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Cycle expansions
18.1 Pseudocycles and shadowing 267 18.2 Construction of cycle expansions 269 18.3 Cycle formulas for dynamical aver- ages 274 18.4 Cycle expansions for finite alpha- bets 277 18.5 Stability ordering of cycle expan- sions 278 18.6 Dirichlet series 282 Summary 283 Further reading 284 Exercises 285 References 287
Recycle... It’s the Law! Poster, New York City Department of Sanitation
The Euler product representations of spectral determinants (17.9) and dynamical zeta functions (17.15) are really only a shorthand notation
- the zeros of the individual factors are not the zeros of the zeta func- tion, and convergence of such objects is far from obvious. Now we shall give meaning to the dynamical zeta functions and spectral deter- minants by expanding them as cycle expansions, series representations ordered by increasing topological cycle length, with products in (17.9), (17.15) expanded as sums over pseudocycles , products of tp’s. The zeros of correctly truncated cycle expansions yield the desired eigenvalues, and the expectation values of observables are given by the cycle aver- aging formulas obtained from the partial derivatives of dynamical zeta functions (or spectral determinants).
18.1 Pseudocycles and shadowing
How are periodic orbit formulas such as (17.15) evaluated? We start by computing the lengths and stability eigenvalues of the shortest cycles. This always requires numerical work, such as the Newton’s method searches for periodic solutions; we shall assume that the numerics is under control, and that all short cycles up to a given (topological) length have been found. Examples of the data required for application of peri- odic orbit formulas are the lists of cycles given in Tables 20.3 and ??. It is important not to miss any short cycles, as the calculation is as accurate as the shortest cycle dropped - including cycles longer than the short- est omitted does not improve the accuracy (more precisely, improves it, but painfully slowly). Expand the dynamical zeta function (17.15) as a formal power series,
1 /ζ =
p
(1 − tp) = 1 −
{p 1 p 2 ...pk }
(−1)k+1tp 1 tp 2... tpk (18.1)
where the prime on the sum indicates that the sum is over all distinct non-repeating combinations of prime cycles. As we shall frequently use such sums, let us denote by tπ = (−1)k+1tp 1 tp 2... tpk an element of the set of all distinct products of the prime cycle weights tp. The formal
268 CHAPTER 18. CYCLE EXPANSIONS
power series (18.1) is now compactly written as
1 /ζ = 1 −
π
tπ. (18.2)
For k > 1 , tπ are weights of pseudocycles ; they are sequences of shorter cycles that shadow a cycle with the symbol sequence p 1 p 2... pk along segments p 1 , p 2 ,.. ., pk.
denotes the restricted sum, for which any given prime cycle p contributes at most once to a given pseudocycle weight tπ. The pseudocycle weight, i.e., the product of weights (17.10) of prime cycles comprising the pseudocycle,
tπ = (−1)k+^
|Λπ | eβAπ^ −sTπ^ znπ^ , (18.3)
depends on the pseudocycle topological length nπ , integrated observ- able Aπ , period Tπ , and stability Λπ
nπ = np 1 +... + npk , Tπ = Tp 1 +... + Tpk Aπ = Ap 1 +... + Apk , Λπ = Λp 1 Λp 2 · · · Λpk. (18.4)
Throughout this text, the terms “periodic orbit” and “cycle” are used interchangeably; while “periodic orbit” is more precise, “cycle” (which has many other uses in mathematics) is easier on the ear than “pseudo- periodic-orbit.” While in Soviet times acronyms were a rage (and in France they remain so), we shy away from acronyms such as UPOs (Unstable Periodic Orbits).
18.1.1 Curvature expansions
The simplest example is the pseudocycle sum for a system described by a complete binary symbolic dynamics. In this case the Euler product (17.15) is given by
1 /ζ = (1 − t 0 )(1 − t 1 )(1 − t 01 )(1 − t 001 )(1 − t 011 ) (18.5) (1 − t 0001 )(1 − t 0011 )(1 − t 0111 )(1 − t 00001 )(1 − t 00011 ) (1 − t 00101 )(1 − t 00111 )(1 − t 01011 )(1 − t 01111 )...
