Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
P. Gaspard and X.-J. Wang, Sporadicity: between periodic and chaotic dynamical behaviors, Proc. Nat. Acad. Sci. USA 85:4591–4595 (1988).
X.-J. Wang, Statistical physics of temporal intermittency, Phys. Rev. A 40:6647 (1989).
P. Collet, A. Galves, and B. Schmitt, Unpredictability of the occurence time of a long laminar period in a model of temporal intermittency, Ann. Inst. Poincaré 57:319–331 (1992).
A. Lambert, S. Siboni, and S. Vaienti, Statistical properties of a non-uniformly hyperbolic map of the interval, J. Stat. Phys. 72:1305–1330 (1993).
M. Mori, On the intermittency of a piecewise linear map, Tokyo J. Math. 16:411–428 (1993).
T. Prellberg, Ph.D. thesis (Virginia Poly. Inst. and State Univ., 1991).
T. Prellberg and J. Slawny, Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions, J. Stat. Phys. 66:503–514 (1992).
A. O. Lopes, The zeta function, non-differentiability of the pressure and the critical exponent of transition, Adv. in Math. 101:133–165 (1993).
Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys. 74:189–197 (1980).
J. Aaronson, The asymptotic distributional behaviour of transformations preserving infinite measures, J. d'Analyse Mathématique 39:203–234 (1981).
M. Campanino and S. Isola, Infinite invariant measures for non-uniformly expanding transformations of [0, 1]: weak law of large numbers with anomalous scaling, Forum Mathem. 8:71–92 (1996).
M. Thaler, Transformations on [0, 1] with infinite invariant measures, Israel J. Math. 46:67–96 (1983).
S. Isola, On the rate of convergence to equilibrium for countable ergodic Markov chains, 1997 preprint.
S. Isola, Dynamical zeta functions and correlation functions for intermittent interval maps, 1997 preprint.
C. Liverani, B. Saussol, and S. Vaienti, A probabilistic approach to intermittency, 1997 preprint.
L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110:153–188 (1999).
M. E. Fisher, The theory of condensation and the critical point, Physics 3:255–283 (1967).
G. Gallavotti, Funzioni zeta e insiemi basilari, Accad. Lincei Rend. Sc. fis. mat. e nat. 61:309–317 (1976).
F. Hofbauer, Examples for the non-uniqueness of the equilibrium states, Trans. AMS 228:133–141 (1977).
J. F. C. Kingman, Regenerative Phenomena (John Wiley, 1972).
H. Hu and L.-S. Young, Nonexistence of SRB measures for some systems that are “almost Anosov,”' Erg. Th. Dyn. Syst. 15:67–76 (1995).
P. Erdoős, W. Feller, and H. Pollard, A property of power series with positive coefficients, Bull. AMS 55:201–204 (1949).
G.H. Hardy, Divergent Series (Oxford, 1949).
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2 (Wiley, New York, 1970).
W. Feller, Fluctuation theory of recurrent events, TAMS 67:99–119 (1949).
P. Dienes, The Taylor Series (Dover, New York, 1957).
N. G. de Bruijn, Asymptotic Methods in Analysis (Dover, New York, 1981).
B. Hu and J. Rudnick, Exact solutions to the Feigenbaum renormalization-group equations for intermittency, Phys. Rev. Lett. 48:1645–1648 (1982).
M. A. Evgrafov, Analytic Functions (Dover, New York, 1966).
D. Ruelle, Thermodynamic Formalism (Addison-Wesley, 1978).
P. Dahlqvist, Do zeta functions for intermittent maps have branch points?, 1997 preprint.
H. H. Rugh, Intermittency and Regularized Fredholm Determinants, Invent. Math. 135:1–24 (1999).
K. L. Chung, Markov Chains with Stationary Transition Probabilities (Springer, 1967).
J. Aaronson, An Introduction to Infinite Ergodic Theory (AMS, 1997).
B. A. Sevast'yanov, Renewal theory, J. Soviet Math. 4(3) (1975).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Isola, S. Renewal Sequences and Intermittency. Journal of Statistical Physics 97, 263–280 (1999). https://doi.org/10.1023/A:1004623303471
Issue Date:
DOI: https://doi.org/10.1023/A:1004623303471