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Concerning the spectral properties of the evolution operator, one gets precise N information on the semiclassical behavior of the eigenvalues e i j , 1 u j u N 1 , N and the corresponding eigenfunctions i j , 1 u j u N 1 , of the unitary operators M ( ) representative of automorphisms : T2 T2 Proposition 162 [[211]] Let a hyperbolic automorphism of the torus Then the eigenvalues of the unitary representative M ( ) are Lebesgue uniformly distributed on the unit circle at the semiclassical limit: lim # { j, N [ 0 , 0 + ]} j N = [ 0 , 0 + ] , (1651)



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The applications program didn t have to change for different hardware! To make the concept of portability a reality, however, a high-level programming language was needed to implement the software needed To that end, Brian Kernighan and Dennis Ritchie created the C programming language In 1973, UNIX was rewritten in C Today, C is still the primary language used to create the UNIX (and Linux) operating system kernels As Ritchie went on to say in his 1980 lecture: Today, the only important UNIX program still written in assembler is the assembler itself; virtually all the utility programs are in C, and so are most of the applications programs, although there are sites with many in Fortran, Pascal, and Algol 68 as well.





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where is the Lebesgue measure This result also holds for the Hamiltonian perturbations of Finally, let us end this chapter by formulating the essential equipartition result in regard to the initial purpose to examine ergodicity on the torus on a semiclassical

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level This amounts to examining the possible limits at N of the matrix elements (and particularly the lower symbols) N |Op k

of the quantized versions of observables for both types of quantizations Theorem 163 Suppose that for all smooth observable f on the torus, that is, f C (T2 ), for all k N, there exists C > 0 so that for all N M ( ) k Op

It seems certain that much of the success of UNIX follows from the readability, modifiability, and portability of its software that in turn follows from its expression in high-level languages If you are a Linux enthusiast and are interested in what features from the early days of Linux have survived, an interesting read is Dennis Ritchie s reprint of the first UNIX programmer s manual (dated November 3, 1971) You can find it at Dennis Ritchie s Web site: http://cmbell-labscom/cm/cs/who/dmr/1stEdmanhtml..

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Here is an area-preserving map on the torus such that N [0, 2 a [0, 2 and the unitary representative M ( ) satis es (1652) b Write M ( ) N = N N j j j for the eigenvalues and eigenfunctions of M ( ) Then there exists a set of indices E (N ) [1, N ] satisfying lim

that, for all functions f C (T ) and for all maps J : N N J (N ) E (N ), we have the ergodicity property of the lower symbols: lim N (N ) Op J

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This is an extension to the sphere S 2 and to the one-sheeted hyperboloid in R3 , the 1 + 1 de Sitter space-time, of the quantization of the unit circle as was described in 14 We show explicitly how the coherent state quantization of these manifolds leads to fuzzy or noncommutative geometries The quantization of the sphere rests upon the unitary irreducible representations D j of SO(3) or SU (2), j N/2 The quantization of the hyperboloid is carried out with unitary irreducible 1 representations of SO 0 (1, 2) or SU (1, 1) in the principal series, namely, U 2 +i , R The limit at in nite values of the representation parameters j and , restores commutativity for the geometries considered It should be stressed that we proceed to a noncommutative reading of a given geometry and not of a given dynamical system This means that we do not consider in our approach any time parameter and related evolution

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