doi: 10.3934/dcds.2020341

Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables

1. 

Dipartimento di Matematica, Via Buonattoti 1, 56127 Pisa, Italy

2. 

Mathematics (CEMPS), Harrison Building (327), North Park Road, EXETER, EX4 4QF, United Kingdom

3. 

Centre for Mathematical Sciences, Lund University, Box 118,221 00 Lund, Sweden

4. 

School of Mathematics and Statistics, Center for Mathematical Sciences, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Sciences and Technology, Wuhan 430074, China

Received  March 2020 Revised  August 2020 Published  October 2020

We establish quantitative results for the statistical behaviour of infinite systems. We consider two kinds of infinite system:

ⅰ) a conservative dynamical system $ (f,X,\mu) $ preserving a $ \sigma $-finite measure $ \mu $ such that $ \mu(X) = \infty $;

ⅱ) the case where $ \mu $ is a probability measure but we consider the statistical behaviour of an observable $ \phi\colon X\to[0,\infty) $ which is non-integrable: $ \int \phi \, d\mu = \infty $.

In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove a general relation between the behaviour of $ \phi $, the local dimension of $ \mu $, and the scaling rate of the growth of Birkhoff sums of $ \phi $ as time tends to infinity. We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings for the power law behaviour of entrance times (also known as logarithm laws), dynamical Borel–Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.

Citation: Stefano Galatolo, Mark Holland, Tomas Persson, Yiwei Zhang. Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020341
References:
[1]

J. Aaronson, On the ergodic theory of non-integrable functions and infinite measure spaces, Israel J. Math., 27 (1977), 163-173.  doi: 10.1007/BF02761665.  Google Scholar

[2]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Volume 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[3]

J. Aaronson and M. Denker, Upper bounds for ergodic sums of infinite measure preserving transformations, Trans. Amer. Math. Soc., 319 (1990), 101-138.  doi: 10.1090/S0002-9947-1990-1024766-3.  Google Scholar

[4]

V. AraújoS. Galatolo and M. J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors, Math. Z., 276 (2014), 1001-1048.  doi: 10.1007/s00209-013-1231-0.  Google Scholar

[5]

R. ArratiaL. Gordon and M. Waterman, The Erdős–Rényi law in distribution, for coin tossing and sequence matching, Ann. Statist., 18 (1990), 539-570.  doi: 10.1214/aos/1176347615.  Google Scholar

[6]

V. Baladi, Decay of correlations, in Smooth Ergodoic Theory and Its Applications, Proc. Sympos. Pure Math., Vol. 69, AMS, Providence, RI, 2001,297–325. doi: 10.1090/pspum/069/1858537.  Google Scholar

[7]

N. Balakrishnan and M. V. Koutras, Runs and Scans with Applications, Wiley-Interscience, New York, 2002.  Google Scholar

[8] A. D. BarbourL. Holst and S. Janson, Poisson Approximation, The Clarendon Press, Oxford University Press, New York, 1992.   Google Scholar
[9]

G. Bateman, On the power function of the longest run as a test for randomness in a sequence of alternatives, Biometrika, 35 (1948), 97-112.  doi: 10.1093/biomet/35.1-2.97.  Google Scholar

[10]

M. Benedicks and L. S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261 (2000), 13-56.   Google Scholar

[11]

H. BruinM. Nicol and D. Terhesiu, On Young towers associated with infinite measure preserving transformations, Stoch. Dyn., 9 (2009), 635-655.  doi: 10.1142/S0219493709002816.  Google Scholar

[12]

Y. Bugeaud and L. Liao, Uniform Diophantine approximation related to $b$-ary and $\beta$-expansions, Ergodic Theory Dynam. Systems, 36 (2016), 1-22.  doi: 10.1017/etds.2014.66.  Google Scholar

[13]

M. Carney and M. Nicol, Dynamical Borel–Cantelli lemmas and the rate of growth of Birkhoff sums of non-integrable observables on chaotic dynamical systems, Nonlinearity, 30 (2017), 2854-2870.  doi: 10.1088/1361-6544/aa72c2.  Google Scholar

[14]

H. CuiL. Fang and Y. Zhang, A note on the run length function for intermittent maps, J. Math. Anal. Appl., 472 (2019), 937-946.  doi: 10.1016/j.jmaa.2018.11.058.  Google Scholar

[15]

M. Denker and Z. Kabluchko, An Erdős–Rényi law for mixing processes, Probab. Math. Statist., 27 (2007), 139-149.   Google Scholar

[16]

