A first approach to the Galois group of chaotic chains
aa r X i v : . [ n li n . C D ] F e b A first approach to theGalois group of chaotic chains Stefan Groote
Loodus- ja T¨appisteaduste valdkond, F¨u¨usika Instituut,Tartu ¨Ulikool, W. Ostwaldi 1, 50411 Tartu, Estonia
Abstract
We explain in detail the definition, construction and generalisation of the Galoisgroup of Chebyshev polynomials of high degree to the Galois group of chaotic chains.The calculations in this paper are performed for Chebyshev polynomials and chaoticchains of degree N = 2. Insides into possible further steps are given. Prepared for the special topic “Mathematical Modelling” of AIMS Mathematics
Introduction
Developed originally for stochastics, coupled map lattices introduced by Kaneko and Kapral[1, 2] examplify spatially extended dynamical systems with a rich structure of complex dy-namical phenomena [3, 4, 5, 6, 7]. Applications for coupled map lattices can be found inmodels for hydrodynamical flows, turbulence, chemical reactions, biological systems, andquantum field theories (cf. Refs. [3, 4] for a review). Coupled map lattices that exhibitspatio-temporal chaotic behaviour are of particular interest. In most of the cases, the in-vestigation of such types of coupled map lattices is mostly restricted to numerical methods,while analytical results are known only for a few exceptional cases. Among these are cou-pled map lattices at small coupling a , consisting locally of hyperbolic maps [8, 9, 10, 11]where a smooth invariant density and an ergodic behaviour can be guaranteed.From the point of view of applications in physics, the nonhyperbolic case is certainlythe more interesting one [12, 13, 14, 15, 16, 17, 18]. Applications include stochasticallyquantised scalar field theories [4, 19], models for vacuum fluctations and dark energy [20],and chemical kinematics [3]. For the one-dimensional case considered throughout thispaper, nonhyperbolicity means that the slope of the local map is allowed to have an absolutevalue smaller than 1 in some regions. In this case, standard techniques from ergodictheory do not apply. However, weakly coupled Chebyshev maps of the order N have beeninvestigated analytically in a series of publications [18, 19, 20, 21, 22, 23, 24, 25, 26].One-dimensional coupled map lattices based on Chebyshev maps with diffusive for-ward coupling, as considered throughout this paper, are given by the simple mathematicalprescription Φ in +1 = T N (Φ in ) + a (cid:0) T N (Φ i − n ) + T N (Φ i +1 n ) (cid:1) , (1)where n is the iterative index and i enumerates an (abstract) location of the fields coupledby a coupling parameter a . The fact that the field at the location i is coupled to the fieldsat the nearest neighbour locations i − i + 1 with cyclicity assumed (Φ L +1 n = Φ n forsome L ∈ N , L ≥
3) generates a one-dimensional structure which will be called a chaotic2hain in the following. It has been shown that a weakly coupled chain of length L = 3exhibits the same behaviour as a weakly coupled longer chain [25], a fact that norishes thehope to determine the Perron–Frobenius operator of this very high-dimensional dynamicalsystem [10, 29, 30, 31, 32] in this special case.Besides power functions, Chebyshev maps T N (Φ) are the only maps with the genuineproperty T N m ( T N n (Φ)) = T N m + n (Φ) (2)for m, n ∈ N [27]. As the uncoupled case a = 0 leads to the iteration Φ in +1 = T N (Φ in ),the deduced property T N ( T N n (Φ)) = T N n +1 (Φ) can be used to understand why Chebyshevmaps are conjugate to a Bernoulli shift of N symbols [28]. This conjugacy is destroyed inthe coupled case where Chebyshev maps of the order N have N − N n , there is some hope to get a deaper insight into the evolution ofchaotic chains by considering the Galois group of these polynomials and their interations.Restricting ourselves to the order N = 2, we start with the uncoupled case. In Sec. 2we construct rational relations between the zeros of the iterative polynomials. Based onthese polynomials, we explain the algorithmic composition of the Galois group. Whilethese constructions are examples for a rigid mathematical modelling, in Sec. 3 we givean assessment on what kind of mathematical modelling will be necessary to generate ageneralised Galois group for a coupled chaotic chain. In the Conclusions we summarise ourresults and give suggestions for further steps of mathematical modelling. N = 2 As the Chebyshev polynomial of degree N n can be written as T N n (Φ) = cos( N n arccos(Φ)) , (3) In previous publications, the term “chaotic string” was used for this object. However, as the term“string” is used for (super)string theories, we avoid confusion by using the different and maybe even moreappropriate term of chaotic chains in the following.
