By Jorge Alberto Barroso

ISBN-10: 0444877274

ISBN-13: 9780444877277

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**Sample text**

The ergodic properties of classic dynamical systems can be conveniently studied in the Hilbert space formulation of dynamics due to Koopman [1,2]. We consider the Hilbert space L2 ¼ L2 ð; S; mÞ of square integrable functions on . The transformations St induce the Koopman evolution operators Vt acting on the functions f 2 L2 as follows: Vt f ðoÞ ¼ f ðSt oÞ; o2 The adjoint operator Ut ¼ Vty is called the Frobenius–Perron operator [2]. As St preserve the measure m, the operators Vt are isometries.

The eigenvalues are & zi ¼ 1 mi ; 1 miþ1 ; i even i odd ð6Þ 40 i. antoniou and z. suchanecki The eigenvectors of the tent maps can be expressed in terms of the Bernoulli and Euler polynomials. For even tent maps m ¼ 2; 4; . . ( ÀÁ Bi 2x ; i ¼ 0; 2; 4; . . fi ðxÞ ¼ iþ1 i ¼ 1; 3; 5; . . 2miþ1 Ei ðxÞ; For odd tent maps m ¼ 3; 5; . . 8 < 1; fi ðxÞ ¼ Ei ðxÞ; : Bi ðxÞ þ EiÀ1 ðxÞ; i¼0 i ¼ 1; 3; 5; . . i ¼ 2; 4; . . The Bernoulli polynomials are defined by the generating function (4). The Euler polynomials are defined by the generating function 1 X 2ext tn ¼ ; E ðxÞ n et þ 1 n¼0 n!

We thus compare the non-Markovian solution jðq; tÞ (30) with the zero-Markovian solution j0 ðq; tÞ (31) and with the full Markovian solution jM ðq; tÞ (32). The former has a simple expression for the standard map in the diffusive regime; using Eq. t 1 1 2 2 2 W0 ðtÞ ¼ ½J0 ðqKÞ % 1 À ðqKÞ % exp À K q t 4 4 t ð35Þ This is precisely of the form of the propagator associated with the diffusion equation (in Fourier representation), with a diffusion coefficient D: W0 ðtÞ % exp½ÀD ð2pqÞ2 t ð36Þ The diffusion coefficient appearing in the zero-Markovian approximation (35) is thus D ¼ DQL ¼ 1 4ð2pÞ2 K2 ð37Þ The latter value is the well-known quasilinear diffusion coefficient [1–7].