(see Table 10.1), and the first few terms of the expansion (18.2) ordered by increasing total pseudocycle length are:
1 /ζ = 1 − t 0 − t 1 − t 01 − t 001 − t 011 − t 0001 − t 0011 − t 0111 −... +t 0 t 1 + t 0 t 01 + t 01 t 1 + t 0 t 001 + t 0 t 011 + t 001 t 1 + t 011 t 1 −t 0 t 01 t 1 −... (18.6)
We refer to such series representation of a dynamical zeta function or a spectral determinant, expanded as a sum over pseudocycles, and or- dered by increasing cycle length and instability, as a cycle expansion. recycle - 30aug2006 ChaosBook.org version11.9.2, Aug 21 2007
270 CHAPTER 18. CYCLE EXPANSIONS
Table 18.1 The binary curvature expansion (18.7) up to length 6 , listed in such way that the sum of terms along the pth horizontal line is the curvature ˆcp associated with a prime cycle p, or a combination of prime cycles such as the
- t 1001 + t 100 t 1 + t 101 t 0 - t 1 t 10 t
- t 10001 + t 1001 t 0 + t 1000 t 1 - t 0 t 100 t
- t 10011 + t 1011 t 0 + t 1001 t 1 - t 0 t 101 t
- t 100001 + t 10001 t 0 + t 10000 t 1 - t 0 t 1000 t
- t 100010 + t 10010 t 0 + t 1000 t 10 - t 0 t 100 t
- t 100011 + t 10011 t 0 + t 10001 t 1 - t 0 t 1001 t
- t 100101 - t 100110 + t 10010 t 1 + t 10110 t
- t 10 t 1001 + t 100 t 101 - t 0 t 10 t 101 - t 1 t 10 t
- t 101110 + t 10110 t 1 + t 1011 t 10 - t 1 t 101 t
- t 100111 + t 10011 t 1 + t 10111 t 0 - t 0 t 1011 t
- recycle - 30aug2006 ChaosBook.org version11.9.2, Aug t 100101 + t 100110 pair.
18.2. CONSTRUCTION OF CYCLE EXPANSIONS 271
length n(i) ≤ n(i+1). The dynamical zeta function 1 /ζN truncated to the np ≤ N cycles is computed recursively, by multiplying
1 /ζ(i) = 1/ζ(i−1)(1 − t(i)zn(i)^ ) , (18.8)
and truncating the expansion at each step to a finite polynomial in zn, n ≤ N. The result is the N th order polynomial approximation
1 /ζN = 1 −
∑^ N
n=
cnzn^. (18.9)
In other words, a cycle expansion is a Taylor expansion in the dummy variable z raised to the topological cycle length. If both the number of cycles and their individual weights grow not faster than exponentially with the cycle length, and we multiply the weight of each cycle p by a factor znp^ , the cycle expansion converges for sufficiently small |z|. If the dynamics is given by iterated mapping, the leading zero of (18.9) as function of z yields the leading eigenvalue of the appropriate evolution operator. For continuous time flows, z is a dummy variable that we set to z = 1, and the leading eigenvalue of the evolution opera- tor is given by the leading zero of (18.9) as function of s.
18.2.2 Evaluation of traces, spectral determinants
Due to the lack of factorization of the full pseudocycle weight,
det ( 1 − Mp 1 p 2 ) = det ( 1 − Mp 1 ) det ( 1 − Mp 2 ) ,
the cycle expansions for the spectral determinant (17.9) are somewhat less transparent than is the case for the dynamical zeta functions. We commence the cycle expansion evaluation of a spectral determin- ant by computing recursively the trace formula (16.10) truncated to all prime cycles p and their repeats such that npr ≤ N :
tr
zL 1 − zL
(i)
= tr
zL 1 − zL
(i−1)
n(i ∑) r≤N
r=
e(β·A(i)−sT(i)^ )r ∣ ∣ ∣
1 − Λr (i),j
zn(i)r
tr
zL 1 − zL
N
∑^ N
n=
Cnzn^ , Cn = tr Ln^. (18.10)
This is done numerically: the periodic orbit data set consists of the list of the cycle periods Tp, the cycle stability eigenvalues Λp, 1 , Λp, 2 ,... , Λp,d, and the cycle averages of the observable Ap for all prime cycles p such that np ≤ N. The coefficient of znpr^ is then evaluated numerically for the given (β, s) parameter values. Now that we have an expansion for the trace formula (16.9) as a power series, we compute the N th order approximation to the spectral determinant (17.3),
det (1 − zL)|N = 1 −
∑^ N
n=
Qnzn^ , Qn = nth cumulant , (18.11)
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18.2. CONSTRUCTION OF CYCLE EXPANSIONS 273
R:a N. det (s − A) 1 /ζ(s) 1 /ζ(s)3-disk
1 0.39 0. 2 0.4105 0.41028 0. 3 0.410338 0.410336 0. 6 4 0.4103384074 0.4103383 0. 5 0.4103384077696 0.4103384 0. 6 0.410338407769346482 0.4103383 0. 7 0.4103384077693464892 0. 8 0. 9 0. 10 0.
1 0. 2 0. 3 0. 4 0. 3 5 0. 6 0. 7 0. 8 0. 9 0. 10 0.
