November  2022, 42(11): 5541-5576. doi: 10.3934/dcds.2022113

Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

1. 

Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

2. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

*Corresponding author: Tanja I. Schindler

Received  October 2021 Revised  June 2022 Published  November 2022 Early access  August 2022

Fund Project: The authors are partially supported by the PRIN Grant 2017S35EHN "Regular and stochastic behaviour in dynamical systems" of the Italian Ministry of University and Research (MUR), Italy. This research is part of the authors' activity within the UMI Group "DinAmicI" www.dinamici.org and of the first author's activity within the Gruppo Nazionale di Fisica Matematica, INdAM, Italy. The second author acknowledges the support of the Centro di Ricerca Matematica Ennio de Giorgi and of UniCredit Bank R & D group for financial support through the 'Dynamics and Information Theory Institute' at the Scuola Normale Superiore

We consider a conservative ergodic measure-preserving transformation $ T $ of a $ \sigma $-finite measure space $ (X, {\mathcal B},\mu) $ with $ \mu(X) = \infty $. Given an observable $ f:X\to \mathbb R $, we study the almost sure asymptotic behaviour of the Birkhoff sums $ S_Nf(x) : = \sum_{j = 1}^N\, (f\circ T^{j-1})(x) $. In infinite ergodic theory it is well known that the asymptotic behaviour of $ S_Nf(x) $ strongly depends on the point $ x\in X $, and if $ f\in L^1(X,\mu) $, then there exists no real valued sequence $ (b(N)) $ such that $ \lim_{N\to\infty} S_Nf(x)/b(N) = 1 $ almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence $ (\alpha(N)) $ and $ m\colon X\times \mathbb N\to \mathbb N $ such that for $ f\in L^1(X,\mu) $ we have $ \lim_{N\to\infty} S_{N+m(x,N)}f(x)/\alpha(N) = 1 $ for $ \mu $-a.e. $ x\in X $. Instead in the case $ f\not\in L^1(X,\mu) $ we give conditions on the induced observable such that there exists a sequence $ (G(N)) $ depending on $ f $, for which $ \lim_{N\to\infty} S_{N}f(x)/G(N) = 1 $ holds for $ \mu $-a.e. $ x\in X $.

Citation: Claudio Bonanno, Tanja I. Schindler. Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (11) : 5541-5576. doi: 10.3934/dcds.2022113
References:
[1]

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

[2]

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.

[3]

J. AaronsonZ. Kosloff and B. Weiss, Symmetric Birkhoff sums in infinite ergodic theory, Ergodic Theory Dynam. Systems, 37 (2017), 2394-2416.  doi: 10.1017/etds.2016.18.

[4]

J. Aaronson and H. Nakada, Trimmed sums for non-negative, mixing stationary processes, Stochastic Process. Appl., 104 (2003), 173-192.  doi: 10.1016/S0304-4149(02)00236-3.

[5]

J. Aaronson and H. Nakada, On the mixing coefficients of piecewise monotonic maps, Israel J. Math., 148 (2005), 1-10.  doi: 10.1007/BF02775429.

[6]

J. AaronsonM. Thaler and R. Zweimüller, Occupation times of sets of infinite measure for ergodic transformations, Ergodic Theory Dynam. Systems, 25 (2005), 959-976.  doi: 10.1017/S0143385704001051.

[7]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.

[8]

C. BonannoA. Del Vigna and S. Munday, A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs, Monatsh. Math., 194 (2021), 1-40.  doi: 10.1007/s00605-020-01500-w.

[9]

C. BonannoP. Giulietti and M. Lenci, Infinite mixing for one-dimensional maps with an indifferent fixed point, Nonlinearity, 31 (2018), 5180-5213.  doi: 10.1088/1361-6544/aadc04.

[10]

C. Bonanno and M. Lenci, Pomeau-Manneville maps are global-local mixing, Discrete Contin. Dyn. Syst., 41 (2021), 1051-1069.  doi: 10.3934/dcds.2020309.

[11]

R. C. Bradley, Basic properties of strong mixing conditions. A survey and some open questions, Probab. Surv., 2 (2005), 107-144.  doi: 10.1214/154957805100000104.

[12]

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

[13]

H. G. Diamond and J. D. Vaaler, Estimates for partial sums of continued fraction partial quotients, Pacific J. Math., 122 (1986), 73-82.  doi: 10.2140/pjm.1986.122.73.

[14]

S. GalatoloM. HollandT. Persson and Y. Zhang, Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables, Discrete Contin. Dyn. Syst., 41 (2021), 1799-1841.  doi: 10.3934/dcds.2020341.

[15]

T. Garrity, On periodic sequences for algebraic numbers, J. Number Theory, 88 (2001), 86-103.  doi: 10.1006/jnth.2000.2608.

[16]

E. Haeusler, A nonstandard law of the iterated logarithm for trimmed sums, Ann. Probab., 21 (1993), 831–860, https://www.jstor.org/stable/2244678.

