Topological entropy, knots and star products

ITERATION THEORY (ECIT ‘02) J. Sousa Ramos, D. Gronau, C. Mira, L. Reich, A. Sharkovsky (Eds.) Grazer Math. Ber., ISSN 1016–7692 Bericht Nr. 346 (2004), 61-72

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K.-F. Cao, C. Zhang, S.-L. Peng

line in space [7]. When a knot is projected to a certain plane, the complete information of over and undercrossings of lines is preserved, while such important information is lost in the permutation of vertex shift matrix in the Poincaré section (cf. Figure 1). If we can recover the three-dimensional information from the two-dimensional one by a suitable means, then the relation between these two topological invariants can be discussed. To this end, we adopt a simplest method, the minimal braid assumption, which can make the first return map of periodic points in the two-dimensional section become a threedimensional flow by a suspension. Due to the simplicity of some dynamical systems themselves, it is possible to make the knot of the flow have the minimal crossings. This paper will set off discussion on this assumption.

Topological entropy, knots and star products Ke-Fei Cao, Chuan Zhang and Shou-Li Peng∗

3D

Abstract The knots of periodic flows of dynamical systems, formed in three-dimensional phase space, are briefly reviewed in this paper. By using the minimal braid assumption, periodic points in the Poincaré section can be transformed by a suspension into a knot of the three-dimensional periodic flow. Thus the connection between knots and sequences of applied symbolic dynamics is established. Furthermore, by means of star products with equal topological entropy and the renormalization of knots, the calculations programed for the Conway polynomials of knots of the period-doubling bifurcation cascade can be carried out. The Conway polynomial provides another topological description of periodic flows in the dynamics, which is different from the Milnor–Thurston polynomial.

1 2 3 3

Poincare section

4 2 1

1D

projection

2D

1

4

Introduction

1

2

3

4

3

4

2

1

seek rule

Consider a smooth dynamical system in three-dimensional space, x˙ = FΛ (x),

(1)

which is autonomous. If its orbits are periodic, then flows of the dynamical system will form some real knots in three-dimensional space in the sense of limit cycles. In general, when a flow is reduced as a map in the Poincaré section, we cannot recognize the real knot only from the section due to the loss of three-dimensional information. However, on the one hand, for the map in the Poincaré section, shifts acting on a symbolic sequence of the dynamics will lead to a vertex shift matrix [1], which is a standard permutation. This permutation is homological to an edge shift matrix [8, 9]. The spectrum of the edge shift matrix will describe topological entropy (h : fλ → R) of the map, which is a topological invariant often used in the dynamics [12]. On the other hand, the knot K : S → R3 is also a topological invariant of the orbit, which describes the topology of the orbital flow ∗ Supported by the Special Funds for Major State Basic Research Projects of China (the “973” Program) (G2000077308) and the Cooperation Research Funds of Yunnan Provincial Government. Mathematics Subject Classification 2000: Primary 37B10, Secondary 57M27, 37B40. Keywords and phrases: knot, star product, Conway polynomial.

Figure 1: Schematic diagram of the knot of a period-4 orbit of the Rössler system, x˙ = −y − z, y˙ = x + ay, z˙ = b + (x − c)z (with parameters a = 0.2, b = 0.2, c = 4.1).

2

Symbolic Dynamics in the Poincar´ e Section

Consider a three-dimensional flow of dynamical system (1), which can form a map f λ on the interval I = [c0 , cN ] in the Poincaré section, i.e., fλ : I → I with N − 1 parameters (λ1 , λ2 , . . . , λN −1 ) ≡ λ ⊂ RN −1 . fλ has N successive subintervals of monotonicity,

