Embedding the torus automorphisms to Hamiltonian flows

Chaos, Solitons and Fractals 12 (2001) 2815±2819

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Embedding the torus automorphisms to Hamiltonian ¯ows q P. Akritas a,b, I. Antoniou a,b,*, G.P. Pronko a,c a

International Solvay Institutes for Physics and Chemistry, CP-231, Campus Plaine ULB, Bd. du Triomphe, 1050 Brussels, Belgium b Theoretische Natuurkunde, Free University of Brussels, Brussels, Belgium c Institute for High Energy Physics, Protvino, 142284 Moscow Region, Russia

Abstract We study two Hamiltonian ¯ows, one resulting from the embedding of the torus automorphisms into R2 and the other restricted on the torus. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Consider the cascade S n ; n 2 Z on the phase space Y , where Y is a standard Lebesque space [1]. Embedding the cascade …S n ; Y † to the ¯ow …Rt ; C†, where C is also a standard Lebesque space means that 1. Rtˆ1 is invariant on a subset Y~ of C which is isomorphic to Y . 2. The cascades …S n ; Y † and …Rtˆ1 ; Y~ † are equivalent. Embedding means therefore that the cascade S n is e€ectively a stroboscopic observation of the ¯ow Rt . The embedding is called [2] restricted if the original space Y is retained, i.e. Y  C and unrestricted if the original space Y is enlarged Y  C. The aim of this paper is to study the Hamiltonian ¯ows which result from the restricted and unrestricted embedding into R2 of the 2-torus automorphisms. Torus automorphism is just a stroboscopic view of the ¯ows at integer multiples of a ®xed period T . In Section 2 we present the torus automorphism and its explicit solution. In Section 3 we discuss the unrestricted embedding of the torus automorphism on R2 and in Section 4 the restricted embedding. We conclude with some remarks on the geodesic ¯ow on the Lobachevski plane, integrability, embeddability and quantization. 2. The torus automorphisms We consider automorphisms of the 2-torus Y ˆ ‰0; 1†  ‰0; 1† given by the formula:     xn xn‡1 ˆA …mod 1†; n 2 Z; S:Y !Y : yn‡1 yn where

 Aˆ

a c

b d

…1†

 …2†

The matrix A belongs to the group SL…2; Z†, i.e. a; b; c; d, are integers and det A ˆ 1:

q *

Taken from the Euro±Russia collaboration project on complexity ESPRIT PROJECT CTIAC 21042. Corresponding author. Tel.: +32-2-650-55-33; fax: +32-2-650-50-28. E-mail address: [email protected] (I. Antoniou).

0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 1 ) 0 0 0 9 5 - 9

…3†

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These area preserving maps result usually as Poincare sections or stroboscopic observations of Hamiltonian systems and they are well-known prototypes for the study of integrability and chaos. The matrix   1 1 Aˆ …4† 1 2 de®nes the Cat map [3] which is a well-known prototype for chaotic behavior due to hyperbolic structure. In fact it is a Bernoulli system in the bottom of the ergodic hierarchy [1,3] with positive Kolmogorov±Sinai entropy equal to its Lyapunov exponent 0.962. We mention in passing that the spectral decomposition of the evolution operators of the Cat map was studied in [4,5]. The explicit solution of the torus automorphism (1) is [6]: xn ˆ yn ˆ

x 0 zn y 0 zn

1 1

‡ …ax0 ‡ by0 †zn mod 1; ‡ …cx0 ‡ dy0 †zn mod 1:

…5†

where zn ˆ

…nX 1†=2 mˆ0

n! …2m ‡ 1†!…n 2m

1 n r 1†! 2n 1

2m 1

…r2

4†m ;

…6†

and r ˆ trace A ˆ a ‡ d:

…7†

The torus automorphisms (1) are actually Hamiltonian systems with discrete time evolution …xn ; yn † ! …xn‡1 ; yn‡1 †, the subscript n represents time. The variables xn ; yn are canonical with symplectic form x ˆ dxn ^ dyn ;

…8†

which is conserved during the evolution (1). It is easy to check that the map (1) has the integral of motion
H ˆ 12 cx2n …a d†xn yn byn2 ;

…9†

where a; b; c; d are the elements of A.

