The kneading theory for unimodal mappings is developed in sect. A14.1. The

Transcription

1 Aendix A14 Symbolic dynamics techniques The kneading theory for unimodal maings is develoed in sect. A14.1. The rime factorization for dynamical itineraries of sect. A14.2 illustrates the sense in which rime cycles are rime - the roduct structure of zeta functions is a consequence of the unique factorization roerty of symbol sequences. A14.1 Toological zeta functions for infinite subshifts (P. Dahlqvist) The transition grah methods outlined in chater 14 are well suited for symbolic dynamics of finite subshift tye. A sequence of well defined rules leads to the answer, the toological zeta function, which turns out to be a olynomial. For infinite subshifts one would have to go through an infinite sequence of grah constructions and it is of course very difficult to make any asymtotic statements about the outcome. Luckily, for some simle systems the goal can be reached by much simler means. This is the case for unimodal mas. We will restrict our attention to the toological zeta function for unimodal mas with one external arameter f Λ (x)=λg(x). As usual, symbolic dynamics is introduced by maing a time series... x i 1 x i x i+1... onto a sequence of symbols... s i 1 s i s i+1... where s i = 0 x i < x c s i = C x i = x c s i = 1 x i > x c (A14.1) and x c is the critical oint of the ma (i.e., maximum of g). In addition to the usual binary alhabet we have added a symbol C for the critical oint. The kneading 897

2 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 898 I(C) ζto 1 (z)/(1 z) 1C 101C C H (1) n=0 (1 z 2n ) 10111C C 101 (1 2z 2 )/(1+z) C C 1011C C 10C (1 z z 2 ) 10010C C I(C) ζto(z)/(1 z) C C 10011C C 100C C 10001C C 1000C C 10000C C 10 (1 2z)/(1 z) Table A14.1: All ordered kneading sequences u to length seven, as well as some longer kneading sequences. Harmonic extension H (1) is defined below. sequence K Λ is the itinerary of the critical oint (14.6). The crucial observation is that no eriodic orbit can have a toological coordinate (see sect. A14.1.1) beyond that of the kneading sequence. The kneading sequence thus inserts a border in the list of eriodic orbits (ordered according to maximal toological coordinate), cycles u to this limit are allowed, all beyond are runed. All unimodal mas (obeying some further constraints) with the same kneading sequence thus have the same set of eriodic orbitsand the same toological zeta function. The toological coordinate of the kneading sequence increases with increasing Λ. The kneading sequence can be of one of three tyes 1. It mas to the critical oint again, after n iterations. If so, we adot the convention to terminate the kneading sequence with a C, and refer to the kneading sequence as finite. 2. Preeriodic, i.e., it is infinite but with a eriodic tail. 3. Aeriodic. As an archetye unimodal ma we will choose the tent ma x f (x)= { Λx x [0, 1/2] Λ(1 x) x (1/2, 1], (A14.2) where the arameterλ (1, 2]. The toological entroy is h = logλ. This follows from the fact any trajectory of the ma is bounded, the escae rate is strictly zero, and so the dynamical zeta function ( ) 1/ζ(z) = 1 zn Λ has its leading zero at z = 1. = ( ( z n ) 1 = 1/ζ to (z/λ) Λ) chater/dahlqvist.tex 30nov2001 ChaosBook.org version15.9, Jun