3t is obvious that the N n zeros are equally distributed over the interval [ − π/ , + π/ T N n (sin θ ) is no longer a polynomial function. Instead, we are lookingat the zeros of the original Chebychev polynomial T N n (Φ). To be definite, in this sectionwe deal with the case N = 2. In this case the zeros are given by nested square roots of 2,for n = 1: ± √ n = 2: ± q √ , ± q − √ n = 3: ± r q √ , ± r − q √ , ± r q − √ , ± r − q − √ . . . (4)In order to classify these square roots, we use a binary code where the digit 1 indicatesthe position of a minus sign in the nested square root, starting from the innermost squareroot. For n = 5 one has e.g.
25 = 11001 : x = − vuut s r − q − √ . (5)The (classical) Galois group of a polynomial is defined as the subgroup of permutationswhich leave invariant rational relations between the zeros of the polynomial. These relationsare obtained by addition, subtraction and stepwise squaring. The simplest relations are x + x = 0 , x + x = 0 , . . . (6)In order to step forth, notice that in squaring a zero the outermost square root will beremoved. Therefore, the sum of x i and x i +2 is equal to 1, while the difference betweenthese squares is the zero x i of the next lowest degree. Using the third Binomial formula,this zero x i can be obtained by two times the product of two zeros of the higher degreewith “distance 2”, i.e. x j − x j +2 . This procedure can be nested, and we obtain n = 1 : x + x = 0 , x + x = 1 . = 2 : x + x = 0 , x + x = 0 ,x + x = 1 , x − x = 2 x x .n = 3 : x + x = 0 , x + x = 0 , x + x = 0 , x + x = 0 ,x + x = 1 , x − x = 2 x x ,x + x = 1 , x − x = 2 x x , ( x − x ) + ( x − x ) = 1 , ( x − x ) − ( x − x ) = 2( x − x )( x − x ) .n = 4 : x + x = 0 , x + x = 0 , . . . x + x = 0 ,x + x = 1 , x − x = 2 x x ,x + x = 1 , x − x = 2 x x ,x + x = 1 , x − x = 2 x x ,x + x = 1 , x − x = 2 x x , ( x − x ) + ( x − x ) = 1 , ( x − x ) − ( x − x ) = 2( x − x )( x − x ) , ( x − x ) + ( x − x ) = 1 , ( x − x ) − ( x − x ) = 2( x − x )( x − x ) , (cid:0) ( x − x ) − ( x − x ) (cid:1) + (cid:0) ( x − x ) − ( x − x ) (cid:1) = 1 , (cid:0) ( x − x ) − ( x − x ) (cid:1) − (cid:0) ( x − x ) − ( x − x ) (cid:1) == 2 (cid:0) ( x − x ) − ( x − x ) (cid:1) (cid:0) ( x − x ) − ( x − x ) (cid:1) . (7)Hoping that these rules are exhaustive, we can start to analyse the rules. For n = 1, therelations does not impose any conditions on the exchange of x and x , i.e. the Galoisgroup is the full permutation group P . However, for n = 2 the first line in Eq. (7) alreadycontains a restriction, as the pairs ( x , x ) and ( x , x ) of independently exchangeable zeroscan be only exchanged in common. While the first relation in the second line does notimpose a condition on the exchange of the pairs, in the last relation this exchange will5hange the sign of the left hand side while the sign is kept on the right hand side. Asinstead of x and x one can use x = − x and/or x = − x as well, the exchange of thepairs ( x , x ) and ( x , x ) will be possible only if either the elements of the first pair or the elements of the second pair are exchanged. As this last condition which turns out to be central for our Galois group occurs as the lastcondition in each of the degrees in nested form, the nestedness of the relations suggests animprovement of the notation. Denoting the two (compound) objects of the last relation by x ( n − and x ( n − , one has x ( n )0 := ( x ( n − ) − ( x ( n − ) = 2 x ( n − x ( n − . (8)As mentioned before, x ( n )0 is a root of the next lowest degree. Accordingly, this root can beidentified with the pair ( x ( n − , x ( n − ), and the interchange is encoded by a sign convention, x ( n )0 = ( x ( n − , x ( n − ) or x ( n )0 = ( − x ( n − , − x ( n − ) , − x ( n )0 = ( x ( n − , − x ( n − ) or − x ( n )0 = ( − x ( n − , x ( n − ) , (9)where x ( n − := ( x ( n − ) − ( x ( n − ) , x ( n − := ( x ( n − ) − ( x ( n − ) . (10)For a given degree n the recursion ends up with x (1)0 = x , − x (1)0 = x , x (1)1 = x , − x (1)1 = x , . . . (11)A general algorithm x ( k +1) i = ( x ( k )2 i ) − ( x ( k )2 i +1 ) → ( x ( k )2 i , x ( k )2 i +1 ) or ( − x ( k )2 i , − x ( k )2 i +1 ) , − x ( k +1) i = − ( x ( k )2 i ) + ( x ( k )2 i +1 ) → ( x ( k )2 i +1 , − x ( k )2 i ) or ( − x ( k )2 i +1 , x ( k )2 i ) ,x (1) i → ( x i , x i +1 ) , − x (1) i → ( x i +1 , x i ) (12)allows to determine the elements of the group by following the branchings by the word“or” in Eq. (12). In detail: 6 = x (2)0 → ( x (1)0 , x (1)1 ) → ( x , x , x , x ) = I → ( − x (1)0 , − x (1)1 ) → ( x , x , x , x ) = A − x (2)0 → ( x (1)1 , − x (1)0 ) → ( x , x , x , x ) = B → ( − x (1)1 , x (1)0 ) → ( x , x , x , x ) = C. (13)The group structure is not the one of the Klein four-group containing the cyclic groupsgenerated by the non-unit three elements as proper subgroups. Instead, this group containsonly a single proper subgroup, namely the cyclic group generated by the element A . n = x (3)0 → ( x (2)0 , x (2)1 ) → ( x (1)0 , x (1)1 , x (1)2 , x (1)3 ) → ( x , x , x , x , x , x , x , x ) → ( x (1)0 , x (1)1 , − x (1)2 , − x (1)3 ) → ( x , x , x , x , x , x , x , x ) → ( − x (1)0 , − x (1)1 , x (1)2 , x (1)3 ) → ( x , x , x , x , x , x , x , x ) → ( − x (1)0 , − x (1)1 , − x (1)2 , − x (1)3 ) → ( x , x , x , x , x , x , x , x ) → ( − x (2)0 , − x (2)1 ) → ( x (1)1 , − x (1)0 , x (1)3 , − x (1)2 ) → ( x , x , x , x , x , x , x , x ) → ( x (1)1 , − x (1)0 , − x (1)3 , x (1)2 ) → ( x , x , x , x , x , x , x , x ) → ( − x (1)1 , x (1)0 , x (1)3 , − x (1)2 ) → ( x , x , x , x , x , x , x , x ) → ( − x (1)1 , x (1)0 , − x (1)3 , x (1)2 ) → ( x , x , x , x , x , x , x , x ) − x (3)0 → ( x (2)1 , − x (2)0 ) → ( x (1)2 , x (1)3 , x (1)1 , − x (1)0 ) → ( x , x , x , x , x , x , x , x ) → ( x (1)2 , x (1)3 , − x (1)1 , x (1)0 ) → ( x , x , x , x , x , x , x , x ) → ( − x (1)2 , − x (1)3 , x (1)1 , − x (1)0 ) → ( x , x , x , x , x , x , x , x ) → ( − x (1)2 , − x (1)3 , − x (1)1 , x (1)0 ) → ( x , x , x , x , x , x , x , x ) → ( − x (2)1 , x (2)0 ) → ( x (1)3 , − x (1)2 , x (1)0 , x (1)1 ) → ( x , x , x , x , x , x , x , x ) → ( x (1)3 , − x (1)2 , − x (1)0 , − x (1)1 ) → ( x , x , x , x , x , x , x , x ) → ( − x (1)3 , x (1)2 , x (1)0 , x (1)1 ) → ( x , x , x , x , x , x , x , x ) → ( − x (1)3 , x (1)2 , − x (1)0 , − x (1)1 ) → ( x , x , x , x , x , x , x , x ) (14)7 .2 Multiplication tables Based on the algorithm explained before, for each degree n the elements of the Galois groupcan be calculated and the group structure can be analysed by considering a multiplicationtable. For n = 1 we obtain the Klein-like four-group V , = I A B CA I C BB C A IC B I A (15)By a systematic procedure, the subgroups of a group represented by a multiplication tablecan be determined. As mentioned before, the group V , has a single subgroup, namely thecyclic group Z generated by the element A , and this subgroup is normal. One obtains { I } ⊳ S = { I, A } ⊳ V , = { I, A, B, C } . (16)8or n = 3 one obtains 16 