Table 18.2 3-disk repeller escape rates computed from the cycle expansions of the spectral determinant (17.6) and the dynamical zeta function (17.15), as function of the maximal cycle length N. The first column indicates the disk-disk center separation to disk radius ratio R:a, the second column gives the maximal cycle length used, and the third the estimate of the classical escape rate from the fun- damental domain spectral determinant cycle expansion. As for larger disk-disk separations the dynamics is more uniform, the convergence is better for R:a = 6 than for R:a = 3. For comparison, the fourth column lists a few estimates from from the fundamental domain dynamical zeta function cycle expansion (18.7), and the fifth from the full 3-disk cycle expansion (18.36). The convergence of the fundamental domain dynamical zeta function is significantly slower than the convergence of the corresponding spectral determinant, and the full (un- factorized) 3-disk dynamical zeta function has still poorer convergence. (P.E. Rosenqvist.)
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274 CHAPTER 18. CYCLE EXPANSIONS
Fig. 18.1 Examples of the complex s plane scans: contour plots of the logarithm of the absolute values of (a) 1 /ζ(s), (b) spec- tral determinant det (s − A) for the 3- disk system, separation a : R = 6, A 1 subspace are evaluated numerically. The eigenvalues of the evolution operator L are given by the centers of elliptic neigh- borhoods of the rapidly narrowing rings. While the dynamical zeta function is an- alytic on a strip s ≥ − 1 , the spectral determinant is entire and reveals further families of zeros. (P.E. Rosenqvist)
plane under investigation and searches for the zeros of spectral deter- minants, Fig. 18.1, reveal complicated patterns of resonances even for something so simple as the 3-disk game of pinball. With a good starting guess (such as a location of a zero suggested by the complex s scan of Fig. 18.1), a zero 1 /ζ(s) = 0 can now be easily determined by standard numerical methods, such as the iterative Newton algorithm (12.4), with the mth Newton estimate given by
sm+1 = sm −
ζ(sm)
∂s
ζ−^1 (sm)
= sm − 1 /ζ(sm) 〈T〉ζ
The dominator 〈T〉ζ required for the Newton iteration is given below, by the cycle expansion (18.22). We need to evaluate it anyhow, as 〈T〉ζ enters our cycle averaging formulas.
18.3 Cycle formulas for dynamical averages
s
β F( ,s( ))=0 lineβ β
__ds dβ
Fig. 18.2 The eigenvalue condition is sat- isfied on the curve F = 0 the (β, s) plane. The expectation value of the observable (15.12) is given by the slope of the curve.
The eigenvalue condition in any of the three forms that we have given so far - the level sum ( ?? ), the dynamical zeta function (18.2), the spec- tral determinant (18.11):
∑^ (n)
i
ti , ti = ti(β, s(β)) , ni = n , (18.14)
π
tπ , tπ = tπ (z, β, s(β)) (18.15)
∑^ ∞
n=
Qn , Qn = Qn(β, s(β)) , (18.16)
is an implicit equation for the eigenvalue s = s(β) of form F (β, s(β)) =
- The eigenvalue s = s(β) as a function of β is sketched in Fig. 18.2; the eigenvalue condition is satisfied on the curve F = 0. The cycle av- eraging formulas for the slope and the curvature of s(β) are obtained as recycle - 30aug2006 ChaosBook.org version11.9.2, Aug 21 2007
276 CHAPTER 18. CYCLE EXPANSIONS
(18.4), and pseudocycle weights tπ = tπ (z, β, s(β)) are evaluated at the eigenvalue s(β). In most applications β = 0, and s(β) of interest is typically the leading eigenvalue s 0 = s 0 (0) of the evolution generator A. For bounded flows the leading eigenvalue (the escape rate) vanishes, s(0) = 0, the exponent βAπ − sTπ in (18.3) vanishes, so the cycle expan- sions take a simple form
〈A〉ζ =
π
(−1)k+^
Ap 1 + Ap 2 · · · + Apk |Λp 1 · · · Λpk |
and similarly for 〈T〉ζ , 〈n〉ζ. For example, for the complete binary sym- bolic dynamics the mean cycle period 〈T〉ζ is given by
〈T〉ζ =
T 0
T 1
T 01
T 0 + T 1
T 001
T 01 + T 0
T 011
T 01 + T 1
Note that the cycle expansions for averages are grouped into the same shadowing combinations as the dynamical zeta function cycle expan- sion (18.7), with nearby pseudocycles nearly cancelling each other. The cycle averaging formulas for the expectation value of the observ- able 〈a〉 follow by substitution into (18.21). Assuming zero mean drift 〈a〉 = 0, the cycle expansion (18.11) for the variance
(A − 〈A〉)^2
ζ is given by
〈 A^2
ζ =^
(−1)k+^
(Ap 1 + Ap 2 · · · + Apk )^2 |Λp 1 · · · Λpk |
18.3.2 Spectral determinant cycle expansions