M. Denker and M. Nicol, Erdős–Rényi laws for dynamical systems, J. Lond. Math. Soc. (2), 87 (2013), 497-508.  doi: 10.1112/jlms/jds060.  Google Scholar

[17]

P. Embrechts, C. Klüpperlberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Applications of Mathematics, Vol. 33, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-33483-2.  Google Scholar

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P. Erdős and A. Rényi, On a new law of large numbers, J. Analyse. Math., 23 (1970), 103-111.  doi: 10.1007/BF02795493.  Google Scholar

[19]

A. H. Fan and B. W. Wang, On the lengths of basic intervals in beta expansions, Nonlinearity, 25 (2012), 1329-1343.  doi: 10.1088/0951-7715/25/5/1329.  Google Scholar

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A. C. M. FreitasJ. M. Freitas and M. Todd, Hitting time statistics and extreme value theory, Probab. Theory Related Fields, 147 (2010), 675-710.  doi: 10.1007/s00440-009-0221-y.  Google Scholar

[21]

A. C. M. Freitas, J. M. Freitas and M. Todd, Extreme value laws in dynamical systems for non-smooth observations, J. Stat. Phys., 142 (2011) 108–126. doi: 10.1007/s10955-010-0096-4.  Google Scholar

[22]

J. Galambos, The Asymptotic Theory of Extreme Order Statistics, John Wiley and Sons, New York-Chichester-Brisbane, 1978.  Google Scholar

[23]

S. Galatolo, Hitting time and dimension in axiom A systems, generic interval exchanges and an application to Birkoff sums, J. Stat. Phys., 123 (2006), 111-124.  doi: 10.1007/s10955-006-9041-y.  Google Scholar

[24]

S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/MRL.2007.v14.n5.a8.  Google Scholar

[25]

S. Galatolo, Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 2477-2487.  doi: 10.1090/S0002-9939-10-10275-5.  Google Scholar

[26]

S. Galatolo and D. H. Kim, The dynamical Borel–Cantelli lemma and the waiting time problems, Indag. Math. (N.S.), 18 (2007), 421-434.  doi: 10.1016/S0019-3577(07)80031-0.  Google Scholar

[27]

S. GalatoloD. H. Kim and K. K. Park, The recurrence time for ergodic systems with infinite invariant measures, Nonlinearity, 19 (2006), 2567-2580.  doi: 10.1088/0951-7715/19/11/004.  Google Scholar

[28]

S. Galatolo and I. Nisoli, Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds, Nonlinearity, 24 (2011), 3099-3113.  doi: 10.1088/0951-7715/24/11/005.  Google Scholar

[29]

S. Galatolo and M. J. Pacifico, Lorenz-like flows: Exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence, Ergodic Theory Dynam. Systems, 30 (2010), 1703-1737.  doi: 10.1017/S0143385709000856.  Google Scholar

[30]

S. Galatolo and P. Peterlongo, Long hitting time, slow decay of correlations and arithmetical properties, Discrete Contin. Dyn. Syst., 27 (2010), 185-204.  doi: 10.3934/dcds.2010.27.185.  Google Scholar

[31]

S. GalatoloJ. Rousseau and B. Saussol, Skew products, quantitative recurrence, shrinking targets and decay of correlations, Ergodic Theory Dynam. Systems, 35 (2015), 1814-1845.  doi: 10.1017/etds.2014.10.  Google Scholar

[32]

S. Gouëzel, A Borel–Cantelli lemma for intermittent interval maps, Nonlinearity, 20 (2007), 1491-1497.  doi: 10.1088/0951-7715/20/6/010.  Google Scholar

[33]

J. Grigull, Groß e Abweichungen und Fluktuationen für Gleichgewichtsmaß e rationaler Abbildungen, Dissertation, Georg-August-Universität Göttingen, 1993. Google Scholar

[34]

C. GuptaM. Nicol and W. Ott, A Borel–Cantelli lemma for nonuniformly expanding dynamical systems, Nonlinearity, 23 (2010), 1991-2008.  doi: 10.1088/0951-7715/23/8/010.  Google Scholar

[35]

N. HaydnM. NicolT. Persson and S. Vaienti, A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems, 33 (2013), 475-498.  doi: 10.1017/S014338571100099X.  Google Scholar

[36]

M. HollandM. Nicol and A. Török, Almost sure convergence of maxima for chaotic dynamical systems, Stochastic Process. Appl., 126 (2016), 3145-3170.  doi: 10.1016/j.spa.2016.04.023.  Google Scholar

[37]

M. HollandR. VitoloP. RabassaA. E. Sterk and H. Broer, Extreme value laws in dynamical systems under physical observables, Phys. D., 241 (2012), 497-513.   Google Scholar

