April  2021, 41(4): 1799-1841. 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  April 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (4) : 1799-1841. 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.

[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.

[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.

[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.

[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.

[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.

[7]

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

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[40]

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

[41]

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

[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.

[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.

[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.

[45]

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

[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.

[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.
[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.

[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.

[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.

[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.

[52]

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

[53]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[40]

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

[41]

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

[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.

[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.

[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.

[45]

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

[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.

[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.
[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.

[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.

[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.

[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.

[52]

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

[53]

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

[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.

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