3. The Hamiltonian ¯ow on R2 Let us consider ®rst the ¯ow on the phase space R2 generated by the Hamiltonian (9)
H ˆ 12 cx2 …a d†xy by 2 :

…10†

As time is continuous we omitted the subscript n in the variables xn ; yn considering them as the time-dependent functions x ˆ x…t†; y ˆ y…t†. The Poisson bracket, corresponding to the symplectic form (8) is fx; yg ˆ 1:

…11†

The equations of the ¯ow, generated by the Hamiltonian (10) are x_ ˆ

a

d 2

y_ ˆ cx

x ‡ by;

a

d 2

…12†

y:

Note that Eq. (12), in contrast to (1), is linear with constant coecients. Therefore the solution is straightforward:     
1 a d a d b tk ‡ k etk k e tk ‡ y…0† e e tk ; x…t† ˆ x…0† 2k 2 2 2k …13†     
1 a d a d c tk tk tk tk e ; y…t† ˆ y…0† ‡ x…0† k e ‡k e e 2k 2 2 2k

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where x…0†; y…0† are the initial data and k is given by 1 k ˆ …r2 2

4†1=2 :

…14†

In order to establish that the torus automorphism (1) is a stroboscopic observation of the Hamiltonian ¯ow (12), the solution (5) should correspond to the solution (13) as follows: xn ˆ x…nT †; yn ˆ y…nT †;

…15†

where T is the period of observation. Let us rewrite Eq. (13) as follows:   sht k a ‡ d sht k ‡ x…0† ‡ chtk ; x…t† ˆ …ax…0† ‡ by…0†† k 2 k   sht k a ‡ d sht k ‡ y…0† ‡ chtk : y…t† ˆ …cx…0† ‡ dy…0†† k 2 k

…16†

Consider the function /…t† ˆ sht k=k. Then /…nT † should be identi®ed with zn (6): zn ˆ /…nT †:

…17†

The term zn (6) satis®es the equation p zn‡1 ‡ zn 1 ˆ zn k2 ‡ 1:

…18†

Then we have /……n ‡ 1†T † ‡ /……n Therefore ch T k ˆ

1†T † ˆ

sh …nT k† ch T k: k

p k2 ‡ 1:

…19†

…20†

By transforming (20) we see that T is uniquely determined by the equation sh kT ˆ k:

…21†

The phase portrait of the ¯ow (13) is shown in Fig. 1. As we can see the generic type of the orbits is the hyberbola. The two separatrixes represent special orbits and the last special orbit is a ®xed point, which in this case is the origin.

Fig. 1. The phase space of the ¯ow described by (13). Horizontal axis is x…t† and vertical is y…t†.

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4. The Hamiltonian ¯ow on the torus Consider the rectangular lattice with the unit square associated with the torus automorphism located at the origin. Each point …x; y† on R2 is speci®ed by the unit cell …m; n† and the coordinates …~x; y~† inside this unit cell. To put the solution (13) on the torus, let us introduce on the plane R2 a coordinate system based on the unit square. Any trajectory …x…t†; y…t†† (13) on the plane may be rewritten in terms of the new coordinates without losing smoothness. The variables u…t† ˆ e2pix…t† ˆ e2pi~x…t† ;

…22†

~ ; v…t† ˆ e2piy…t† ˆ e2piy…t†

where x…t† and y…t† given by (13), actually correspond to the factorization of R2 by the unit cell. The Hamiltonian written in terms of the new variables f…~x; y~†; …m; n†g does depend however on …m; n†. So for conservation of energy we need these ``hidden'', from the point of view of torus variables …m; n†, which enter also the equations of motion for the variables ~x…t†; y~…t†. In other words we can say that the Hamiltonian (10) is a multi-valued function of the cell variables ~x…t†; y~…t† in the following sense. H is conserved during some period of time, then jumps as the orbit enters another cell, and again is conserved, etc. In fact there is no function of f~x; y~g which is conserved globally. Indeed, during some period, when H is conserved this new integral should be a function f …H † of H (10) as H is conserved. For global conservation the function f …H † should compensate all jumps of the Hamiltonian, therefore H should be periodic with ®nite period. This is of course impossible, because these jumps are not rationally dependent, therefore f …H † cannot be periodic. Although the Hamiltonian ¯ow (13) is integrable, as it has the integral of motion (10), the factorized system with respect to the unit square is a non-integrable time-dependent Hamiltonian system because expression (10) for the Hamiltonian becomes a multi-valued function and the di€erent values correspond to invariant subspaces only for ®nite durations. In fact as the ¯ow on the torus is a restricted embedding of the Cat map, we conclude following [1] that, as the Cat cascade is a Kolmogorov system [3] so the Cat ¯ow on the torus is a Kolmogorov ¯ow.