[38]

D. H. Kim, The dynamical Borel–Cantelli lemma for interval maps, Discrete Contin. Dyn. Syst., 17 (2007), 891-900.  doi: 10.3934/dcds.2007.17.891.  Google Scholar

[39]

D. H. Kim and B. K. Seo, The waiting time for irrational rotations, Nonlinearity, 16 (2003), 1861-1868.  doi: 10.1088/0951-7715/16/5/318.  Google Scholar

[40]

M. Lenci and S. Munday, Pointwise convergence of Birkhoff averages for global observables, Chaos, 28 (2018), 083111. doi: 10.1063/1.5036652.  Google Scholar

[41]

C. LiveraniB. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.  Google Scholar

[42]

V. Lucarini, D. Faranda, A. Freitas, J. Freitas, M. Holland, T. Kuna, M. Nicol, M. Todd and S. Vaienti, Extremes and Recurrence in Dynamical Systems, Pure and Applied Mathematics, John Wiley & Sons, Inc., Hoboken, NJ, 2016. doi: 10.1002/9781118632321.  Google Scholar

[43]

P. Manneville and Y. Pomeau, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197.  doi: 10.1007/BF01197757.  Google Scholar

[44]

I. Melbourne and D. Terhesiu, Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math., 189 (2012), 61-110.  doi: 10.1007/s00222-011-0361-4.  Google Scholar

[45]

A. de Moivre, The Doctrine of Chances, H. Woodfall, London, 1738. Google Scholar

[46]

M. Muselli, New improved bounds for reliability of consecutive-$k$-out-of-$n$:$F$ systems, J. Appl. Probab., 37 (2000), 1164-1170.  doi: 10.1239/jap/1014843097.  Google Scholar

[47] Y. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar
[48]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.  doi: 10.1007/BF02788928.  Google Scholar

[49]

X. TongY. Yu and Y. Zhao, On the maximal length of consecutive zero digits of $\beta$-expensions, Int. J. Number Theory, 12 (2016), 625-633.  doi: 10.1142/S1793042116500408.  Google Scholar

[50]

C. Ulcigrai, Mixing of asymmetric logarithmic suspension flows over interval exchange transformations, Ergodic Theory Dynam. Systems, 27 (2007), 991-1035.  doi: 10.1017/S0143385706000836.  Google Scholar

[51]

C. Ulcigrai, Absence of mixing in area-preserving flows on surfaces, Ann. of Math., 173 (2011), 1743-1778.  doi: 10.4007/annals.2011.173.3.10.  Google Scholar

[52]

L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

[53]

L. S. Young, Recurrence times and rates of mixing, Israel J Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.  Google Scholar

[54]

R. Zweimüller, Invariant measures for general(ized) induced transformations, Proc. Amer. Math. Soc., 133 (2005), 2283-2295.  doi: 10.1090/S0002-9939-05-07772-5.  Google Scholar

show all references

References:
[1]

J. Aaronson, On the ergodic theory of non-integrable functions and infinite measure spaces, Israel J. Math., 27 (1977), 163-173.  doi: 10.1007/BF02761665.  Google Scholar

[2]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Volume 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[3]

J. Aaronson and M. Denker, Upper bounds for ergodic sums of infinite measure preserving transformations, Trans. Amer. Math. Soc., 319 (1990), 101-138.  doi: 10.1090/S0002-9947-1990-1024766-3.  Google Scholar

[4]

V. AraújoS. Galatolo and M. J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors, Math. Z., 276 (2014), 1001-1048.  doi: 10.1007/s00209-013-1231-0.  Google Scholar

[5]

R. ArratiaL. Gordon and M. Waterman, The Erdős–Rényi law in distribution, for coin tossing and sequence matching, Ann. Statist., 18 (1990), 539-570.  doi: 10.1214/aos/1176347615.  Google Scholar

[6]

V. Baladi, Decay of correlations, in Smooth Ergodoic Theory and Its Applications, Proc. Sympos. Pure Math., Vol. 69, AMS, Providence, RI, 2001,297–325. doi: 10.1090/pspum/069/1858537.  Google Scholar

[7]

N. Balakrishnan and M. V. Koutras, Runs and Scans with Applications, Wiley-Interscience, New York, 2002.  Google Scholar

[8] A. D. BarbourL. Holst and S. Janson, Poisson Approximation, The Clarendon Press, Oxford University Press, New York, 1992.   Google Scholar
[9]

