Continued fraction representation of temporal multiscaling in turbulence

It was shown recently that the anomalous scaling of simultaneous correlatio n functions in turbulence is intimately related to the breaking of temporal scale invariance, which is equivalent to the appearance of infinitely many times scales in the time-dependence of time-correlation functions. In this paper we derive a continued fraction representation of turbulent time corr elation functions which is exact and in which the multiplicity of time scal es is explicit. We demonstrate that this form yields precisely the same sca ling laws for time derivatives and time integrals as the "multi-fractal" re presentation that was used before. Truncating the continued fraction repres entation yields the ''best'' estimates of time correlation functions if the given information is limited to the scaling exponents of the simultaneous correlation functions up to a certain, finite order. It is worth noting tha t the derivation of a continued fraction representation obtained here for a time evolution operator which is not Hermitian or anti-Hermitian may be of independent interest. [S1063-651X(99)05511-7].