5. Concluding remarks 1. Although the Hamiltonian (10) is known [7], the Hamiltonian ¯ows on R2 and on the torus have not been studied to our knowledge. 2. The Cat ¯ow on the torus is of the same type with the geodesic ¯ow on the compacti®ed Lobachevski plane [8]. Indeed let us use the unit disk model of the Lobachevski plane [9]: D ˆ f…x; y† 2 R2 : jxj2 ‡ jyj2 < 1g;

ds2 ˆ

4…dx2 ‡ dy 2 † …1

x2

y 2 †2

:

…23†

The geodesic ¯ow on D corresponds to the Hamiltonian system with Hamiltonian H ˆ 18…1

x2

y 2 †2 …px2 ‡ py2 †:

…24†

This Hamiltonian system is integrable in the sense of Liouville [10]. The second integral of motion is the angular momentum J ˆ ypx

xpy :

…25†

There exists [9] a discrete subgroup C of the group G of isometries of D (G is isomorphic to SL…2; R†) such that the Riemannian manifold [8] M ˆ D=C is di€eomorphic to the sphere with two handles and has constant Gauss curvature equal to 1. The geodesic ¯ow on M is not however a Hamiltonian system because the Hamiltonian (24) is not compatible with the action of the subgroup C. Therefore expression (24) de®nes a multi-valued function on the phase space (the cotangent bundle T  …M† of M). According to [7] the restriction of the geodesic ¯ow on M to the unit cotangent bundle T  …M† of M is a K-¯ow (with respect to the natural measure, generated by the Riemannian structure). However this K-¯ow is a non-integrable time-dependent Hamiltonian system, as is also the case of the ¯ow on the torus associated to the Cat map. 3. We saw that the Hamiltonian ¯ows (10) on the torus and (24) on the compacti®ed Lobachevski plane are speci®c examples of non-integrable time-dependent Hamiltonian systems which are Kolmogorov systems. We were surprised

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not to have seen any other example of non-integrable, in the sense of Liouville, Hamiltonian system in the literature except of the exotic example [11]: H ˆ Re ‰…p1 ‡ iq2 †2

…q1 ‡ ip2 †5

1Š:

…26†

4. Contrary to the Cat automorphism, the Baker automorphism on the torus: B:Y !Y : xn‡1 ˆ 2xn …mod 1†;  yn =2; yn‡1 ˆ …yn ‡ 1†=2;

0 6 xn < 1=2; 1=2 6 xn < 1

is not embeddable to a ¯ow because of the existence of ®nite cycles [12]. 5. The quantization of the Hamiltonian ¯ow (9) on the plane R2 gives the operators X ; Y with ‰X ; Y Š ˆ

iI:

…27†

The Hamiltonian operator has the same form with the classical case (9). However as the representation of the canonical commutation relation (23) is di€erent from the representation of the Quantum Cat map there is no direct way to relate the ¯ow with the cascade as we have done in the classical one. The quantization of the torus automorphisms is discussed in [6,7].

Acknowledgements We would like to thank Prof. Ilya Prigogine for his interest and support, and Prof. Shkarin, Prof. S. Ichtiaroglou and Mr. G. Pavliotis for helpful discussions which contributed to the clari®cation of questions addressed in the paper. The ®nancial support of the Belgian Government through the Interuniversity Attraction Poles and the National Lotterie is gratefully aknowledged. References [1] Cornfeld I, Fomin S, Sinai Ya. Ergodic theory. Berlin: Springer; 1982. [2] Foland N, Utz W. The embedding of discrete ¯ows in continuous ¯ows. In: Wright F, editor. Ergodic Theory Symposium, New Orleans, 1961. New York: Academic Press; 1963. p. 121±34. [3] Arlond VI, Avez A. Ergodic problems of classical mechanics. New York: Benjamin; 1968. [4] Antoniou I, Qiao B, Suchanecki Z. Generalized spectral decomposition and intrinsic irreversibility of the Arnold Cat map. Chaos, Solitons & Fractals 1997;8:77±90. [5] Antoniou I, Qiao B, Suchanecki Z. Generalized spectral decomposition and intrinsic irreversibility of the Arnold Cat map. Chaos, Solitons & Fractals 2000;11:1475±7. [6] Akritas P, Antoniou IE, Pronko G. On the torus automorphisms: analytic solutions, computability and quantization. Chaos, Solitons & Fractals 2001;12(14±15):2805±14. [7] Keating JP. Nonlinearity 1991;4:309±61. [8] Anosov D. Geodesic ¯ows on closed Riemanism manifolds with negative curvature. In: Proc. Steklov Math. Inst., 1967, Moscow, vol. 10. Providence, RJ: American Mathematical Society; 1969. [9] Dubrovin B, Fomenko A, Novikov S. Modern geometry: methods and applications, II The geometry and topology of manifolds. Berlin: Springer; 1985. [10] Arnold VI. Mathematical methods of classical mechanics. Berlin: Springer; 1978. [11] Zakharov V, editor. What is integrability? Berlin: Springer; 1991. [12] Schweiser B, Sklar A. Found Phys 1990;20:873±9.