[17]

E. Haeusler and D. M. Mason, Laws of the iterated logarithm for sums of the middle portion of the sample, Math. Proc. Cambridge Philos. Soc., 101 (1987), 301-312.  doi: 10.1017/S0305004100066676.

[18]

A. Haynes, Quantitative ergodic theorems for weakly integrable functions, Ergodic Theory Dynam. Systems, 34 (2014), 534-542.  doi: 10.1017/etds.2012.146.

[19]

T. Inoue, Ergodic theorems for piecewise affine Markov maps with indifferent fixed points, Hiroshima Math. J., 24 (1994), 447-471.  doi: 10.32917/hmj/1206127920.

[20]

T. Inoue, Sojourn times in small neighborhoods of indifferent fixed points of one-dimensional dynamical systems, Ergodic Theory Dynam. Systems, 20 (2000), 241-257.  doi: 10.1017/S0143385700000110.

[21]

T. Inoue, Ergodic sums of non-integrable functions under one-dimensional dynamical systems with indifferent fixed points, Ergodic Theory Dynam. Systems, 24 (2004), 525-545.  doi: 10.1017/S0143385703000506.

[22]

M. Kesseböhmer and T. Schindler, Strong laws of large numbers for intermediately trimmed sums of i.i.d. random variables with infinite mean, J. Theoret. Probab., 32 (2019), 702-720.  doi: 10.1007/s10959-017-0802-0.

[23]

M. Kesseböhmer and T. Schindler, Strong laws of large numbers for intermediately trimmed Birkhoff sums of observables with infinite mean, Stochastic Process. Appl., 130 (2020), 7019.  doi: 10.1016/j.spa.2018.11.015.

[24]

M. Kesseböhmer and T. I. Schindler, Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type, Dyn. Syst., 35 (2020), 275-305.  doi: 10.1080/14689367.2019.1667305.

[25]

M. Kesseböhmer and T. I. Schindler, Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails, Nonlinearity, 33 (2020), 5543-5566.  doi: 10.1088/1361-6544/ab9585.

[26]

H. Kesten and R. A. Maller, Ratios of trimmed sums and order statistics, Ann. Probab., 20 (1992), 1805-1842.  doi: 10.1214/aop/1176989530.

[27]

Z. Kosloff, A universal divergence rate for symmetric Birkhoff sums in infinite ergodic theory, Trans. Amer. Math. Soc., 369 (2017), 6373-6388.  doi: 10.1090/tran/6867.

[28]

M. Lenci and S. Munday, Pointwise convergence of Birkhoff averages for global observables, Chaos, 28 (2018), 083111, 16 pp. doi: 10.1063/1.5036652.

[29]

F. Móricz, On the harmonic averages of numerical sequences, Arch. Math., (Basel) 86 (2006), 375–384. doi: 10.1007/s00013-005-1588-3.

[30]

H. Nakada and R. Natsui, On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi-Perron algorithm, Monatsh. Math., 138 (2003), 267-288.  doi: 10.1007/s00605-002-0473-4.

[31]

T. I. Schindler, Trimmed sums for observables on the doubling map, preprint, 2018, arXiv: 1810.03223.

[32]

F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

[33]

M. Thaler, Transformations on $[0, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.

[34]

R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergodic Theory Dynam. Systems, 20 (2000), 1519-1549.  doi: 10.1017/S0143385700000821.

[35]

R. Zweimüller, Surrey Notes on Infinite Ergodic Theory, Lecture notes, 2009. Available from: http://mat.univie.ac.at/%7Ezweimueller/MyPub/SurreyNotes.pdf

show all references

References:
[1]

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

[2]

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.

[3]

J. AaronsonZ. Kosloff and B. Weiss, Symmetric Birkhoff sums in infinite ergodic theory, Ergodic Theory Dynam. Systems, 37 (2017), 2394-2416.  doi: 10.1017/etds.2016.18.

[4]

J. Aaronson and H. Nakada, Trimmed sums for non-negative, mixing stationary processes, Stochastic Process. Appl., 104 (2003), 173-192.  doi: 10.1016/S0304-4149(02)00236-3.

[5]

J. Aaronson and H. Nakada, On the mixing coefficients of piecewise monotonic maps, Israel J. Math., 148 (2005), 1-10.  doi: 10.1007/BF02775429.

[6]

J. AaronsonM. Thaler and R. Zweimüller, Occupation times of sets of infinite measure for ergodic transformations, Ergodic Theory Dynam. Systems, 25 (2005), 959-976.  doi: 10.1017/S0143385704001051.

[7]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.

[8]

C. BonannoA. Del Vigna and S. Munday, A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs, Monatsh. Math., 194 (2021), 1-40.  doi: 10.1007/s00605-020-01500-w.

[9]

C. BonannoP. Giulietti and M. Lenci, Infinite mixing for one-dimensional maps with an indifferent fixed point, Nonlinearity, 31 (2018), 5180-5213.  doi: 10.1088/1361-6544/aadc04.