The dynamical zeta function cycle expansions have a particularly sim- ple structure, with the shadowing apparent already by a term-by-term inspection of Table 18.2. For “nice” hyperbolic systems the shadowing ensures exponential convergence of the dynamical zeta function cycle expansions. This, however, is not the best achievable convergence. As has been explained in Chapter ?? , for such systems the spectral det- erminant constructed from the same cycle data base is entire, and its cycle expansion converges faster than exponentially. In practice, the best convergence is attained by the spectral determinant cycle expan- sion (18.16) and its derivatives. The ∂/∂s, ∂/∂β derivatives are in this case computed recursively, by taking derivatives of the spectral deter- minant cycle expansion contributions (18.12) and (18.10). The cycle averaging formulas are exact, and highly convergent for nice hyperbolic dynamical systems. An example of its utility is the cy- cle expansion formula for the Lyapunov exponent of Example 18.1. Fur- ther applications of cycle expansions will be discussed in Chapter ??. recycle - 30aug2006 ChaosBook.org version11.9.2, Aug 21 2007
18.4. CYCLE EXPANSIONS FOR FINITE ALPHABETS 277
18.3.3 Continuous vs. discrete mean return time
Sometimes it is convenient to compute an expectation value along a flow, in continuous time, and sometimes it might be easier to compute it in discrete time, from a Poincar´e return map. Return times (3.1) might vary wildly, and it is not at all clear that the continuous and discrete time averages are related in any simple way. The relationship turns on to be both elegantly simple, and totally general. The mean cycle period 〈T〉ζ fixes the normalization of the unit of time; it can be interpreted as the average near recurrence or the average first return time. For example, if we have evaluated a billiard expecta- tion value 〈a〉 in terms of continuous time, and would like to also have the corresponding average 〈a〉dscr measured in discrete time, given by the number of reflections off billiard walls, the two averages are related by 〈a〉dscr = 〈a〉 〈T〉ζ / 〈n〉ζ , (18.26)
where 〈n〉ζ is the average of the number of bounces np along the cycle p.
Example 18.1 Cycle expansion formula for Lyapunov exponents:
In Section 15.3 we defined the Lyapunov exponent for a 1-d mapping, re- lated it to the leading eigenvalue of an evolution operator and promised to evaluate it. Now we are finally in position to deliver on our promise. The cycle averaging formula (18.23) yields an exact explict expression for the Lyapunov exponent in terms of prime cycles:
λ = 1 〈n〉ζ
∑′ (−1)k+^ log |Λp 1 | + · · · + log |Λpk | |Λp 1 · · · Λpk |
. (18.27)
For a repeller, the 1 /|Λp | weights are replaced by normalized measure ( ?? ) exp(γnp)/|Λp|, where γ is the escape rate.
We mention here without proof that for 2-d Hamiltonian flows such as our game of pinball there is only one expanding eigenvalue and (18.27) applies as it stands.
in depth: Chapter ?? , p. ??
18.4 Cycle expansions for finite alphabets
A finite Markov graph like the one given in Fig. 13.3.1 (d) is a compact encoding of the transition or the Markov matrix for a given subshift. It is a sparse matrix, and the associated determinant (13.17) can be written down by inspection: it is the sum of all possible par- titions of the graph into products of non-intersecting loops, with each loop carrying a minus sign:
det (1 − T ) = 1 − t 0 − t 0011 − t 0001 − t 00011 + t 0 t 0011 + t 0011 t 0001 (18.28) ChaosBook.org version11.9.2, Aug 21 2007 recycle - 30aug
18.5. STABILITY ORDERING OF CYCLE EXPANSIONS 279
converge slowly. For such systems schemes which collect the pseudo- cycle terms according to some criterion other than the topology of the flow may converge more quickly than expansions based on the topo- logical length. All chaotic systems exhibit some degree of shadowing, and a good truncation criterion should do its best to respect the shadowing at least approximately. If a long cycle is shadowed by two or more shorter cycles and the flow is smooth, the period and the action will be addi- tive in sense that the period of the longer cycle is approximately the sum of the shorter cycle periods. Similarly, stability is multiplicative, so shadowing is approximately preserved by including all terms with pseudocycle stability
|Λp 1 · · · Λpk | ≤ Λmax (18.30)
and ignoring all more unstable pseudocycles. Two such schemes for ordering cycle expansions which approximately respect shadowing are truncations by the pseudocycle period (or ac- tion) and the stability ordering that we shall discuss here. In these schemes a dynamical zeta function or a spectral determinant is expanded keeping all terms for which the period, action or stability for a combi- nation of cycles (pseudocycle) is less than a given cutoff. The two settings in which the stability ordering may be preferable to the ordering by topological cycle length are the cases of bad grammar and of intermittency.