elements, and the multiplication table reads I A B C D E F G H J K L M N P QA I C B E D G F J H L K N M Q PB C I A F G D E K L H J P Q M NC B A I G F E D L K J H Q P N MD E F G C B A I N M Q P K L H JE D G F B C I A M N P Q L K J HF G D E A I C B Q P N M H J K LG F E D I A B C P Q M N J H L KH K J L N Q M P D F E G A C I BJ L H K M P N Q E G D F I B A CK H L J Q N P M F D G E C A B IL J K H P M Q N G E F D B I C AM P N Q K H L J B I C A D F E GN Q M P L J K H C A B I E G D FP M Q N H K J L I B A C F D G EQ N P M J L H K A C I B G E F D (17)The subgroups are no longer in a one-dimensional sequence but constitute a network whichis shown in Fig. 1. All subgroups are normal subgroups of the previous ones. V , The algorithm finds its manifestation in the group V , which is central in the Galois groupsof Chebychev polynomials of degree n ≥
1. A representation in terms of 2 × In order to avoid confusions, for ordinary elements I omitted the capital letters I and O which couldbe mixed up with the unit elements of the multiplicative and additive group, respectively. ,4 2,24 (1,A,B,C,D,E,F,G) = V(1,A,B,C) = V (1,C) = Z (1,C,D,G,H,L,N,P) = V (1,C,D,G,J,K,M,Q) = V(1,C,E,F) = V (1,C,D,G) = V(1,B) = Z (1) = Z(1,A,B,C,D,E,F,G,H,J,K,L,M,N,P,Q) = V (1,A) = Z Figure 1: Galois group of T (Φ) and (normal) subgroups. Ideals marked in green.the algorithm. Using the notation given for n = 2, one obtains I = ! , A = − − ! , B = − ! , C = −
11 0 ! . (18)As A = − I and C = − B , both { I, A } and { B, C } are images of mappings of the cyclicgroup Z . But while { I, A } is a subgroup of the central group V , , { B, C } is a coset.Because of this, only the first one can be considered as (reducible) representation of Z . Observables on the chaotic chain for vanishing coupling are determined by an infinitelyinterated Chebyshev map, and a smooth transition to an infinitely iterated map for in-creasing coupling is investigated in detail in Refs. [23, 24]. An insight into the behaviour ofsuch maps can be obtained by considering iterations of high but finite degree. Such a finitedegree can be taken as regularisation in the framework of a renormalisation approach, asthe limit n → ∞ is well understood by now and under control [26]. The zeros charac-terising the Galois group of the high-degree Chebyshev polynomial are traded to zeros ofthe highly iterated map of a coupled chaotic chain. The zeros define a partition of the Note that in case of n = 3, he elements I , A , B and C constitute the Klein four-group. n max = 6 and a = 0 . − , If the coupling increases, starting from a Chebyshev polynomial of high degree 2 n max , theiterated map of the same iterative degree n max to the base N = 2 will be deformed smoothly.Starting with the three maxima at Φ = ± − n ≥ πt/
2) with t = ± m/ n − , m = 0 , , . . . n − . The situation is shownin Fig. 2 for the iterative degree n max = 6 and for the coupling a = 0 . n = 4 with eight maxima, the sixteen zeros surrounding these maxima meet and vanishat different values of the coupling a . Using the same classification as for the zeros of theunperturbed Chebyshev polynomial in the previous section, the maxima between the zeros x , x , x , x disappear at a = 0 . × − , between x , x , x , x at a = 0 . × − ,between x , x , x , x at a = 0 . × − , and between x , x , x , x at a = 0 . × − .The zeros will vanish slightly earlier (i.e. for lower values of a ) but in the same order. Inparallel to this, the structure of the Galois group will