3 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 899 The set of eriodic oints of the tent ma is countable. A consequence of this fact is that the set of arameter values for which the kneading sequence (14.6) is eriodic or reeriodic are countable and thus of measure zero and consequently the kneading sequence is aeriodic for almost all Λ. For general unimodal mas the corresonding statement is that the kneading sequence is aeriodic for almost all toological entroies. For a given eriodic kneading sequence of eriod n, K Λ = PC= s 1 s 2... s n 1 C there is a simle exansion for the toological zeta function. Then the exanded zeta function is a olynomial of degree n and a 0 = 1. n 1 1/ζ to (z)= (1 z n )=(1 z) a i z i, a i = i=0 i ( 1) s j j=1 (A14.3) Aeriodic and reeriodic kneading sequences are accounted for by simly relacing n by. Examle. Consider as an examle the kneading sequence K Λ = 10C. From (A14.3) we get the toological zeta function 1/ζ to (z)=(1 z)(1 z z 2 ), see table A14.1. This can also be realized by redefining the alhabet. The only forbidden subsequence is 100. All allowed eriodic orbits, excet 0, can can be built from a alhabet with letters 10 and 1. We write this alhabet as{10, 1; 0}, yielding the toological zeta function 1/ζ to (z)=(1 z)(1 z z 2 ). The leading zero is the inverse golden mean z 0 = ( 5 1)/2. Examle. As another examle we consider the reeriodic kneading sequence K Λ = 101. From (A14.3) we get the toological zeta function 1/ζ to (z)= (1 z)(1 2z 2 )/(1+z), see table A14.1. This can again be realized by redefining the alhabet. There are now an infinite number of forbidden subsequences, namely 101 2n 0 where n 0. These runing rules are resected by the alhabet {01 2n+1 ; 1, 0}, yielding the toological zeta function above. The ole in the zeta functionζto 1 (z) is a consequence of the infinite alhabet. An imortant consequence of (A14.3) is that the sequence{a i } has a eriodic tail if and only if the kneading sequence has one (however, their eriod may differ by a factor of two). We know already that the kneading sequence is aeriodic for almost allλ. The analytic structure of the function reresented by the infinite series a i z i with unity as radius of convergence, deends on whether the tail of{a i } is eriodic or not. If the eriod of the tail is N we can write 1/ζ to (z)= (z)+q(z)(1+ z N + z 2N...)= (z)+ q(z) 1 z N, for some olynomials (z) and q(z). The result is a set of oles sread out along the unit circle. This alies to the reeriodic case. An aeriodic sequence of chater/dahlqvist.tex 30nov2001 ChaosBook.org version15.9, Jun

4 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 900 coefficients would formally corresond to infinite N and it is natural to assume that the singularities will fill the unit circle. There is indeed a theorem ensuring that this is the case [15.58], rovided the a i s can only take on a finite number of values. The unit circle becomes a natural boundary, already aarent in a finite olynomial aroximations to the toological zeta function, as in figure A function with a natural boundary lacks an analytic continuation outside it. To conclude: The toological zeta function 1/ζ to for unimodal mas has the unit circle as a natural boundary for almost all toological entroies and for the tent ma (A14.2), for almost allλ. Let us now focus on the relation between the analytic structure of the toological zeta function and the number of eriodic orbits, or rather (18.6), the number N n of fixed oints of f n (x). The trace formula is (see sect. 18.4) N n = tr T n = 1 2πi γ r dz z n d dz logζ 1 to whereγ r is a (circular) contour encircling the origin z=0in clockwise direction. Residue calculus turns this into a sum over zeros z 0 and oles z ofζ 1 to N n = z 0 :r< z 0 <R z n 0 z :r< z <R z n dz z n d 2πi γ R dz logζ 1 to and a contribution from a large circleγ R. For meromorhic toological zeta functions one may let R with vanishing contribution fromγ R, and N n will be a sum of exonentials. The leading zero is associated with the toological entroy, as discussed in chater 18. We have also seen that for reeriodic kneading there will be oles on the unit circle. To areciate the role of natural boundaries we will consider a (very) secial examle. Cascades of eriod doublings is a central concet for the descrition of unimodal mas. This motivates a close study of the function Ξ(z)= (1 z 2n ). n=0 (A14.4) This function will aear again when we derive (A14.3). The exansion ofξ(z) begins asξ(z)=1 z z 2 + z 3 z 4 + z 5... The radius of convergence is obviously unity. The simle rule governing the exansion will effectively rohibit any eriodicity among the coefficients making the unit circle a natural boundary. It is easy to see thatξ(z)=0 if z=ex(2πm/2 n ) for any integer m and n. (Strictly seaking we mean thatξ(z) 0 when z ex(2πm/2 n ) from inside). chater/dahlqvist.tex 30nov2001 ChaosBook.org version15.9, Jun