G. Bateman, On the power function of the longest run as a test for randomness in a sequence of alternatives, Biometrika, 35 (1948), 97-112.  doi: 10.1093/biomet/35.1-2.97.  Google Scholar

[10]

M. Benedicks and L. S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261 (2000), 13-56.   Google Scholar

[11]

H. BruinM. Nicol and D. Terhesiu, On Young towers associated with infinite measure preserving transformations, Stoch. Dyn., 9 (2009), 635-655.  doi: 10.1142/S0219493709002816.  Google Scholar

[12]

Y. Bugeaud and L. Liao, Uniform Diophantine approximation related to $b$-ary and $\beta$-expansions, Ergodic Theory Dynam. Systems, 36 (2016), 1-22.  doi: 10.1017/etds.2014.66.  Google Scholar

[13]

M. Carney and M. Nicol, Dynamical Borel–Cantelli lemmas and the rate of growth of Birkhoff sums of non-integrable observables on chaotic dynamical systems, Nonlinearity, 30 (2017), 2854-2870.  doi: 10.1088/1361-6544/aa72c2.  Google Scholar

[14]

H. CuiL. Fang and Y. Zhang, A note on the run length function for intermittent maps, J. Math. Anal. Appl., 472 (2019), 937-946.  doi: 10.1016/j.jmaa.2018.11.058.  Google Scholar

[15]

M. Denker and Z. Kabluchko, An Erdős–Rényi law for mixing processes, Probab. Math. Statist., 27 (2007), 139-149.   Google Scholar

[16]

M. Denker and M. Nicol, Erdős–Rényi laws for dynamical systems, J. Lond. Math. Soc. (2), 87 (2013), 497-508.  doi: 10.1112/jlms/jds060.  Google Scholar

[17]

P. Embrechts, C. Klüpperlberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Applications of Mathematics, Vol. 33, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-33483-2.  Google Scholar

[18]

P. Erdős and A. Rényi, On a new law of large numbers, J. Analyse. Math., 23 (1970), 103-111.  doi: 10.1007/BF02795493.  Google Scholar

[19]

A. H. Fan and B. W. Wang, On the lengths of basic intervals in beta expansions, Nonlinearity, 25 (2012), 1329-1343.  doi: 10.1088/0951-7715/25/5/1329.  Google Scholar

[20]

A. C. M. FreitasJ. M. Freitas and M. Todd, Hitting time statistics and extreme value theory, Probab. Theory Related Fields, 147 (2010), 675-710.  doi: 10.1007/s00440-009-0221-y.  Google Scholar

[21]

A. C. M. Freitas, J. M. Freitas and M. Todd, Extreme value laws in dynamical systems for non-smooth observations, J. Stat. Phys., 142 (2011) 108–126. doi: 10.1007/s10955-010-0096-4.  Google Scholar

[22]

J. Galambos, The Asymptotic Theory of Extreme Order Statistics, John Wiley and Sons, New York-Chichester-Brisbane, 1978.  Google Scholar

[23]

S. Galatolo, Hitting time and dimension in axiom A systems, generic interval exchanges and an application to Birkoff sums, J. Stat. Phys., 123 (2006), 111-124.  doi: 10.1007/s10955-006-9041-y.  Google Scholar

[24]

S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/MRL.2007.v14.n5.a8.  Google Scholar

[25]

S. Galatolo, Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 2477-2487.  doi: 10.1090/S0002-9939-10-10275-5.  Google Scholar

[26]

S. Galatolo and D. H. Kim, The dynamical Borel–Cantelli lemma and the waiting time problems, Indag. Math. (N.S.), 18 (2007), 421-434.  doi: 10.1016/S0019-3577(07)80031-0.  Google Scholar

[27]

S. GalatoloD. H. Kim and K. K. Park, The recurrence time for ergodic systems with infinite invariant measures, Nonlinearity, 19 (2006), 2567-2580.  doi: 10.1088/0951-7715/19/11/004.  Google Scholar

[28]

S. Galatolo and I. Nisoli, Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds, Nonlinearity, 24 (2011), 3099-3113.  doi: 10.1088/0951-7715/24/11/005.  Google Scholar

[29]

S. Galatolo and M. J. Pacifico, Lorenz-like flows: Exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence, Ergodic Theory Dynam. Systems, 30 (2010), 1703-1737.  doi: 10.1017/S0143385709000856.  Google Scholar

[30]

S. Galatolo and P. Peterlongo, Long hitting time, slow decay of correlations and arithmetical properties, Discrete Contin. Dyn. Syst., 27 (2010), 185-204.  doi: 10.3934/dcds.2010.27.185.  Google Scholar