18.5.1 Stability ordering for bad grammars
For generic flows it is often not clear what partition of the state space generates the “optimal” symbolic dynamics. Stability ordering does not require understanding dynamics in such detail: if you can find the cycles, you can use stability ordered cycle expansions. Stability trun- cation is thus easier to implement for a generic dynamical system than the curvature expansions (18.7) which rely on finite subshift approxi- mations to a given flow. Cycles can be detected numerically by searching a long trajectory for near recurrences. The long trajectory method for detecting cycles pref- erentially finds the least unstable cycles, regardless of their topological length. Another practical advantage of the method (in contrast to New- ton method searches) is that it only finds cycles in a given connected ergodic component of state space, ignoring isolated cycles or other er- godic regions elsewhere in the state space. Why should stability ordered cycle expansion of a dynamical zeta function converge better than the rude trace formula ( ?? )? The argu- ment has essentially already been laid out in Section 13.7: in truncations that respect shadowing most of the pseudocycles appear in shadowing combinations and nearly cancel, while only the relatively small subset affected by the longer and longer pruning rules is not shadowed. So the error is typically of the order of 1 /Λ, smaller by factor ehT^ than ChaosBook.org version11.9.2, Aug 21 2007 recycle - 30aug
280 CHAPTER 18. CYCLE EXPANSIONS
the trace formula ( ?? ) error, where h is the entropy and T typical cycle length for cycles of stability Λ.
18.5.2 Smoothing
The breaking of exact shadowing cancellations deserves further comment. Partial shadowing which may be present can be (partially) restored by smoothing the stability ordered cycle expansions by re- placing the 1 /Λ weight for each term with pseudocycle stability Λ = Λp 1 · · · Λpk by f (Λ)/Λ. Here, f (Λ) is a monotonically decreasing func- tion from f (0) = 1 to f (Λmax) = 0. No smoothing corresponds to a step function. A typical “shadowing error” induced by the cutoff is due to two pseudocycles of stability Λ separated by ΔΛ, and whose contribution is of opposite signs. Ignoring possible weighting factors the magnitude of the resulting term is of order 1 /Λ− 1 /(Λ+ΔΛ) ≈ ΔΛ/Λ^2. With smooth- ing there is an extra term of the form f ′(Λ)ΔΛ/Λ, which we want to minimise. A reasonable guess might be to keep f′(Λ)/Λ constant and as small as possible, that is
f (Λ) = 1 −
Λmax
The results of a stability ordered expansion (18.30) should always be tested for robustness by varying the cutoff Λmax. If this introduces significant variations, smoothing is probably necessary.
18.5.3 Stability ordering for intermittent flows
Longer but less unstable cycles can give larger contributions to a cycle expansion than short but highly unstable cycles. In such situ- ation truncation by length may require an exponentially large number of very unstable cycles before a significant longer cycle is first included in the expansion. This situation is best illustrated by intermittent maps that we shall study in detail in Chapter ?? , the simplest of which is the Farey map
f (x) =
f 0 = x/(1 − x) 0 ≤ x ≤ 1 / 2 f 1 = (1 − x)/x 1 / 2 ≤ x ≤ 1 ,
a map which will reappear in the intermittency Chapter ?? , and in Chapter ?? , in context of circle maps. For this map the symbolic dynamics is of complete binary type, so lack of shadowing is not due to lack of a finite grammar, but rather to the intermittency caused by the existence of the marginal fixed point x 0 = 0, for which the stability equals Λ 0 = 1. This fixed point does not participate directly in the dynamics and is omitted from cycle expan- sions. Its presence is felt in the stabilities of neighboring cycles with recycle - 30aug2006 ChaosBook.org version11.9.2, Aug 21 2007
282 CHAPTER 18. CYCLE EXPANSIONS
18.6 Dirichlet series
The most patient reader will thank me for compressing so much nonsense and falsehood into a few lines. Gibbon
A Dirichlet series is defined as
f (s) =
∑^ ∞
j=
aj e−λj^ s^ (18.32)
where s, aj are complex numbers, and {λj } is a monotonically increas- ing series of real numbers λ 1 < λ 2 < · · · < λj < · · ·. A classical exam- ple of a Dirichlet series is the Riemann zeta function for which aj = 1, λj = ln j. In the present context, formal series over individual pseudo- cycles such as (18.2) ordered by the increasing pseudocycle periods are often Dirichlet series. For example, for the pseudocycle weight (18.3), the Dirichlet series is obtained by ordering pseudocycles by increasing periods λπ = Tp 1 + Tp 2 +... + Tpk , with the coefficients
aπ = eβ·(Ap^1 +Ap^2 +...+Apk^ ) |Λp 1 Λp 2... Λpk |
dπ ,
where dπ is a degeneracy factor, in the case that dπ pseudocycles have the same weight. If the series
|aj | diverges, the Dirichlet series is absolutely conver- gent for s > σa and conditionally convergent for s > σc, where σa is the abscissa of absolute convergence
