Skip to main content
Log in

Distributional Behavior of Time Averages of Non-\(L^1\) Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript


In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of \(L^1(m)\) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-\(L^1(m)\) functions. Here, we provide another distributional behavior of time averages of non-\(L^1(m)\) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-\(L^1(m)\) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-\(L^1(m)\) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-\(L^1(m)\) function in the one-dimensional intermittent maps.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others


  1. Aaronson, J.: The asymptotic distributional behavior of transformations preserving infinite measures. J. D’Anal. Math. 39, 203–234 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aaronson, J.: An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence (1997)

    Book  MATH  Google Scholar 

  3. Aizawa, Y.: Non-stationary chaos revisited from large deviation theory. Prog. Theor. Phys. Suppl. 99, 149–164 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  4. Akimoto, T.: Generalized arcsine law and stable law in an infinite measure dynamical system. J. Stat. Phys. 132, 171–186 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Akimoto, T.: Distributional response to biases in deterministic superdiffusion. Phys. Rev. Lett. 108, 164101 (2012)

  6. Akimoto, T., Aizawa, Y.: New aspects of the correlation functions in non-hyperbolic chaotic systems. J. Korean Phys. Soc. 50, 254–260 (2007)

    Article  Google Scholar 

  7. Akimoto, T., Aizawa, Y.: Subexponential instability in one-dimensional maps implies infinite invariant measure. Chaos 20, 033110 (2010)

  8. Akimoto, T., Barkai, E.: Aging generates regular motions in weakly chaotic systems. Phys. Rev. E 87, 032915 (2013)

  9. Akimoto, T., Hasumi, T., Aizawa, Y.: Characterization of intermittency in renewal processes: application to earthquakes. Phys. Rev. E 81, 031133 (2010)

  10. Akimoto, T., Miyaguchi, T.: Role of infinite invariant measure in deterministic subdiffusion. Phys. Rev. E 82, 030102(R) (2010)

  11. Akimoto, T., Miyaguchi, T.: Distributional ergodicity in stored-energy-driven lévy flights. Phys. Rev. E 87, 062134 (2013)

  12. Akimoto, T., Miyaguchi, T.: Phase diagram in stored-energy-driven lévy flights. J. Stat. Phys. 157, 515–530 (2014)

  13. Barkai, E.: Aging in subdiffusion generated by a deterministic dynamical system. Phys. Rev. Lett. 90, 104101 (2003)

  14. Birkhoff, G.D.: Proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA 17, 656–660 (1931)

    Article  ADS  Google Scholar 

  15. Brokmann, X., et al.: Statistical aging and nonergodicity in the fluorescence of single nanocrystals. Phys. Rev. Lett. 90, 120601 (2003)

  16. Cox, D.R.: Renewal Theory. Methuen, London (1962)

    MATH  Google Scholar 

  17. Darling, D.A., Kac, M.: On occupation times for markov processes. Trans. Am. Math. Soc. 84, 444–458 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dynkin, E.: Some limit theorems for sums of independent random variables with infinite mathematical expectations. In: Selected Translations in Mathematical Statistics and Probability, vol. 1, p. 171. American Mathematical Society, Providence (1961)

  19. Gaspard, P., Wang, X.J.: Sporadicity: between periodic and chaotic dynamical behaviors. Proc. Natl. Acad. Sci. USA 85, 4591–4595 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Geisel, T., Thomae, S.: Anomalous diffusion in intermittent chaotic systems. Phys. Rev. Lett. 52, 1936–1939 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  21. Golding, I., Cox, E.C.: Physcial nature of bacterial cytoplasm. Phys. Rev. Lett. 96, 098102 (2006)

  22. He, Y., Burov, S., Metzler, R., Barkai, E.: Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett. 101, 058101 (2008)

  23. Korabel, N., Barkai, E.: Pesin-type identity for intermittent dynamics with a zero lyapunov exponent. Phys. Rev. Lett. 102, 050601 (2009)

  24. Korabel, N., Barkai, E.: Infinite invariant density determines statistics of time averages for weak chaos. Phys. Rev. Lett. 108, 060604 (2012)

