By V. I Arnold, A. Avez
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21) we deduce: 93 (see I\ppen- 45 ERGODIC PROPERTIES n=oo The sequence ¢-1{3, ¢-1{3 V ¢-2{3, ... 9) we get, h (¢) 2: h ({31 V ¢k k<-1 (3) . Now, it is sufficient to show thath ({31 Vk <-1 ¢k(3) > O. 6); {3 ~ V ¢k{3 (mod 0) . k~-1 Consequently: that is which. 32). REMARK (Q. E. 33 Guirsanov [1] has constructed a nonclassical dynamical system with zero entropy and denumerably infinite Lebesgue spectrum. Hence it i~ not a K-system. Gourevitch [1] proved that the horocyc1ic flow on a compact surface with constant negative curvature has denumerably infinite Lebesgue spectrum and zero entropy.
L 2 (M. ). This is equivalent to: lim n+ oo for every I. ~ < Un I I ~ > = 0 orthogonal to 1. It is sufficient to prove this when I and ~ are basis vectors. l/ J . k ,r > which is null for n large enough. 5 The automorphism if> (x. y) = (x+ y. x+ 2y)(mod 1) of the torus M = I (x. 1). Then. 4). 6 The Bernouillischemes have countable Lebes~ue spectrum. In par- ticular they are spectrally equivalent. Prool: We prove it for B(~. • Pn) UI to minor modifications. 11. 31 ERGODIC PROPERTIES The function 1 and the function j -1 Yn(x) = if x = 0 if x = 1 1+ 1 form an· orthonormal basis of the space L 2(Z2.
The differential ¢ * is dilating in the direction X and is contracting in the direction Y. To be precise, let Xm and Ym be the subs paces of TM m, respectively parallel to X and Y. Then: II¢*(II ? A1 '1I(1I if (( X m • ( \ > 1) ; II¢"(II ::; A2 '11(11 if (( Ym , (0 < A2 < 1) This is a characteristic example of the C-systems that we define next. 3 Let ¢ be a C 2 -differentiable diffeomorphism of a compact, connected, smooth manifold M. We denote the differential of ¢ by ¢*. (M, ¢) is called a C-diffeomorphism if there exist two fi~/ds of tangent planes Xm and Ym 'such that: (1) TM m falls into the direct sum of X m and Y : m TM m =Xm mY, dimX m =kIO, dimYm m (2) = 110.