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Extra info for Aspects of mathematics and its applications, no TOC and Index
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 . 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].