5 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 901 Consequently, zeros are dense on the unit circle. One can also show that singular oints are dense on the unit circle, for instance Ξ(z) when z ex(2πm/3 n ) for any integer m and n. As an examle, the toological zeta function at the accumulation oint of the first Feigenbaum cascade isζto 1 (z)=(1 z)ξ(z). Then N n = 2 l+1 if n= 2 l, otherwise N n = 0. The growth rate in the number of cycles is anything but exonential. It is clear that N n cannot be a sum of exonentials, the contourγ R cannot be ushed away to infinity, R is restricted to R 1 and N n is entirely determined by which icks u its contribution from the natural boundary. γ R We have so far studied the analytic structure for some secial cases and we know that the unit circle is a natural boundary for almost allλ. But how does it look out there in the comlex lane for some tyical arameter values? To exlore that we will imagine a journey from the origin z=0out towards the unit circle. While traveling we let the arameterλchange slowly. The tri will have a distinct science fiction flavor. The first zero we encounter is the one connected to the toological entroy. Obviously it moves smoothly and slowly. When we move outward to the unit circle we encounter zeros in increasing densities. The closer to the unit circle they are, the wilder and stranger they move. They move from and back to the horizon, where they are created and destroyed through bizarre bifurcations. For some secial values of the arameter the unit circle suddenly gets transarent and and we get (infinitely) short glimses of another world beyond the horizon. We end this section by deriving eqs (A14.5) and (A14.6). The imenetrable rose is hoefully exlained by the accomanying tables. We know one thing from chater 14, namely for that finite kneading sequence of length n the toological olynomial is of degree n. The grah contains a node which is connected to itself only via the symbol 0. This imlies that a factor (1 z) may be factored out andζ to (z)=(1 z) n 1 i=0 a iz i. The roblem is to find the coefficients a i. eriodic orbits P1=A (P) P0 P0P1 H (P) finite kneading sequences PC P0PC P0P1P0PC H (P) Table A14.2: Relation between eriodic orbits and finite kneading sequences in a harmonic cascade. The string P is assumed to contain an odd number of 1 s. The ordered list of (finite) kneading sequences table A14.1 and the ordered list of eriodic orbits (on maximal form) are intimately related. In table A14.2 we indicate how they are nested during a eriod doubling cascade. Every finite chater/dahlqvist.tex 30nov2001 ChaosBook.org version15.9, Jun

6 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 902 kneading sequence PC is bracketed by two eriodic orbits, P1 and P0. We have P1<PC<P0 if P contains an odd number of 1 s, and P0<PC<P1 otherwise. From now on we will assume that P contains an odd number of 1 s. The other case can be worked out in comlete analogy. The first and second harmonic of PC are dislayed in table A14.2. The eriodic orbit P1 (and the corresonding infinite kneading sequence) is sometimes referred to as the antiharmonic extension of PC (denoted A (P)) and the accumulation oint of the cascade is called the harmonic extension of PC [A1.8] (denoted H (P)). A central result is the fact that a eriod doubling cascade of PC is not interfered by any other sequence. Another way to exress this is that a kneading sequence PC and its harmonic are adjacent in the list of kneading sequences to any order. I(C) ζto 1 P 1 = 100C 1 z z 2 z 3 H (P 1 ) = z z 2 z 3 z 4 + z 5 + z 6 + z 7 z 8... P = 10001C 1 z z 2 z 3 z 4 + z 5 A (P 2 ) = z z 2 z 3 z 4 + z 5 z 6 z 7 z 8... P 2 = 1000C 1 z z 2 z 3 z 4 Table A14.3: Examle of a ste in the iterative construction of the list of kneading sequences PC. Table A14.3 illustrates another central result in the combinatorics of kneading sequences. We suose that P 1 C and P 2 C are neighbors in the list of order 5 (meaning that the shortest finite kneading sequence P C between P 1 C and P 2 C is longer than 5.) The imortant result is that P (of length n = 6) has to coincide with the first n 1 letters of both H (P 1 ) and A (P 2 ). This is exemlified in the left column of table A14.3. This fact makes it ossible to generate the list of kneading sequences in an iterative way. The zeta function at the accumulation oint H (P 1 ) is ζ 1 P 1 (z)ξ(z n 1 ), and just before A (P 2 ) ζ 1 P 2 (z)/(1 z n 2 ). (A14.5) (A14.6) A short calculation shows that this is exactly what one would obtain by alying (A14.3) to the antiharmonic and harmonic extensions directly, rovided that it alies toζp 1 1 (z) andζp 1 2 (z). This is the key observation. Recall now the roduct reresentation of the zeta functionζ 1 = (1 z n ). We will now make use of the fact that the zeta function associated with P C is a olynomial of order n. There is no eriodic orbit of length shorter than n + 1 between H (P 1 ) and A (P 2 ). It thus follows that the coefficients of this olynomial coincides with those of (A14.5) and (A14.6), see Table A14.3. We can thus conclude that our rule can be alied directly to P C. This can be used as an induction ste in roving that the rule can be alied to every finite and infinite kneading sequences. chater/dahlqvist.tex 30nov2001 ChaosBook.org version15.9, Jun