σa = lim N →∞
sup
λN
ln
∑^ N
j=
|aj | , (18.33)
and σc is the abscissa of conditional convergence
σc = lim N →∞ sup
λN
ln
∑^ N
j=
aj
We shall encounter another example of a Dirichlet series in the semi- classical quantization Chapter ?? , where the inverse Planck constant is a complex variable s = i/ℏ, λπ = Sp 1 + Sp 2 +... + Spk is the pseudo- cycle action, and aπ = 1/
|Λp 1 Λp 2... Λpk | (times possible degeneracy and topological phase factors). As the action is in general not a lin- ear function of energy (except for billiards and for scaling potentials, where a variable s can be extracted from Sp), semiclassical cycle expan- sions are Dirichlet series in variable s = i/ℏ but not in E, the complex energy variable. recycle - 30aug2006 ChaosBook.org version11.9.2, Aug 21 2007
18.6. DIRICHLET SERIES 283
Summary
A cycle expansion is a series representation of a dynamical zeta func- tion, trace formula or a spectral determinant, with products in (17.15), ( ?? ) expanded as sums over pseudocycles , products of the prime cycle weights tp. If a flow is hyperbolic and has a topology of a Smale horseshoe (a subshift of finite type), the dynamical zeta functions are holomorphic, the spectral determinants are entire, and the spectrum of the evolu- tion operator is discrete. The situation is considerably more reassuring than what practitioners of quantum chaos fear; there is no “abscissa of absolute convergence” and no “entropy barier”, the exponential pro- liferation of cycles is no problem, spectral determinants are entire and converge everywhere, and the topology dictates the choice of cycles to be used in cycle expansion truncations. In that case, the basic observation is that the motion in dynamical systems of few degrees of freedom is in this case organized around a few fundamental cycles, with the cycle expansion of the Euler product
1 /ζ = 1 −
f
tf −
n
c ˆn,
regrouped into dominant fundamental contributions tf and decreasing curvature corrections ˆcn. The fundamental cycles tf have no shorter approximants; they are the “building blocks” of the dynamics in the sense that all longer orbits can be approximately pieced together from them. A typical curvature contribution to ˆcn is a difference of a long cycle {ab} minus its shadowing approximation by shorter cycles {a} and {b}: tab − tatb = tab(1 − tatb/tab)
The orbits that follow the same symbolic dynamics, such as {ab} and a “pseudocycle” {a}{b}, lie close to each other, have similar weights, and for longer and longer orbits the curvature corrections fall off rapidly. Indeed, for systems that satisfy the “axiom A” requirements, such as the 3-disk billiard, curvature expansions converge very well. Once a set of the shortest cycles has been found, and the cycle pe- riods, stabilities and integrated observable computed, the cycle aver- aging formulas such as the ones associated with the dynamical zeta function
〈a〉 = 〈A〉ζ / 〈T〉ζ
〈A〉ζ = −
∂β
ζ
Aπ tπ , 〈T〉ζ =
∂s
ζ
Tπtπ
yield the expectation value (the chaotic, ergodic average over the non- wandering set) of the observable a(x).
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Exercises 285
Exercises
(18.1) Cycle expansions. Write programs that imple- ment binary symbolic dynamics cycle expansions for (a) dynamical zeta functions, (b) spectral deter- minants. Combined with the cycles computed for a 2-branch repeller or a 3-disk system they will be useful in problem that follow. (18.2) Escape rate for a 1- d repeller. (Continuation of Exercise 17.1 - easy, but long) Consider again the quadratic map (17.31) f (x) = Ax(1 − x) on the unit interval, for definitiveness take A = 6. Describing the itinerary of any trajectory by the bi- nary alphabet { 0 , 1 } (’0’ if the iterate is in the first half of the interval and ’1’ if is in the second half), we have a repeller with a complete binary symbolic dynamics. (a) Sketch the graph of f and determine its two fixed points 0 and 1 , together with their stabil- ities. (b) Sketch the two branches of f−^1. Determine all the prime cycles up to topological length 4 using your pocket calculator and backwards iteration of f (see Section 12.2.1). (c) Determine the leading zero of the zeta func- tion (17.15) using the weigths tp = znp^ /|Λp | where Λp is the stability of the p cycle. (d) Show that for A = 9 / 2 the escape rate of the repeller is 0. 361509... using the spectral determinant, with the same cycle weight. If you have taken A = 6 , the escape rate is in 0. 83149298.. ., as shown in Solution 18.2. Compare the coefficients of the spectral det- erminant and the zeta function cycle expan- sions. Which expansion converges faster? (Per Rosenqvist)
(18.3) Escape rate for the Ulam map. (medium) We will try to compute the escape rate for the Ulam map (12.22) f (x) = 4x(1 − x), using the method of cycle expansions. The answer should be zero, as nothing escapes. (a) Compute a few of the stabilities for this map. Show that Λ 0 = 4 , Λ 1 = − 2 , Λ 01 = − 4 , Λ 001 = − 8 and Λ 011 = 8.