Topological entropy, knots and star products

63

[c0 , c1 ], [c1 , c2 ], . . ., [cN −1 , cN ]. The N − 1 critical points and the N subintervals are numbered in their natural order in symbolic space, I1 ≺ C1 ≺ I2 ≺ C2 ≺ . . . ≺ IN −1 ≺ CN −1 ≺ IN , where ≺ is the Metropolis–Stein–Stein (MSS) order [11]. Thus each point x ∈ I must have a unique symbolic address A(x) ∈ ΣN according to the symbol of the set to which x belongs, x ∈ A(x). So the symbolic order and the real order are consistent, namely, A(x) ¹ A(x0 ) if x < x0 . Each point x can be assigned an itinerary [12], I(x) = A0 (x)A1 (x) . . . An (x) . . ., defined as the sequence of addresses An (x) = A(fλn (x)) ∈ ΣN , here n ∈ Z+ . The itinerary I(x) is an injection from the interval I to the sequence space (ΣN )Z+ which is a set of symbolic sequences (or words) of all itineraries. If a shift map ϕ : (ΣN )Z+ → (ΣN )Z+ is defined as ϕ(I(x))i = (I(x))i+1 or ϕ(I(x)) = A1 (x)A2 (x) . . . An (x) . . ., then the map fλ on the interval I corresponds to the shift map ϕ on the symbolic space (ΣN )Z+ . For an arbitrary admissible N − 1 multiple superstable kneading sequence W = C1 X1 C2 X2 . . . CN −1 XN −1 = w1 w2 . . . w|W | where |W | denotes the length of sequence W , shifts ϕi (W ) = (W )i+1 = wi+1 wi+2 . . . w|W | (i = 1, 2, . . . , |W |) form |W | vertices. On the set C0 = {ϕi (W )} of vertices, a basic shift matrix, |W | × |W | 0-1 matrix ω, is constructed by the natural shift order: ω((W )i , (W )i+1 ) = 1, which is a simple periodic matrix. The vertex set C0 is ordered by the MSS order ≺. The ordering procedure can be completed by a permutation π : Z+ → Z+ which makes ϕπ(1) (W ) ≺ ϕπ(2) (W ) ≺ . . . ≺ ϕπ(|W |) (W ) hold [14]. The permutation π can be written as a |W | × |W | 0-1 matrix. Note that these ordered vertices form a line graph G, while shift maps from vertex ϕπ(i) (W ) to vertex ϕπ(i)+1 (W ) make a directed graph. Then the adjacency matrix Sv of the directed graph is constructed such that for every π(i), Sv ((W )π(i) , (W )π(i)+1 ) = 1, which is another |W | × |W | 0-1 matrix. These two vertex matrices are similar [8]: Sv = πωπ −1 .

(2)

The shift on the symbolic space ΣN is transformed into a finite walk through the graph, where the itineraries of sequence are recorded by the vertices traversed. Of course, if there is no arc from k to k 0 , then (Sv )k,k0 = 0. The case of non-arc is represented by a list of forbidden words. The shift restricted to ΣN is a shift of finite type, so a vertex shift is isomorphic to it [1].

3 3.1

Knot as a Suspension Minimal braid assumption

Let us first give an example to show how the braid of a period-4 sequence RLRC of unimodal maps is formed [5]. The dynamical permutation of orbital points of RLRC is [2] ¶ µ 1 2 3 4 . 3 4 2 1

64


In Figure 2 four lines 1 → 3, 2 → 4, 3 → 2 and 4 → 1 illustrate this permutation clearly.

1

2

3

4

1

2

3

4

Figure 2: Schematic diagram of the minimal braid for sequence RLRC = RC ∗ RC All five crossings are arranged in the following way of weaving: a new line should be drawn under the previous one(s) according to the order in the permutation. Thus we obtain the braid as in Figure 2. This method can be summarized as the minimal braid assumption [23]: (a) The weaving rule for the type of negative crossings: according to the order of appearances of the oriented lines inside the inner region between the two horizontal lines, the last line is always under the previous one(s). (b) The forbidden rule for the twist of lines: there is no more than half-twist for any two lines. This assumption is simple. It is true for some smooth dynamical systems and has been numerically verified [23] for various periodic sequences of the Rössler system [20]. Although this assumption strictly restrict the types of knots, it is an intuitive simple method to construct three-dimensional flow by a suspension. On this basis, we may gradually relax the restriction on the types of knots, for instance, we may use twists higher than half-twist and the local rule of more complex types of crossings. This would be the work in the future.