7 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 903 Remark A14.1 How to rove things. The exlicit relation between the kneading sequence and the coefficients of the toological zeta function is not commonly seen in the literature. The result can roven by combining some theorems of Milnor and Thurston [A1.9]. That aroach is hardly instructive in the resent context. Our derivation was insired by Metroolis, Stein and Stein classical aer [A1.8]. For further details, consult ref. [18.14]. A Periodic orbits of unimodal mas A eriodic oint (cycle oint) x k belonging to a cycle of eriod n is a real solution of f n (x k )= f ( f (... f (x k )...))= x k, k=0, 1, 2,...,n 1. (A14.7) The nth iterate of a unimodal ma has at most 2 n monotone segments, and therefore there will be 2 n or fewer eriodic oints of length n. Similarly, the backward and the forward Smale horseshoes intersect at most 2 n times, and therefore there will be 2 n or fewer eriodic oints of length n. A eriodic orbit of length n corresonds to an infinite reetition of a length n=n symbol string, customarily indicated by a line over the string: S = (s 1 s 2 s 3... s n ) = s 1 s 2 s 3... s n. As all itineraries are infinite, we shall adot convention that a finite string itinerary S = s 1 s 2 s 3... s n stands for infinite reetition of a finite block, and routinely omit the overline. x 0, its cyclic ermutation s k s k+1... s n s 1... s k 1 corresonds to the oint x k 1 in the same cycle. A cycle is called rime if its itinerary S cannot be written as a reetition of a shorter block S. Each cycle is a set of n rational-valued full tent ma eriodic ointsγ. It follows from (14.4) that if the reeating string s 1 s 2... s n contains an odd number 1 s, the string of well ordered symbols w 1 w 2...w 2n has to be of the double length before it reeats itself. The cycle-oint γ is a geometrical sum which we can rewrite as the fraction γ(s 1 s 2... s n )= 22n 2 2n 1 2n t=1 w t /2 t (A14.8) Using this we can calculate the ˆγ(S ) for all short cycles. For orbits u to length 5 this is done in table Here we give exlicit formulas for the toological coordinate of a eriodic oint, given its itinerary. For the urose of what follows it is convenient to comactify the itineraries by relacing the binary alhabet s i ={0, 1} by the infinite alhabet {a 1, a 2, a 3, a 4, ; 0}={1, 10, 100, 1000,... ; 0}. (A14.9) chater/aendsymb.tex 23mar98 ChaosBook.org version15.9, Jun