[31]

S. GalatoloJ. Rousseau and B. Saussol, Skew products, quantitative recurrence, shrinking targets and decay of correlations, Ergodic Theory Dynam. Systems, 35 (2015), 1814-1845.  doi: 10.1017/etds.2014.10.  Google Scholar

[32]

S. Gouëzel, A Borel–Cantelli lemma for intermittent interval maps, Nonlinearity, 20 (2007), 1491-1497.  doi: 10.1088/0951-7715/20/6/010.  Google Scholar

[33]

J. Grigull, Groß e Abweichungen und Fluktuationen für Gleichgewichtsmaß e rationaler Abbildungen, Dissertation, Georg-August-Universität Göttingen, 1993. Google Scholar

[34]

C. GuptaM. Nicol and W. Ott, A Borel–Cantelli lemma for nonuniformly expanding dynamical systems, Nonlinearity, 23 (2010), 1991-2008.  doi: 10.1088/0951-7715/23/8/010.  Google Scholar

[35]

N. HaydnM. NicolT. Persson and S. Vaienti, A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems, 33 (2013), 475-498.  doi: 10.1017/S014338571100099X.  Google Scholar

[36]

M. HollandM. Nicol and A. Török, Almost sure convergence of maxima for chaotic dynamical systems, Stochastic Process. Appl., 126 (2016), 3145-3170.  doi: 10.1016/j.spa.2016.04.023.  Google Scholar

[37]

M. HollandR. VitoloP. RabassaA. E. Sterk and H. Broer, Extreme value laws in dynamical systems under physical observables, Phys. D., 241 (2012), 497-513.   Google Scholar

[38]

D. H. Kim, The dynamical Borel–Cantelli lemma for interval maps, Discrete Contin. Dyn. Syst., 17 (2007), 891-900.  doi: 10.3934/dcds.2007.17.891.  Google Scholar

[39]

D. H. Kim and B. K. Seo, The waiting time for irrational rotations, Nonlinearity, 16 (2003), 1861-1868.  doi: 10.1088/0951-7715/16/5/318.  Google Scholar

[40]

M. Lenci and S. Munday, Pointwise convergence of Birkhoff averages for global observables, Chaos, 28 (2018), 083111. doi: 10.1063/1.5036652.  Google Scholar

[41]

C. LiveraniB. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.  Google Scholar

[42]

V. Lucarini, D. Faranda, A. Freitas, J. Freitas, M. Holland, T. Kuna, M. Nicol, M. Todd and S. Vaienti, Extremes and Recurrence in Dynamical Systems, Pure and Applied Mathematics, John Wiley & Sons, Inc., Hoboken, NJ, 2016. doi: 10.1002/9781118632321.  Google Scholar

[43]

P. Manneville and Y. Pomeau, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197.  doi: 10.1007/BF01197757.  Google Scholar

[44]

I. Melbourne and D. Terhesiu, Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math., 189 (2012), 61-110.  doi: 10.1007/s00222-011-0361-4.  Google Scholar

[45]

A. de Moivre, The Doctrine of Chances, H. Woodfall, London, 1738. Google Scholar

[46]

M. Muselli, New improved bounds for reliability of consecutive-$k$-out-of-$n$:$F$ systems, J. Appl. Probab., 37 (2000), 1164-1170.  doi: 10.1239/jap/1014843097.  Google Scholar

[47] Y. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar
[48]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.  doi: 10.1007/BF02788928.  Google Scholar

[49]

X. TongY. Yu and Y. Zhao, On the maximal length of consecutive zero digits of $\beta$-expensions, Int. J. Number Theory, 12 (2016), 625-633.  doi: 10.1142/S1793042116500408.  Google Scholar

[50]

C. Ulcigrai, Mixing of asymmetric logarithmic suspension flows over interval exchange transformations, Ergodic Theory Dynam. Systems, 27 (2007), 991-1035.  doi: 10.1017/S0143385706000836.  Google Scholar

[51]

C. Ulcigrai, Absence of mixing in area-preserving flows on surfaces, Ann. of Math., 173 (2011), 1743-1778.  doi: 10.4007/annals.2011.173.3.10.  Google Scholar

[52]

L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

[53]

L. S. Young, Recurrence times and rates of mixing, Israel J Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.  Google Scholar

[54]

R. Zweimüller, Invariant measures for general(ized) induced transformations, Proc. Amer. Math. Soc., 133 (2005), 2283-2295.  doi: 10.1090/S0002-9939-05-07772-5.  Google Scholar

Figure 1.  An intermittent map, with induced map in top right quadrant
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