Renewal Sequences and Intermittency

In this paper we examine the generating function Φ(z) of a renewal sequence arising from the distribution of return times in the “turbulent” region for a class of piecewise affine interval maps introduced by Gaspard and Wang and studied by several authors. We prove that it admits a meromorphic continuation to the entire complex z-plane with a branch cut along the ray (1, +∞). Moreover, we compute the asymptotic behavior of the coefficients of its Taylor expansion at z=0. From this, we obtain the exact polynomial asympotics for the rate of mixing when the invariant measure is finite and of the scaling rate when it is infinite.

[1]  Kai Lai Chung,et al.  Markov Chains with Stationary Transition Probabilities , 1961 .

[2]  P. Gaspard,et al.  Sporadicity: Between periodic and chaotic dynamical behaviors. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Jon Aaronson,et al.  An introduction to infinite ergodic theory , 1997 .

[4]  Wang.  Statistical physics of temporal intermittency. , 1989, Physical review. A, General physics.

[5]  P. Dienes,et al.  The Taylor Series. , 1932 .

[6]  L. Young.  Recurrence times and rates of mixing , 1999 .

[7]  Y. Pomeau,et al.  Intermittent transition to turbulence in dissipative dynamical systems , 1980 .

[8]  H. H. Rugh.  Intermittency and regularized Fredholm determinants , 1996, chao-dyn/9610011.

[9]  J. Slawny,et al.  Maps of intervals with indifferent fixed points: Thermodynamic formalism and phase transitions , 1992 .

[10]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[11]  J. Aaronson.  The asymptotic distributional behaviour of transformations preserving infinite measures , 1981 .

[12]  Makoto Mori.  ON THE INTERMITTENCY OF A PIECEWISE LINEAR MAP , 1993 .

[13]  Michael E. Fisher,et al.  The theory of condensation and the critical point , 1967 .

[14]  N. D. Bruijn.  Asymptotic methods in analysis , 1958 .

[15]  Infinite invariant measures for non-uniformly expanding transformations of [0,1]: weak law of large numbers with anomalous scaling , 1996 .

[16]  A. Lopes.  The Zeta Function, Non-differentiability of Pressure, and the Critical Exponent of Transition , 1993 .

[17]  S. Vaienti,et al.  Statistical properties of a nonuniformly hyperbolic map of the interval , 1993 .

[18]  M. Thaler.  Transformations on [0, 1] with infinite invariant measures , 1983 .

[19]  W. Feller.  Fluctuation theory of recurrent events , 1949 .

[20]  Paul Erdös,et al.  A property of power series with positive coefficients , 1949 .

[21]  C. Liverani,et al.  A probabilistic approach to intermittency , 1999, Ergodic Theory and Dynamical Systems.

[22]  Pierre Collet,et al.  Unpredictability of the occurrence time of a long laminar period in a model of temporal intermittency , 1992 .

[23]  Bambi Hu,et al.  Exact Solutions to the Feigenbaum Renormalization-Group Equations for Intermittency , 1982 .