  25. Korabel, N., Barkai, E.: Distributions of time averages for weakly chaotic systems: the role of infinite invariant density. Phys. Rev. E 88, 032114 (2013)

  26. Lamperti, J.: An occupation time theorem for a class of stochastic processes. Trans. Am. Math. Soc. 88, 380–387 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  27. Manneville, P.: Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems. J. Phys. (Paris) 41(11), 1235–1243 (1980)

    Article  MathSciNet  Google Scholar 

  28. Margolin, G., Barkai, E.: Nonergodicity of a time series obeying lévy statistics. J. Stat. Phys. 122, 137–167 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Miyaguchi, T., Akimoto, T.: Intrinsic randomness of transport coefficient in subdiffusion with static disorder. Phys. Rev. E 83, 031926 (2011)

  30. Miyaguchi, T., Akimoto, T.: Ergodic properties of continuous-time random walks: finite-size effects and ensemble dependences. Phys. Rev. E 87, 032130 (2013)

  31. Shinkai, S., Aizawa, Y.: The lempel-ziv complexity in infinite ergodic systems. J. Korean Phys. Soc. 50, 261–266 (2007)

    Article  Google Scholar 

  32. Shlesinger, M., Klafter, J., Wong, Y.: Random walks with infinite spatial and temporal moments. J. Stat. Phys. 27(3), 499–512 (1982)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Tabei, S.A., Burov, S., Kim, H.Y., Kuznetsov, A., Huynh, T., Jureller, J., Philipson, L.H., Dinner, A.R., Scherer, N.F.: Intracellular transport of insulin granules is a subordinated random walk. Proc. Natl. Acad. Sci. USA 110(13), 4911–4916 (2013)

    Article  ADS  Google Scholar 

  34. Thaler, M.: Transformations on [0,1] with infinite invariant measures. Isr. J. Math. 46, 67–96 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  35. Thaler, M.: The dynkin-lamperti arc-sine laws for measure preserving transformations. Trans. Am. Math. Soc. 350, 4593–4607 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  36. Thaler, M.: The asymptotics of the perron-frobenius operator of a class of interval maps preserving infinite measures. Studia Math 143(2), 103–119 (2000)

    MATH  MathSciNet  Google Scholar 

  37. Thaler, M.: A limit theorem for sojourns near indifferent fixed points of one dimensional maps. Ergod. Theory Dyn. Syst. 22, 1289–1312 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. Thaler, M., Zweimüller, R.: Distributional limit theorems in infinite ergodic theory. Probab. Theory Relat. Fields 135, 15–52 (2006)

    Article  MATH  Google Scholar 

  39. Weigel, A., Simon, B., Tamkun, M., Krapf, D.: Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking. Proc. Natl. Acad. Sci. USA 108(16), 6438 (2011)

    Article  ADS  Google Scholar 

Download references


This work was inspired by the conference of “Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics.” We are indebted to T. Miyaguchi and H. Takahashi for his helpful comments. This work was partially supported by Grant-in-Aid for Young Scientists (B) (Grant No. 26800204 to TA) and by the MEXT, Japan (Platform for Dynamic Approaches to Living System; KAKENHI 23115007 to SS).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Takuma Akimoto.


Appendix 1: First Moment

The Laplace transform of \(\langle X_t \rangle \) is given by

$$\begin{aligned} \langle X_s \rangle&= \left. -\frac{\partial {\hat{P}}(k,s)}{\partial k}\right| _{k=0} \nonumber \\&= -\frac{\hat{\psi }'(0,s) \int _0^\infty W(\tau ) e^{-s\tau } d\tau }{ [1 - {\hat{\psi }}(0,s)]^2} + \frac{\int _0^\infty w(\tau ) [\int _0^\tau I(t,\tau )e^{-st}dt] d\tau }{ 1 - {\hat{\psi }}(0,s)} \nonumber \\&=-\frac{\hat{\psi }'(0,s)}{ s[1 - \hat{w}(s)]} +\frac{\int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )\right] e^{-st}}{ 1 - {\hat{w}}(s)}. \end{aligned}$$