(b) Show that Λ 1 ...n = ± 2 n and determine a rule for the sign. (c) (hard) Compute the dynamical zeta function for this system
ζ−^1 = 1 − t 0 − t 1 − (t 01 − t 0 t 1 ) − · · ·
You might note that the convergence as func- tion of the truncation cycle length is slow. Try to fix that by treating the Λ 0 = 4^ cycle sepa- rately.
(18.4) Pinball escape rate, semi-analytical. Estimate the 3-disk pinball escape rate for R : a = 6 by substi- tuting analytical cycle stabilities and periods (Exer- cise 9.3 and Exercise 9.4) into the appropriate binary cycle expansion. Compare with the numerical esti- mate Exercise 15. (18.5) Pinball escape rate, from numerical cycles. Com- pute the escape rate for R : a = 6 3-disk pinball by substituting list of numerically computed cycle sta- bilities of Exercise 12.5 into the binary cycle expan- sion. (18.6) Pinball resonances, in the complex plane. Plot the logarithm of the absolute value of the dynam- ical zeta function and/or the spectral determinant cycle expansion (18.5) as contour plots in the com- plex s plane. Do you find zeros other than the one corresponding to the complex one? Do you see ev- idence for a finite radius of convergence for either cycle expansion? (18.7) Counting the 3-disk pinball counterterms. Ver- ify that the number of terms in the 3-disk pinball curvature expansion (18.35) is given by
∏
p
(1 + tp) = 1 − 3 z^4 − 2 z^6 1 − 3 z^2 − 2 z^3 = 1 + 3z^2 + 2z^3 + z^4 (6 + 12z + 2z^2 ) 1 − 3 z^2 − 2 z^3 = 1 + 3z^2 + 2z^3 + 6z^4 + 12z^5 + 20z^6 + 48z^7 + 84z^8 + 184z
This means that, for example, c 6 has a total of 20 terms, in agreement with the explicit 3-disk cycle expansion (18.36). ChaosBook.org version11.9.2, Aug 21 2007 exerRecyc - 15nov
286 Exercises
(18.8) 3–disk unfactorized zeta cycle expansions. Check that the curvature expansion (18.2) for the 3-disk pinball, assuming no symmetries between disks, is given by 1 /ζ = (1 − z^2 t 12 )(1 − z^2 t 13 )(1 − z^2 t 23 ) (1 − z^3 t 123 )(1 − z^3 t 132 )(1 − z^4 t 1213 ) (1 − z^4 t 1232 )(1 − z^4 t 1323 )(1 − z^5 t 12123 ) · · · = 1 − z^2 t 12 − z^2 t 23 − z^2 t 31 − z^3 (t 123 + t 132 ) −z^4 [(t 1213 − t 12 t 13 ) + (t 1232 − t 12 t 23 ) + (t 1323 − t 13 t 23 )] −z^5 [(t 12123 − t 12 t 123 ) + · · ·] − · · · (18.35) The symmetrically arranged 3-disk pinball cycle ex- pansion of the Euler product (18.2) (see Table 13. and Fig. 9.3) is given by: 1 /ζ = (1 − z^2 t 12 )^3 (1 − z^3 t 123 )^2 (1 − z^4 t 1213 )^3 (1 − z^5 t 12123 )^6 (1 − z^6 t 121213 )^6 (1 − z^6 t 121323 )^3... = 1 − 3 z^2 t 12 − 2 z^3 t 123 − 3 z^4 (t 1213 − t^212 ) − 6 z^5 (t 12123 − t 12 t 123 ) −z^6 (6 t 121213 + 3 t 121323 + t^312 − 9 t 12 t 1213 − t^2123 ) − 6 z^7 (t 1212123 + t 1212313 + t 1213123 + t^212 t 123 − 3 t 12 t 12123 − t 123 t 1213 ) − 3 z^8 (2 t 12121213 + t 12121313 + 2 t 12121323 + 2 t 12123123