[10]

C. Bonanno and M. Lenci, Pomeau-Manneville maps are global-local mixing, Discrete Contin. Dyn. Syst., 41 (2021), 1051-1069.  doi: 10.3934/dcds.2020309.

[11]

R. C. Bradley, Basic properties of strong mixing conditions. A survey and some open questions, Probab. Surv., 2 (2005), 107-144.  doi: 10.1214/154957805100000104.

[12]

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

[13]

H. G. Diamond and J. D. Vaaler, Estimates for partial sums of continued fraction partial quotients, Pacific J. Math., 122 (1986), 73-82.  doi: 10.2140/pjm.1986.122.73.

[14]

S. GalatoloM. HollandT. Persson and Y. Zhang, Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables, Discrete Contin. Dyn. Syst., 41 (2021), 1799-1841.  doi: 10.3934/dcds.2020341.

[15]

T. Garrity, On periodic sequences for algebraic numbers, J. Number Theory, 88 (2001), 86-103.  doi: 10.1006/jnth.2000.2608.

[16]

E. Haeusler, A nonstandard law of the iterated logarithm for trimmed sums, Ann. Probab., 21 (1993), 831–860, https://www.jstor.org/stable/2244678.

[17]

E. Haeusler and D. M. Mason, Laws of the iterated logarithm for sums of the middle portion of the sample, Math. Proc. Cambridge Philos. Soc., 101 (1987), 301-312.  doi: 10.1017/S0305004100066676.

[18]

A. Haynes, Quantitative ergodic theorems for weakly integrable functions, Ergodic Theory Dynam. Systems, 34 (2014), 534-542.  doi: 10.1017/etds.2012.146.

[19]

T. Inoue, Ergodic theorems for piecewise affine Markov maps with indifferent fixed points, Hiroshima Math. J., 24 (1994), 447-471.  doi: 10.32917/hmj/1206127920.

[20]

T. Inoue, Sojourn times in small neighborhoods of indifferent fixed points of one-dimensional dynamical systems, Ergodic Theory Dynam. Systems, 20 (2000), 241-257.  doi: 10.1017/S0143385700000110.

[21]

T. Inoue, Ergodic sums of non-integrable functions under one-dimensional dynamical systems with indifferent fixed points, Ergodic Theory Dynam. Systems, 24 (2004), 525-545.  doi: 10.1017/S0143385703000506.

[22]

M. Kesseböhmer and T. Schindler, Strong laws of large numbers for intermediately trimmed sums of i.i.d. random variables with infinite mean, J. Theoret. Probab., 32 (2019), 702-720.  doi: 10.1007/s10959-017-0802-0.

[23]

M. Kesseböhmer and T. Schindler, Strong laws of large numbers for intermediately trimmed Birkhoff sums of observables with infinite mean, Stochastic Process. Appl., 130 (2020), 7019.  doi: 10.1016/j.spa.2018.11.015.

[24]

M. Kesseböhmer and T. I. Schindler, Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type, Dyn. Syst., 35 (2020), 275-305.  doi: 10.1080/14689367.2019.1667305.

[25]

M. Kesseböhmer and T. I. Schindler, Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails, Nonlinearity, 33 (2020), 5543-5566.  doi: 10.1088/1361-6544/ab9585.

[26]

H. Kesten and R. A. Maller, Ratios of trimmed sums and order statistics, Ann. Probab., 20 (1992), 1805-1842.  doi: 10.1214/aop/1176989530.

[27]

Z. Kosloff, A universal divergence rate for symmetric Birkhoff sums in infinite ergodic theory, Trans. Amer. Math. Soc., 369 (2017), 6373-6388.  doi: 10.1090/tran/6867.

[28]

M. Lenci and S. Munday, Pointwise convergence of Birkhoff averages for global observables, Chaos, 28 (2018), 083111, 16 pp. doi: 10.1063/1.5036652.

[29]

F. Móricz, On the harmonic averages of numerical sequences, Arch. Math., (Basel) 86 (2006), 375–384. doi: 10.1007/s00013-005-1588-3.

[30]

H. Nakada and R. Natsui, On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi-Perron algorithm, Monatsh. Math., 138 (2003), 267-288.  doi: 10.1007/s00605-002-0473-4.

[31]

T. I. Schindler, Trimmed sums for observables on the doubling map, preprint, 2018, arXiv: 1810.03223.

[32]

F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

[33]

M. Thaler, Transformations on $[0, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.

[34]

R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergodic Theory Dynam. Systems, 20 (2000), 1519-1549.  doi: 10.1017/S0143385700000821.

[35]

R. Zweimüller, Surrey Notes on Infinite Ergodic Theory, Lecture notes, 2009. Available from: http://mat.univie.ac.at/%7Ezweimueller/MyPub/SurreyNotes.pdf

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