Because the asymptotic behavior of \(w(\tau ) I(t,\tau )\) is given by \(w(\tau ) I(t,\tau )\sim AB\gamma t\tau ^{-2+\gamma -\alpha }/|\Gamma (-\alpha )|\) for \(t/\tau \ll 1\), the integral in the second term can be calculated as follows:

$$\begin{aligned} \int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )\right] e^{-st}&\sim \frac{AB\gamma }{|\Gamma (-\alpha )|} \int _0^\infty t\left[ \int _t^\infty \tau ^{-2-\alpha +\gamma }d\tau \right] e^{-st} dt \nonumber \\&= \frac{AB\gamma }{|\Gamma (-\alpha )| (1+\alpha -\gamma )} \int _0^\infty t^{\gamma -\alpha }e^{-st}dt \nonumber \\&\sim \frac{AB\gamma \Gamma (\gamma -\alpha +1) }{|\Gamma (-\alpha )| (1+\alpha -\gamma )} \frac{1}{s^{\gamma -\alpha +1}}. \end{aligned}$$

Using the asymptotic behavior of \(\hat{\psi }'(0,s)\),

$$\begin{aligned} \hat{\psi }'(0,s) \sim - \frac{AB \Gamma (\gamma -\alpha )}{|\Gamma (-\alpha )|}\frac{1}{s^{\gamma -\alpha }}\quad (\gamma >\alpha ), \end{aligned}$$

we have

$$\begin{aligned} \langle X_s \rangle&= \frac{B \Gamma (\gamma -\alpha )}{|\Gamma (-\alpha )|} \frac{1}{s^{1+\gamma }} +\frac{B\gamma (\gamma -\alpha ) \Gamma (\gamma -\alpha ) }{|\Gamma (-\alpha )| (1+\alpha -\gamma )} \frac{1}{s^{1+\gamma }} \nonumber \\&= \frac{B \Gamma (\gamma -\alpha )}{|\Gamma (-\alpha )|} \left[ 1+\frac{\gamma (\gamma -\alpha )}{1+\alpha -\gamma }\right] \frac{1}{s^{1+\gamma }}. \end{aligned}$$

Appendix 2: Second Moment

The Laplace transform for the second moment of \(X_t\) is given by

$$\begin{aligned} \langle X^{2}_s \rangle&= \left. \frac{\partial ^{2} {\hat{P}}(k,s)}{\partial k^{2}} \right| _{k=0} \nonumber \\&= \frac{\int _0^\infty w(\tau ) \int _0^\tau I(t,\tau )^2 e^{-st} dt d\tau }{1-\hat{w}(s)} - \frac{2\hat{\psi }'(0,s) \int _0^\infty w(\tau ) \int _0^\tau I(t,\tau ) e^{-st}dt d\tau }{[1-\hat{w}(s)]^2}\nonumber \\&\quad + \frac{2 \{\hat{\psi }'(0,s)\}^2}{s[1-\hat{w}(s)]^2} + \frac{\hat{\psi }''(0,s)}{s[1-\hat{w}(s)]} \nonumber \\&= \frac{\int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )^2\right] e^{-st}}{1-\hat{w}(s)} - \frac{2\hat{\psi }'(0,s) \int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )\right] e^{-st} }{[1-\hat{w}(s)]^2}\nonumber \\&\quad + \frac{2 \{\hat{\psi }'(0,s)\}^2}{s[1-\hat{w}(s)]^2} + \frac{\hat{\psi }''(0,s)}{s[1-\hat{w}(s)]}. \end{aligned}$$