8 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 904 In this notation the itinerary S = a i a j a k a l and the corresonding toological coordinate (14.4) are related byγ(s )=.1 i 0 j 1 k 0 l. For examle: S = = a 1 a 1 a 2 a 1 a 1 a 2 a 3 a 4... γ(s ) = = Cycle oints whose itineraries start with w 1 = w 2 = =w i = 0, w i+1 = 1 remain on the left branch of the tent ma for i iterations, and satisfyγ(0...0s )=γ(s )/2 i. Periodic oints corresond to rational values of γ, but we have to distinguish even and odd cycles. The even (odd) cycles contain even (odd) number of a i in the reeating block, with eriodic oints given by 2 n 2 γ(a i a j a k a l )= n 1.1i 0 j 1k even 1 ( 1+2 n.1 i 0 j 1l) odd, (A14.10) 2 n +1 where n=i+ j+ +k+l is the cycle eriod. The maximal value eriodic oint is given by the cyclic ermutation of S with the largest a i as the first symbol, followed by the smallest available a j as the next symbol, and so on. For examle: ˆγ(1) = γ(a 1 ) = =.10 = 2/3 ˆγ(10) = γ(a 2 ) = =.1100 = 4/5 ˆγ(100) = γ(a 3 ) = = = 8/9 ˆγ(101) = γ(a 2 a 1 ) = =.110 = 6/7 An examle of a cycle where only the third symbol determines the maximal value eriodic oint is ˆγ( )=γ(a 2 a 1 a 2 a 1 a 1 )= = 100/129. Maximal values of all cycles u to length 5 are given in table!? A14.2 Prime factorization for dynamical itineraries The Möbius function is not only a number-theoretic function, but can be used to maniulate ordered sets of noncommuting objects such as symbol strings. Let P={ 1, 2, 3, } be an ordered set of rime strings, and { } N={n}= k 1 1 k 2 2 k 3 3 k j j, j N, k i Z +, be the set of all strings n obtained by the ordered concatenation of the rimes i. By construction, every string n has a unique rime factorization. We say that a string has a divisor d if it contains d as a substring, and define the string division n/d as n with the substring d deleted. Now we can do things like this: defining t n := t k 1 1 t k 2 2 t k j j we can write the inverse dynamical zeta function (23.3) as (1 t )= µ(n)t n, n chater/aendsymb.tex 23mar98 ChaosBook.org version15.9, Jun

9 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 905 factors string factors string factors string factors string Table A14.4: Factorization of all eriodic oints strings u to length 5 into ordered concatenations k 1 1 k 2 2 k n n of rime strings 1 = 0, 2 = 1, 3 = 10, 4 = 100,..., 14 = and, if we care (we do in the case of the Riemann zeta function), the dynamical zeta function as. 1 = t n (A14.11) 1 t n A striking asect of this formula is its resemblance to the factorization of natural numbers into rimes: the relation of the cycle exansion (A14.11) to the roduct over rime cycles is analogous to the Riemann zeta (exercise 22.9) reresented as a sum over natural numbers vs. its Euler roduct reresentation. We now imlement this factorization exlicitly by decomosing recursively binary strings into ordered concatenations of rime strings. There are 2 strings of length 1, both rime: 1 = 0, 2 = 1. There are 4 strings of length 2: 00, 01, 11, 10. The first three are ordered concatenations of rimes: 00= 2 1, 01= 1 2, 11= 2 2 ; by ordered concatenations we mean that 1 2 is legal, but 2 1 is not. The remaining string is the only rime of length 2, 3 = 10. Proceeding by discarding the strings which are concatenations of shorter rimes k 1 1 k 2 2 k j j, with rimes lexically ordered, we generate the standard list of rimes, in agreement with table 18.1: 0, 1, 10, 101, 100, 1000, 1001, 1011, 10000, 10001, 10010, 10011, 10110, 10111, , , , , , , , , ,... This factorization is illustrated in table A14.4. A Prime factorization for sectral determinants Following sect. A14.2, the sectral determinant cycle exansions is obchater/aendsymb.tex 23mar98 ChaosBook.org version15.9, Jun

