0.40840 0.40682 0.40676 0.4068 1 0.40682 0.40683 0.40682 0.40682 0.40682 0.40682
0.60616 0.61412 0.61751 0.61925 0.6203 1 0.62101 0.62152 0.62190 0.66219 0.62243 0.62262 0.62279
0.60616 0.62209 0.62595 0.62458 0.62459 0.62458 0.62456 0.62457 0.62457 0.62458 0.62457 0.62456
Table 2 For three values of p, we give the Hausdorff dimension after n = 11and n = 12 backward the lower bound (D; ) and upper bounds (Dg ) given by eq. (25) and (26).
3 10 30
lated sequences DAi” Dj,l”
0.62262 0.38198 0.29069
0.62457 0.38223 0.29074
0.62279 0.3820 0.29069
for 1~ n < 12. In table 2 we compare and the extrapolated values Dk’l’iext), DA”‘” (ext) with the bounds given by (25 ) and (26 ). Finally for a comparison with refs. [7,24] we quote @t-‘2) (ext) for cr=2.4725, 3.75, 20, 48.75, 90, which reads kp$“) (ext) =0.263796, 0.552765, 1.47904, 1.93565, 2.24574. In fig. 3 we show DA12’ (ext), &pf”) (ext) and &4(12)(ext) for 2qp~4.5 and we see that the limits agree with 1, 0 and log 2 respectively which are the values for the Julia set with p= 2. This set is no-longer a Cantor but the closed interval [ - 2, 2 1. In the same limit a behavior like (p- 2) “’ of pT and 1 -DH is observed. For p> 2 the curves are smooth and suggest a regular dependence on p of the relevant dynamical and fractal variables. The divergence abscissa has a poorer convergence to 1 when p-t2 which can be explained as follows. The pressure we denote P(p; p) =P( T, /3 log 1T’ (x; p) ) ) has forp-t2 a limit which can be analytically computed [ 25 ] and reads P(/3;2)=2/?log2for/3>1,P(~;2)=(1+/Qlog2for p< 1. The divergence abscissa cr is a solution of the
0.62456 0.38223 0.29074
0.60744 0.38125 0.29059
iterations,
escape rate, Hausdorff
aO’l 0.60616 0.59904 0.59847 0.59867 0.59888 0.59904 0.59915 0.59923 0.59930 0.59935 0.59939 0.59943
dimension,
ffr:xL 0.60616 0.59193 0.59779 0.60196 0.59996 0.59964 0.59978 0.59986 0.59982 0.59982 0.59982 0.59982
with their Thiele extrapolations
0.8707 0.41841 0.30224
and
0.63015 0.38370 0.29109
equation P( cu; 2) = 2 log 2 exactly at the point where the derivative of the pressure is discontinuous. Our sequence of approximations provides a limit curve for the pressure P(P; 2) where the edge at j?= 1 is rounded off and precisely at that point the convergence will be the worst.
2
IP 4.5
Fig. 3. We show the dependence on p of the escape rate (line a), Hausdorff dimension (line b) and Lyapunov exponent (line c) for 2-cpS4.5.
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Volume 128, number
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PHYSICS
The present method can be extended to conformal expanding maps of the plane with a disconnected repeller. For example if the invariant set J is included in a closed disc D with the same diameter (this is the case for the Julia sets), then the scales of the approximate linear Cantors are again given by eq. ( 14) where I; denote the preimages by the map T” of the disk D. To conclude we notice that a procedure to determine the Hausdorff dimension of an attractor based on the generation of resealed Cantor sets was also recently developed in ref. [ 26 1.
References [ 1 ] D. Ruelle, Ergod. Theory Dynam. Syst. 2 ( 1982) 99. [ 21 H. Kantr and P. Grassberger, Physica D 17 ( 1985) 75. [ 3) S. Takesue, Prog. Theor. Phys. 76 ( 1986) 11. [4] P. Szepfalusy and T. Tel, Phys. Rev. A 34 (1986) 2520. [ 5 ] S. McDonald, C. Grebogi, E. Ott and J.A. Yorke, Physica D 17 (1985) 125. [ 61 L.P. Kadanoff and C. Tang, Proc. Natl. Acad. Sci. USA 8 1 (1984) 1276. [ 71 M. Widom, D. Bensimon, L.P. Kadanoff and S.J. Shenker, J. Stat. Phys. 32, (1983) 443. [ 81 D. Bessis, G. Servizi, G. Turchetti and S. Vaienti, Phys. Lett. A 119 (1987) 345.
348
LETTERS
A
I 1April 1988
[ 91 S. Pelikan. Trans. Am. Math. Sot. 292 ( 1985 ) 695. [ lo] P. Grassberger and I. Procaccia, Physica D 9 ( 1983) 189. [ 111 H. Brolin, Ark. Math. 6 (1965) 103. [ 121 D. Ruelle, Thermodynamic formalism (Addison-Wesley, Reading, I978 ). [ 131 J.P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57 ( I985 ) 617. [ 141 K.J. Falconer, The geometry of fractal sets (Cambrrdge Univ. Press, Cambridge, 1985 ) [ 151 D. Bessis. J.D. Fournier. G. Servizr, Ci. Turchetti and S. Vaienti. Phys. Rev. A 36 (1987) 920. [ 161 M.Ju. Ljubich, Ergod. Theory Dynam. Syst. 3 ( 1983) 35 1. [ 171 M.F. Barnsley and S. Demko. Proc. R. Sot. A 399 ( 1985) 243. [ 181 S. Vaienti, Some properties of mixing repellers, to be published in J. Phys. A. [ 191 L.S. Young, Ergod. Theory Dynam. Syst. 2 ( I982 ) 109. [ 201 F. Ledrappier and M. Misiurewiez, Ergod. Theory Dynam. Syst. 5 (1985) 595. [ 211 T.S. Pitcher and J.R. Kinney, Ark. Mat. 8 ( 1968) 25. [22] S. Vaienti. Nuovo Cimento B 99 (1987) 77. [ 231 W. Van Assche, G. Turchetti and D. Bessis, J. Stat. Phys. 43 (1986) 85. [ 24 ] T. Tel, preprint ICTP, Trieste ( 1986). [25] D. Bessis, G. Paladin. G. Turchetti and S. Vaientr, Gcneralized dimensions, entropies and Lyapunov exponents from the pressure for strange sets, to be published m J. Stat. Phys. [26] K.J. Tsang, Phys. Rev. Lett. 57 (1986) 1390.