- 2 t 12123213 + t 12132123 + 3 t^212 t 1213 + t 12 t^2123 − 6 t 12 t 121213 − 3 t 12 t 121323 − 4 t 123 t 12123 − t^21213 ) − · · · (18.36) Unsymmetrized cycle expansions. The above 3-disk cycle expansions might be useful for cross- checking purposes, but, as we shall see in Chap- ter ?? , they are not recommended for actual compu- tations, as the factorized zeta functions yield much better convergence. (18.9) 4–disk unfactorized dynamical zeta function cycle expansions For the symmetriclly arranged 4-disk pinball the symmetry group is C 4 v , of order 8. The degenerate cycles can have multiplicities 2, 4 or 8 (see Table 13.2): 1 /ζ = (1 − z^2 t 12 )^4 (1 − z^2 t 13 )^2 (1 − z^3 t 123 )^8 (1 − z^4 t 1213 )^8 (1 − z^4 t 1214 )^4 (1 − z^4 t 1234 )^2 (1 − z^4 t 1243 )^4 (1 − z^5 t 12123 )^8 (1 − z^5 t 12124 )^8 (1 − z^5 t 12134 )^8 (1 − z^5 t 12143 )^8 (1 − z^5 t 12313 )^8 (1 − z^5 t 12413 )^8 · · · (18.37) and the cycle expansion is given by 1 /ζ = 1 − z^2 (4 t 12 + 2 t 13 ) − 8 z^3 t 123 −z^4 (8 t 1213 + 4 t 1214 + 2 t 1234 + 4 t 1243 − 6 t^212 − t^213 − 8 t 12 t 13 ) − 8 z^5 (t 12123 + t 12124 + t 12134 + t 12143 + t 12313 + t 12413 − 4 t 12 t 123 − 2 t 13 t 123 ) − 4 z^6 (2 S 8 + S 4 + t^312 + 3 t^212 t 13 + t 12 t^213 − 8 t 12 t 1213 − 4 t 12 t 1214 − 2 t 12 t 1234 − 4 t 12 t 1243 − 4 t 13 t 1213 − 2 t 13 t 1214 − t 13 t 1234 − 2 t 13 t 1243 − 7 t^2123 ) − · · · (18.38)
where in the coefficient to z^6 the abbreviations S 8 and S 4 stand for the sums over the weights of the 12 orbits with multiplicity 8 and the 5 orbits of mul- tiplicity 4 , respectively; the orbits are listed in Ta- ble 13.4.
(18.10) Tail resummations. A simple illustration of such tail resummation is the ζ function for the Ulam map (12.22) for which the cycle structure is exceptionally simple: the eigenvalue of the x 0 = 0 fixed point is 4, while the eigenvalue of any other n-cycle is ± 2 n. Typical cycle weights used in thermodynamic av- eraging are t 0 = 4τ^ z, t 1 = t = 2τ^ z, tp = tnp^ for p = 0. The simplicity of the cycle eigenvalues en- ables us to evaluate the ζ function by a simple trick: we note that if the value of any n-cycle eigenvalue were tn, (17.21) would yield 1 /ζ = 1 − 2 t. There is only one cycle, the x 0 fixed point, that has a dif- ferent weight (1 − t 0 ), so we factor it out, multiply the rest by (1 − t)/(1 − t), and obtain a rational ζ function
1 /ζ(z) = (1 − 2 t)(1 − t 0 ) (1 − t) (18.39)
Consider how we would have detected the pole at z = 1/t without the above trick. As the 0 fixed point is isolated in its stability, we would have kept the factor (1 − t 0 ) in (18.7) unexpanded, and noted that all curvature combinations in (18.7) which include the t 0 factor are unbalanced, so that the cycle ex- pansion is an infinite series: ∏
p
(1 − tp) = (1 − t 0 )(1 − t − t^2 − t^3 − t^4 −.. .)
(18.40) (we shall return to such infinite series in Chap- ter ?? ). The geometric series in the brackets sums up to (18.39). Had we expanded the (1 − t 0 )^ factor, we would have noted that the ratio of the succes- sive curvatures is exactly cn+1/cn = t; summing we would recover the rational ζ function (18.39).
(18.11) Escape rate for the R ¨ossler system. (continuation of Exercise 12.7) Try to compute the escape rate for the R ¨ossler system (2.15) using the method of cycle expansions. The answer should be zero, as nothing escapes. Ideally you should already have computed the cycles and have an approximate grammar, but failing that you can cheat a bit and peak into Ta- ble ??.
recycle - 30aug2006 ChaosBook.org version11.9.2, Aug 21 2007