10 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 906 tained by exanding F as a multinomial in rime cycle weights t F= C kt k = k=0 τ k 1 1 k 2 2 k 3 3 = i=1 k 1 k 2 k 3 =0 τ k 1 1 k 2 2 k 3 3 (A14.12) where the sum goes over all seudo-cycles. In the above we have defined C i k i t k i i. (A14.13) A striking asect of the sectral determinant cycle exansion is its resemblance to the factorization of natural numbers into rimes: as we already noted in sect. A14.2, the relation of the cycle exansion (A14.12) to the roduct formula (22.8) is analogous to the Riemann zeta reresented as a sum over natural numbers vs. its Euler roduct reresentation. exercise 22.9 This is somewhat unexected, as the cycle weights factorize exactly with resect to r reetitions of a rime cycle, t... = t r, but only aroximately (shadowing) with resect to subdividing a string into rime substrings, t 1 2 t 1 t 2. The coefficients C k have a simle form only in 1-dimensional given by the Euler formula (28.5). In higher dimensions C k can be evaluated by exanding (22.8), F(z)= F, where t r F = 1 rd + 1 t r 2 r=1,r 2 rd... r=1,r Exanding and recollecting terms, and suressing the cycle label for the moment, we obtain F = D k = C k t k, C k = ( ) k c k /D k, r=1 k d r = r=1 a=1 r=1 d k (1 u r a) (A14.14) where evaluation of c k requires a certain amount of not too luminous algebra: c 0 = 1 c 1 = 1 c 2 = 1 ( ) d2 d 1 = 1 d 2 d 1 2 (1+u a ) a=1 c 3 = 1 d 2d 3 3! d d 1 d 2 3d 3 = 1 d 6 (1+2u a + 2u 2 a + u3 a ) +2 a=1 d (1 u a ) a=1 d d (1 u a u 2 a + u3 a ) 3 (1 u 3 a ) a=1 a=1 chater/aendsymb.tex 23mar98 ChaosBook.org version15.9, Jun

11 APPENDIX A14. SYMBOLIC DYNAMICS TECHNIQUES 907 etc.. For examle, for a general 2-dimensional ma we have F = 1 1 t+ u 1+ u 2 t 2 u 1u 2 (1+u 1 )(1+u 2 )+u u3 2 t (A14.15) D 1 D 2 We discuss the convergence of such cycle exansions in sect. A20.4. Withτ k 1 1 k 2 defined as above, the rime factorization of symbol strings is 2 kn n unique in the sense that each symbol string can be written as a unique concatenation of rime strings, u to a convention on ordering of rimes. This factorization is a nontrivial examle of the utility of generalized Möbius inversion, sect. A14.2. How is the factorization of sect. A14.2 used in ractice? Suose we have comuted (or erhas even measured in an exeriment) all rime cycles u to length n, i.e., we have a list of t s and the corresonding Jacobian matrix eigenvaluesλ,1,λ,2,...λ,d. A cycle exansion of the Selberg roduct is obtained by generating all strings in order of increasing length j allowed by the symbolic dynamics and constructing the multinomial F= (A14.16) n τ n where n= s 1 s 2 s j, s i range over the alhabet, in the resent case{0, 1}. Factorizing every string n= s 1 s 2 s j = k 1 1 k 2 2 k j j as in table A14.4, and substitutingτ k 1 1 k 2 we obtain a multinomial aroximation to F. For examle, 2 τ =τ =τ 001 2τ 01 3, andτ 01 3,τ are known functions of the corresonding cycle eigenvalues. The zeros of F can now be easily determined by standard numerical methods. The fact that as far as the symbolic dynamics is concerned, the cycle exansion of a Selberg roduct is simly an average over all symbolic strings makes Selberg roducts rather retty. To be more exlicit, we illustrate the above by exressing binary strings as concatenations of rime factors. We start by comuting N n, the number of terms in the exansion (A14.12) of the total cycle length n. Setting C kt k = z n k in (A14.12), we obtain N n z n = z nk = n=0 k=0 1 (1 z n ). So the generating function for the number of terms in the Selberg roduct is the toological zeta function. For the comlete binary dynamics we have N n = 2 n contributing terms of length n: D 3 ζ to = 1 (1 z n ) = 1 1 2z = 2 n z n n=0 Hence the number of distinct terms in the exansion (A14.12) is the same as the number of binary strings, and conversely, the set of binary strings of length n suffices to label all terms of the total cycle length n in the exansion (A14.12). chater/aendsymb.tex 23mar98 ChaosBook.org version15.9, Jun