3.2

Entropy of knot

Under the minimal braid assumption, for arbitrary periodic flows we can obtain an edge shift matrix Se from the vertex shift matrix Sv . The topological entropy of the minimal knot is determined by the spectrum of the edge shift matrix Se . The edges are order pairs (πW (i), πW (i + 1)) = IπW (i) ≡ Ik in the line graph G with the vertices {πW (i)}. By the homological language [8, 9], the vector space of 0-chains, C0 , is spanned by {ϕπ(i) (W )}, and that of 1-chains, C1 , by the edge set {Ik } (k = {0, 1, . . . , |W | − 1}). According to [8], the vertex shift matrix Sv induces an isomorphism Sp and leads to an edge shift matrix Se = |Sp | (i.e. the Stefan matrix) which reads as: Se = |µSv µ−1 |,

(3)


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where µ is a nonsquare matrix and the absolute value of matrix S = [sαβ ] is denoted by |S| = [|sαβ |]. Thus the entropy of the knot in the dynamical system can be defined as (4)

h = log γmax ,

(a)

where γmax is the largest eigenvalue of Se .

4

(b)

1

1

Star Products and Knots

4.1

The way of weaving

A periodic sequence is a cycle, the star products [3, 10, 2, 16, 17, 24] indicate the way of weaving of one cycle with another. When we endow the weaving of cycles with the over or undercrossing information given from the minimal braid assumption, the star product would be a special knot. We now illustrate such an idea in Figure 3. The orbital iterative graphs of superstable periodic sequences RC, RLC and RC ∗ RLC are displayed in Figure 3(a)–(c), respectively. From the minimal braid assumption, we can easily obtain the weaving of orbital line of RC ∗ RLC as in Figure 3(d).

4.2

0

0

1

(c)

Star product

0

0

1

(d)

1

C1 X11 C2 X21

. . . CN −1 XN1 −1

For two arbitrary periodic sequences W1 = and W2 in (N −1)modal maps, the star product ∗ is defined as a star direct product ~ of their two vertex shift matrices Sv (W1 ) and Sv (W2 ) [21]: Sv (W1 ∗ W2 ) = Sv (W1 ) ~ Sv (W2 ), where the star direct product ~ is obtained by a set of substitutions from the symbols {Ij , Ci } in the symbolic matrix of W1 to the unit matrix E and angle matrices {Sv (Ci )} with parity operations:   0 → [0] (|W2 | × |W2 | zreo matrix), Ij → E τ (Ij ) (j = 1, 2, . . . , N ), Sv (W1 )~ : (5) 1  Ci → Sv (Ci )τ (Xi ) (i = 1, 2, . . . , N − 1). where

E τ (Ij ) =

½ 1

Sv (Ci )τ (Xi )

unit matrix E, flipped unit matrix E, ½ Sv (Ci ), = S v (Ci ) = ESv (Ci ),

if parity τ (Ij ) = +, if parity τ (Ij ) = −; if parity τ (Xi1 ) = +, if parity τ (Xi1 ) = −.

The single angle matrices Sv (Ci ) satisfy a decomposition of product factors for the vertex shift matrix Sv (W2 ) of the second word W2 . The order of product factors in

0

0

1

Figure 3: (a)–(c) Orbits of superstable periodic sequences RC, RLC and RC ∗ RLC (for the logistic map fλ (x) = λx(1 − x)); (d) Schematic diagram of the weaving of orbital line of RC ∗ RLC.


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the decomposition would be in accord with the order of critical points in the first word W1 = C1 X11 C2 X21 . . . CN −1 XN1 −1 as follows Sv (W2 ) = Sv (C1 )Sv (C2 ) . . . Sv (CN −1 ). Thus the star product of symbolic sequences is transformed into a direct product of matrices, which is a very prefect algebraic form to theory and application of star product. We notice that an arbitrary vertex shift matrix can be made up of three kinds of submatrices: (i) angle matrix; (ii) unit and flipped unit matrices; (iii) zero matrix. It is not difficult to verify Sv (RC ∗ RLC) = Sv (RC) ~ Sv (RLC). Here, we take the period-tripling bifurcation (RLC)∗n as a simple example. The dynamical permutations and vertex shift matrices Sv are listed for n = 1, 2, 3, respectively: RLC: µ

¶ 1 2 3 , 2 3 1   0 1 0 Sv (RLC) =  0 0 1  ; 1 0 0 (RLC)∗2 = RLL RLR RLC: µ

1 2 3 4 5 6 7 8 9 4 5 6 7 9 8 3 2 1 

      Sv ((RLC)∗2 ) =       

0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0 0

1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

¶

,

0 0 0 1 0 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 1 0 0 0 0



      ;      

(RLC)∗3 = RLL RLR RLR RLL RLR RLL RLL RLR RLC: µ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 10 11 12 13 14 15 16 17 18 19 20 21 26 27 25 24 23 22 19 20 21 22 23 24 25 26 27 9 8 7 6 5 4 3 2 1

¶

,

Sv ((RLC)∗3 ) =  0 0 0 0 0  0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0    0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 1   0 0 0 1 0   0 0 1 0 0   0 1 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0



                         .                        