Because the asymptotic behavior of \(w(\tau ) I(t,\tau )^2\) is given by \(w(\tau ) I(t,\tau )^2\sim A(B\gamma t)^2\tau ^{-3+2\gamma -\alpha }/|\Gamma (-\alpha )|\) for \(t/\tau \ll 1\), the integral in the first term can be calculated as follows:

$$\begin{aligned} \int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )^2 \right] e^{-st}&\sim \frac{A(B\gamma )^2}{|\Gamma (-\alpha )|} \int _0^\infty t^2\left[ \int _t^\infty \tau ^{-3-\alpha +2\gamma }d\tau \right] e^{-st} dt \nonumber \\&= \frac{A(B\gamma )^2 }{|\Gamma (-\alpha )| (2+\alpha -2\gamma )} \int _0^\infty t^{2\gamma -\alpha }e^{-st}dt \nonumber \\&\sim \frac{A(B\gamma )^2 \Gamma (2\gamma -\alpha +1) }{|\Gamma (-\alpha )| (2+\alpha - 2\gamma )} \frac{1}{s^{2\gamma -\alpha +1}}. \end{aligned}$$

Using Eq. (49) and

$$\begin{aligned} \hat{\psi }''(0,s) \sim \frac{AB^2 \Gamma (2\gamma -\alpha )}{|\Gamma (-\alpha )|}\frac{1}{s^{2\gamma -\alpha }}, \quad (\gamma >\alpha ) \end{aligned}$$

we have

$$\begin{aligned} \langle X^2_s \rangle&\sim \left[ \frac{(B\gamma )^2 \Gamma (2\gamma -\alpha +1) }{|\Gamma (-\alpha )| (2+\alpha - 2\gamma )} + \frac{2B \Gamma (\gamma -\alpha )}{|\Gamma (-\alpha )|}\frac{B\gamma \Gamma (\gamma -\alpha +1) }{|\Gamma (-\alpha )| (1+\alpha -\gamma )}\right. \nonumber \\&\left. \quad +\,\, \frac{2B^2 \Gamma (\gamma -\alpha )^2}{|\Gamma (-\alpha )|^2} +\; \frac{B^2 \Gamma (2\gamma -\alpha )}{|\Gamma (-\alpha )|} \right] \frac{1}{s^{2\gamma +1}} \nonumber \\&= \frac{B^2 }{|\Gamma (-\alpha )|} \left[ \frac{\gamma ^2 (2\gamma -\alpha ) \Gamma (2\gamma -\alpha ) }{2+\alpha - 2\gamma } +\; \frac{2 \gamma (\gamma -\alpha ) \Gamma (\gamma -\alpha )^2 }{|\Gamma (-\alpha )| (1+\alpha -\gamma )}\right. \nonumber \\&\left. \quad +\,\, \frac{2 \Gamma (\gamma -\alpha )^2}{|\Gamma (-\alpha )|} + \Gamma (2\gamma -\alpha ) \right] \frac{1}{s^{2\gamma +1}} \nonumber \\&= \frac{B^2 }{|\Gamma (-\alpha )|} \left[ \left\{ 1+\frac{\gamma ^2 (2\gamma -\alpha ) }{2+\alpha - 2\gamma }\right\} \Gamma (2\gamma -\alpha )\right. \nonumber \\&\left. \quad +\,\, \left\{ 1 + \frac{\gamma (\gamma -\alpha ) }{1+\alpha -\gamma } \right\} \frac{2 \Gamma (\gamma -\alpha )^2}{|\Gamma (-\alpha )|} \right] \frac{1}{s^{2\gamma +1}}. \end{aligned}$$

The inverse Laplace transform reads

$$\begin{aligned} \langle X^2_t \rangle&\sim \frac{B^2 }{|\Gamma (-\alpha )|\Gamma (1+2\gamma )} \left[ \left\{ 1+\frac{\gamma ^2 (2\gamma -\alpha ) }{2+\alpha - 2\gamma }\right\} \Gamma (2\gamma -\alpha )\right. \nonumber \\&\left. \quad + \left\{ 1 + \frac{\gamma (\gamma -\alpha ) }{1+\alpha -\gamma } \right\} \frac{2 \Gamma (\gamma -\alpha )^2}{|\Gamma (-\alpha )|} \right] t^{2\gamma }. \end{aligned}$$

Appendix 3: \(n\)th (\(n>1\)) Moment and Its Coefficient \(M_n(\alpha ,\gamma )\)