change. Passing these thresholds, rational relations in (7) including disappearing zeros will beskipped, leading to a smaller set of relations. In case of n = 4, if x , x , x and x disappear, the disappearence of x + x = x + x = 0 and x + x = 1 has no consequenceson the remaining zeros. However, in skipping also x − x = 2 x x and x − x = 2 x x ,the zeros x , x , x and x will obtain an additional degree of freedom, as there are nolonger restrictions to the interchange of these zeros. Therefore, the four zeros are subjectto the permutative group P while the remaining zeros x , . . . , x obey the relations of thedegree n = 3. One obtains the transition V , , , → P ⊗ V , . Passing the next thresholdwhere x , x , x and x disappear, the group structure will be totally resolved to P ⊗ P ,as these four zeros amount to half of the related zeros. Finally, at the last two thresholdsthe freely permutating zeros will be removed in turn. Therefore, the cascade process ofdegradation for the Galois group induced by the changes of the topology is given by V , , , → P ⊗ V , → P ⊗ P → P → . (19)This is only the first step in analysing the topological changes in the group structure.Further considerations have to and will follow in future publications. By convention, the maxima at Φ = ± .3 Generalisation of the Galois group In applying results of the previous section to chaotic chains we have lost sight of theGalois group as the subgroup of permutations of the zeros of a polynomial, leaving rationalrelations between these zeros invariant. Neither the zeros are related by rational relationsnor the iterated coupled map is a polynomial. The only statement we can give is thatboth holds in the limit a →
0. Due to this, a mathematical modelling for a generalisationof the Galois group applicable to chaotic chains has to use a wider understanding of theconcepts of both polynomial and rationality, relating both to the definition in the limitingcase a = 0, as we have done it intuitively in this section. The hope for obtaining suchkind of generalisation is norished by the fact that in increasing the coupling, zeros willdisappear but certainly not (re)appear. Therefore, this limit is always possible to perform. By keeping into touch with the uncoupled case, i.e. by performing the limit a → N = 2. Similar considerations can be performed for N = 3. This will be subject for a future publication. In order to see the large picture of thecascade decay of degradation for the Galois group for rising coupling, a much faster code hasto be written to analyse the subgroup structure for a given degree n . Based on this, there ishope that similar renormalisation methods as those developed in Ref. [26] can be found alsofor the Galois group, enabling one to perform the limit n → ∞ and seeing something like acontinuum limit for group and algebra. As graded Z N algebras have relations to space-timemetric and tetrads just in case of N = 2 and N = 3 [33, 34, 35, 36, 37, 38, 39, 40], the samecases for which chaotic chains are non-trivial, a relation of chaotic chains to space-time viageneralised Galois groups is expected to exist and will be investigated in the future.13 cknowledgements The author acknowledges fruitful discussions about this subject with C. Beck, R. Kerner,V. Abramov and M. Menert. The work on this subject is supported by the EstonianScience Foundation via the Centre of Excellence TK133 “Dark Side of the Universe”.
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