When we obtain star products from direct products of matrices, and construct knots of the star products from the minimal braid assumption, an important phenomenon is that all the knots are of equal entropy in the sense of (4). These equal entropy knots have a characteristic, that is, when flow lines are renormalized from p lines (e.g., p = 2, 3 for period-doubling and period-tripling respectively) to one line, knots after renormalization reduce naturally to the shapes before bifurcations. This is the form of the Kadanoff renormalization [19, 15]. This characteristic would make their Conway polynomials have very regular expression. We will show this in the next section.


5

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The Conway Polynomial of the Minimal Braid

5.1

The Kauffman state model of braid

On the basis of the minimal braid assumption, we can calculate the algebraic characteristic polynomials of knots, among which the Conway polynomial is often used. It is convenient to calculate it by using the Kauffman state model [6, 7]. However, to carry out programable calculation of the Kauffman state model, there still need some technical work. The detail of the program of the state model is presented in [22]. Here we only introduce several key steps of programing. The computer needs to deal with the following items: (i) recognize types of crossings; (ii) recognize topological regions; (iii) calculate the Alexander determinant. In the practical calculation, one may meet with the following difficulties: (i) The computer recognizes the knots by the ordered weaving method; (ii) The number N of crossings increase rapidly with the length p of the sequence (N ∼ pk ); for example, for the period-doubling cascade, corresponding to p = 2, 4, 8, 16, 32, . . ., we have N = 1, 5, 23, 97, 399, . . .; (iii) The order of the Alexander determinant increases exponentially with the length p of the sequence (∼ C p , where C is a constant).

5.2

The Conway polynomial

We obtain the Conway polynomials of the period-doubling bifurcation cascade (i.e. (RC)∗n ) of unimodal maps as follows: ∇K(RC) = 1,

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Concluding remarks

Finally we briefly summarize some interesting results: (a) The normal star products satisfy: (i) For the period-doubling bifurcation cascade, all the coefficients of the Conway polynomials equal to +1 or −1; (ii) ∇K((RC)∗n ) (1) = ±1; (iii) The Conway polynomials can be factorized in the meaning of (ii). (b) There are infinitely many kinds of knots in an arbitrary equal entropy class (including zero and positive entropy), which is in accord with the result of [4]. They are a kind of knot series which approaches infinity. (c) The dynamical knots are topological description of lines of real orbits. The minimal braid assumption provides us with a direct method to connect one-dimensional applied symbolic dynamics [5] with three-dimensional dynamics. To compare the dynamics of periodic flows with the minimal knot method, we summarize their relations in Table 1. Of course, for different systems, there may appear cases of going beyond the minimal braid assumption. Therefore, more complex assumptions still need to be developed. Table 1. Topological characteristics of dynamics and knot for periodic sequences Symbolic dynamics Knot Symbolic space Period Number of lines in a braid Type of periodic Single, double, triple, . . . Link type of 1,2,3, . . . knots sequences superstable kneading sequences Star products Permutation of orbital lines Weaving pattern of knots Renormalizations Kadanoff model Reidemeister moves Characteristic Milnor–Thurston polynomials Conway polynomials polynomials Entropy Equal value Möbius ribbon formed by lines

1 ∇K((RC)∗2 ) = − 2 (1 − t2 + t4 ), t ∇K((RC)∗3 ) = −

1 (1 − t2 + t8 − t10 + t12 − t14 + t16 − t18 + t20 − t22 + t24 − t30 + t32 ), t16

∇K((RC)∗4 ) = −

1 (1 − t2 + t16 − t18 + t24 − t26 + t32 − t34 + t40 − t42 + t48 t82 −t50 + t52 − t54 + t56 − t58 + t64 − t66 + t68 − t70 + t72 −t74 + t76 − t78 + t80 − t82 + t84 − t86 + t88 − t90 + t92 −t94 + t96 − t98 + t100 − t106 + t108 − t110 + t112 − t114 + t116 −t122 + t124 − t130 + t132 − t138 + t140 − t146 + t148 − t162 + t164 ).

Similarly, the Conway polynomials of the period-doubling cascade of Lorenz maps [5, 13, 18] can also be calculated [22]. In principle, the above method for the calculation of the Conway polynomials is suitable for multimodal maps (with N = 3, 4, 5, . . .) [22]. Multimodal maps would lead to more complex types of knots and links.

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[4] Franks, J., Williams, R.F., Entropy and knots, Trans. Amer. Math. Soc. 291 (1985), 241–253.

[19] Reichl, L.E., A Modern Course in Statistical Physics, University of Texas Press, 1980.

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[20] Rössler, O.E., An equation for continuous chaos, Phys. Lett. A 57 (1976), 397–398.

[6] Kauffman, L.H., On Knots, in: Browder, W., Langlands, R.P., Milnor, J., Stein, E.M. (eds.), Annals of Mathematics Studies 115, Princeton University Press, 1987. [7] Kauffman, L.H., Knots and Physics (2nd Edition), in: Series on Knots and Everything 1, World Scientific, Singapore 1993. [8] Lampreia, J.P., Sousa Ramos, J., Trimodal maps, Int. J. Bifurcation and Chaos 3 (1993), 1607–1617. [9] Lampreia, J.P., Sousa Ramos, J., Symbolic dynamics of bimodal maps, Portugaliae Math. 54 (1997), 1–18. [10] Llibre, J., Mumbr´ u, P., Extending the ∗-product operator, in: Mira, C., Netzer, N., Simó, C., Targonsky, G. (eds.), Proceedings of the European Conference on Iteration Theory – 1989, 199–214, World Scientific, Singapore 1991. [11] Metropolis, N., Stein, M.L., Stein, P.R., On finite limit sets for transformations on the unit interval, J. Comb. Theory A 15 (1973), 25–44. [12] Milnor, J., Thurston, W., On iterated maps of the interval, I and II, Princeton preprints (1977); in: Alexander, J.C. (ed.), Dynamical Systems — Proceedings, University of Maryland 1986–87, Lecture Notes in Math. 1342, 465–563, SpringerVerlag, Berlin 1988. [13] Peng, S.-L., Du, L.-M., Dual star products and symbolic dynamics of Lorenz maps with the same entropy, Phys. Lett. A 261 (1999), 63–73. [14] Peng, S.-L., Luo, L.-S., The ordering of critical periodic points in coordinate space by symbolic dynamics, Phys. Lett. A 153 (1991), 345–352. [15] Peng, S.-L., Xu, C.-Y., The spliting of states with the same entropy in symbolic dynamics, YNU-CNLCS preprint (2003). [16] Peng, S.-L., Zhang, X.-S., The generalized Milnor–Thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters, Commun. Math. Phys. 213 (2000), 381–411. [17] Peng, S.-L., Zhang, X.-S., Cao, K.-F., Dual star products and metric universality in symbolic dynamics of three letters, Phys. Lett. A 246 (1998), 87–96. [18] Procaccia, I., Thomae, S., Tresser, C., First return maps as a unified renormalization scheme for dynamical systems, Phys. Rev. A 35 (1987), 1884–1900.

[21] Xu, C.-Y., Peng, S.-L., The Star direct product of vertex shift matrices in symbolic dynamics, YNU-CNLCS preprint (2002). [22] Zhang, C., Calculation of the Conway polynomials of the minimal braids, YNUCNLCS preprint (2002). [23] Zhang, C., Zhang, Y.-G., Peng, S.-L., Minimal braid in applied symbolic dynamics, Chin. Phys. Lett. 20 (2003), 1444–1447. [24] Zhou, Z., Peng, S.-L., Cyclic star products and universalities in symbolic dynamics of trimodal maps, Physica D 140 (2000), 213–226.

Ke-Fei Cao, Chuan Zhang, Shou-Li Peng Center for Nonlinear Complex Systems Department of Physics Yunnan University Kunming, Yunnan 650091 China e-mail: [email protected] (K.-F. Cao), [email protected] (S.-L. Peng)