The \(n\)-th (\(n>1\)) differentiation of \(\hat{P}(k,s)\) is given by the recursion relation:

$$\begin{aligned} \hat{P}^{(n)}(k,s)&= \frac{1}{1-\hat{\psi }(k,s)} \left[ \sum _{i=1}^{n-1} c_{n,i}\hat{P}^{(i)}(k,s) \hat{\psi }^{(n-i)}(k,s) + \hat{P}(k,s) \hat{\psi }^{(n)}(k,s)\right. \nonumber \\&\left. \quad + \int _0^\infty d\tau w(\tau ) \hat{\Psi }^{(n)}(k,s;\tau ) \right] , \end{aligned}$$

where \(c_{n,i}=c_{n-1,i}+c_{n-1,i-1}\) (\(i=2, \ldots , n-2\)) and \(c_{n,n-1}=c_{n,1}=n\).


$$\begin{aligned} \int _0^\infty d\tau w(\tau ) \hat{\Psi }^{(n)}(0,s;\tau )&= \int _0^\infty d\tau w(\tau ) \int _0^\tau I(t,\tau )^n e^{-st}dt \nonumber \\&= \int _0^\infty dt \int _t^\infty d\tau w(\tau ) I(t,\tau )^n e^{-st} \nonumber \\&\sim \frac{A(B\gamma )^n}{|\Gamma (-\alpha )|} \int _0^\infty dt t^n \int _t^\infty \tau ^{-(n+1) -\alpha +n \gamma } d\tau e^{-st} \nonumber \\&\sim \frac{AB^n \gamma ^n \Gamma (n\gamma -\alpha +1)}{|\Gamma (-\alpha )|(n+\alpha -n\gamma )}\frac{1}{s^{1-\alpha + n\gamma }}. \end{aligned}$$

We assume

$$\begin{aligned} \hat{P}^{(i)}(0,s) \sim (-1)^i\frac{B^i M_i(\alpha ,\gamma )}{|\Gamma (-\alpha )|} \frac{1}{s^{1+i\gamma }}, \end{aligned}$$

for \(i<n\). It follows that

$$\begin{aligned} \hat{P}^{(n)}(0,s)&= \left[ \sum _{i=1}^{n-1} c_{n,i} \frac{M_i(\alpha ,\gamma )}{|\Gamma (-\alpha )|} \frac{\Gamma ((n-i)\gamma - \alpha )}{|\Gamma (-\alpha )|} +\; \frac{\Gamma (n\gamma - \alpha )}{|\Gamma (-\alpha )|}\right. \nonumber \\&\left. \quad +\,\, \frac{\gamma ^n\Gamma (n\gamma - \alpha +1)}{|\Gamma (-\alpha )|(n+\alpha -n\gamma )} \right] \frac{(-B)^n}{s^{1 + n\gamma }} \nonumber \\&= \left[ \sum _{i=1}^{n-1} c_{n,i} M_i(\alpha ,\gamma ) \frac{\Gamma ((n-i)\gamma - \alpha )}{|\Gamma (-\alpha )|}\right. \nonumber \\&\left. \quad + \left\{ 1+ \frac{\gamma ^n(n\gamma -\alpha )}{n+\alpha -n\gamma }\right\} \Gamma (n\gamma - \alpha ) \right] \frac{(-B)^n}{|\Gamma (-\alpha )| s^{1 + n\gamma }}. \end{aligned}$$


$$\begin{aligned} M_n (\alpha ,\gamma ) = \sum _{i=1}^{n-1} c_{n,i} M_i(\alpha ,\gamma ) \frac{\Gamma ((n-i)\gamma - \alpha )}{|\Gamma (-\alpha )|} + \left\{ 1+ \frac{\gamma ^n(n\gamma -\alpha )}{n+\alpha -n\gamma }\right\} \Gamma (n\gamma - \alpha ). \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Akimoto, T., Shinkai, S. & Aizawa, Y. Distributional Behavior of Time Averages of Non-\(L^1\) Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures. J Stat Phys 158, 476–493 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: