Automatic sequences as good weights for ergodic theorems

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August 2018, 38(8): 4087-4115. doi: 10.3934/dcds.2018178

Automatic sequences as good weights for ergodic theorems

Tanja Eisner ^1, and Jakub Konieczny ^2,3,

1.	Institute of Mathematics, University of Leipzig, P.O. Box 100 920, 04009 Leipzig, Germany
2.	Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
3.	Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Łojasiewicza 6, 30-348 Kraków, Poland

Received November 2017 Revised March 2018 Published May 2018

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems. We show that automatic sequences are good weights in $ L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $ L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $ L^r$, $ r>1$.

Keywords: Automatic sequence, ergodic theorem, Wiener–Wintner.

Mathematics Subject Classification: Primary: 11B85, 37A30; Secondary: 47A35, 68Q45.

Citation: Tanja Eisner, Jakub Konieczny. Automatic sequences as good weights for ergodic theorems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4087-4115. doi: 10.3934/dcds.2018178

References:

[1]	J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoret. Comput. Sci., 98 (1992), 163–197. doi: 10.1016/0304-3975(92)90001-V.
[2]	J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563.
[3]	I. Assani, A weighted pointwise ergodic theorem, Ann. Inst. H. Poincaré Probab. Statist., 34 (1998), 139–150. doi: 10.1016/S0246-0203(98)80021-6.
[4]	I. Assani, Wiener Wintner Ergodic Theorems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/4538.
[5]	I. Assani and R. Moore, A good universal weight for multiple recurrence averages with commuting transformations in norm, Ergodic Theory Dynam. Systems, 37 (2017), 1009–1025, available at https://arXiv.org/abs/1506.06730. doi: 10.1017/etds.2015.76.
[6]	I. Assani and K. Presser, A survey of the return times theorem, in Ergodic Theory and Dynamical Systems, De Gruyter Proc. Math., De Gruyter, Berlin, 2014, 19–58.
[7]	J. P. Bell, M. Coons and K. G. Hare, The minimal growth of a k-regular sequence, Bull. Aust. Math. Soc., 90 (2014), 195–203. doi: 10.1017/S0004972714000197.
[8]	J. P. Bell, M. Coons and K. G. Hare, Growth degree classification for finitely generated semigroups of integer matrices, Semigroup Forum, 92 (2016), 23–44. doi: 10.1007/s00233-015-9725-1.
[9]	A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Trans. Amer. Math. Soc., 288 (1985), 307–345, URL http://dx.doi.org/10.2307/2000442. doi: 10.1090/S0002-9947-1985-0773063-8.
[10]	D. Berend, M. Lin, J. Rosenblatt and A. Tempelman, Modulated and subsequential ergodic theorems in Hilbert and Banach spaces, Ergodic Theory Dynam. Systems, 22 (2002), 1653–1665. doi: 10.1017/S0143385702000846.
[11]	J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ. Math., Inst. Hautes Études Sci., 69 (1985), 5–45, URL http://www.numdam.org/item?id=PMIHES_1989__69__5_0, With an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein.
[12]	J. Bourgain, H. Furstenberg, Y. Katznelson and D. S. Ornstein, Appendix on return-time sequences, Publ. Math., Inst. Hautes Études Sci., 69 (1985), 42–45, URL http://www.numdam.org/item?id=PMIHES_1989__69__42_0.
[13]	Z. Buczolich and R. D. Mauldin, Divergent square averages, Ann. of Math. (2), 171 (2010), 1479–1530. doi: 10.4007/annals.2010.171.1479.
[14]	Q. Chu, Convergence of weighted polynomial multiple ergodic averages, Proc. Amer. Math. Soc., 137 (2009), 1363–1369. doi: 10.1090/S0002-9939-08-09614-7.
[15]	D. Cömez, M. Lin and J. Olsen, Weighted ergodic theorems for mean ergodic L₁- contractions, Trans. Amer. Math. Soc., 350 (1998), 101–117. doi: 10.1090/S0002-9947-98-01986-2.
[16]	C. Cuny and M. Weber, Ergodic theorems with arithmetical weights, Israel J. Math., 217 (2017), 139-180. doi: 10.1007/s11856-017-1441-y.
[17]	S. Drappeau and C. Müllner, Exponential sums with automatic sequences, 2017, Preprint, available at https://arXiv.org/abs/1710.01091.
[18]	M. Drmota and J. F. Morgenbesser, Generalized Thue-Morse sequences of squares, Israel J. Math., 190 (2012), 157–193. doi: 10.1007/s11856-011-0186-2.
[19]	P. Dumas, Joint spectral radius, dilation equations, and asymptotic behavior of radix-rational sequences, Linear Algebra Appl., 438 (2013), 2107–2126. doi: 10.1016/j.laa.2012.10.013.
[20]	P. Dumas, Asymptotic expansions for linear homogeneous divide-and-conquer recurrences: algebraic and analytic approaches collated, Theoret. Comput. Sci., 548 (2014), 25–53. doi: 10.1016/j.tcs.2014.06.036.
[21]	M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, vol. 259 of Graduate Texts in Mathematics, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.
[22]	T. Eisner, A polynomial version of Sarnak's conjecture, C. R. Math. Acad. Sci. Paris, 353 (2015), 569–572. doi: 10.1016/j.crma.2015.04.009.
[23]	T. Eisner, Linear sequences and weighted ergodic theorems, Abstr. Appl. Anal., (2013), Art. ID 815726, 5 pp.
[24]	T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, vol. 272 of Graduate Texts in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.
[25]	T. Eisner and B. Krause, (Uniform) convergence of twisted ergodic averages, Ergodic Theory Dynam. Systems, 36 (2016), 2172–2202. doi: 10.1017/etds.2015.6.
[26]	T. Eisner and P. Zorin-Kranich, Uniformity in the Wiener-Wintner theorem for nilsequences, Discrete Contin. Dyn. Syst., 33 (2013), 3497–3516. doi: 10.3934/dcds.2013.33.3497.
[27]	E. H. El Abdalaoui, J. Ku laga-Przymus, M. Lemańczyk and T. de la Rue, The Chowla and the Sarnak conjectures from ergodic theory point of view, Discrete Contin. Dyn. Syst., 37 (2017), 2899–2944. doi: 10.3934/dcds.2017125.
[28]	A.-H. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory and Dynamical Systems, (2017), 1-15. doi: 10.1017/etds.2017.81.
[29]	N. Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061–1071. doi: 10.1017/S0143385706000204.
[30]	A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., 13 (1967/1968), 259-265. doi: 10.4064/aa-13-3-259-265.
[31]	B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465–540. doi: 10.4007/annals.2012.175.2.2.
[32]	B. Host and B. Kra, Uniformity seminorms on $ \ell^∞$ and applications, J. Anal. Math., 108 (2009), 219–276. doi: 10.1007/s11854-009-0024-1.
[33]	J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, 2017, Preprint, available at https://arXiv.org/abs/1611.09985.
[34]	B. Krause and P. Zorin-Kranich, A random pointwise ergodic theorem with Hardy field weights, Illinois J. Math., 59 (2015), 663–674, URL http://projecteuclid.org/euclid.ijm/1475266402.
[35]	P. LaVictoire, Universally L¹-bad arithmetic sequences, J. Anal. Math., 113 (2011), 241–263. doi: 10.1007/s11854-011-0006-y.
[36]	E. Lesigne, Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner, Ergodic Theory Dynam. Systems, 10 (1990), 513– 521. doi: 10.1017/S014338570000571X.
[37]	E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784.
[38]	E. Lesigne and C. Mauduit, Propriétés ergodiques des suites q-multiplicatives, Compositio Math., 100 (1996), 131–169, URL http://www.numdam.org/item?id=CM_1996__100_2_131_0.
[39]	E. Lesigne, C. Mauduit and B. Mossé, Le théorème ergodique le long d'une suite q-multiplicative, Compositio Math., 93 (1994), 49–79, URL http://www.numdam.org/item?id=CM_1994__93_1_49_0.
[40]	M. Lin, J. Olsen and A. Tempelman, On modulated ergodic theorems for Dunford-Schwartz operators, in Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, IL, 1997), 43 (1999), 542–567, URL http://projecteuclid.org/euclid.ijm/1255985110.
[41]	B. Martin, C. Mauduit and J. Rivat, Théorème des nombres premiers pour les fonctions digitales, Acta Arith., 165 (2014), 11–45. doi: 10.4064/aa165-1-2.
[42]	C. Mauduit, Automates finis et ensembles normaux, Ann. Inst. Fourier (Grenoble), 36 (1986), 1–25, URL http://www.numdam.org/item?id=AIF_1986__36_2_1_0. doi: 10.5802/aif.1044.
[43]	C. Mauduit, Propriétés arithmétiques des substitutions et automates infinis, Ann. Inst. Fourier (Grenoble), 56 (2006), 2525–2549, URL http://aif.cedram.org/item?id=AIF_200__56_7_2525_0, Numération, pavages, substitutions. doi: 10.5802/aif.2248.
[44]	C. Mauduit and A. Sárközy, On finite pseudorandom binary sequences. Ⅱ. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences, a further construction, J. Number Theory, 73 (1998), 256–276. doi: 10.1006/jnth.1998.2286.
[45]	C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219–3290. doi: 10.1215/00127094-2017-0024.
[46]	C. Müllner, Exponential Sum Estimates and Fourier Analytic Methods for Digitally Based Dynamical Systems, PhD thesis, Technische Universit at Wien, 2017.
[47]	K. Petersen, Ergodic Theory, vol. 2 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989, Corrected reprint of the 1983 original.
[48]	M. Queffélec, Substitution Dynamical Systems—Spectral Analysis, vol. 1294 of Lecture Notes in Mathematics, 2nd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.
[49]	T. Tao, Poincaré's Legacies, Pages from Year two of a Mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009.
[50]	P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.
[51]	N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415–426. doi: 10.2307/2371534.
[52]	M. Wierdl, Pointwise ergodic theorems along the prime numbers, Israel J. Math., 64 (1988), 315-336. doi: 10.1007/BF02882425.
[53]	P. Zorin-Kranich, A double return times theorem, Israel J. Math., 204 (2014), 85–96, available at https://arXiv.org/abs/1506.05748. doi: 10.1007/s11856-014-1112-1.

show all references

References:

[1]	J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoret. Comput. Sci., 98 (1992), 163–197. doi: 10.1016/0304-3975(92)90001-V.
[2]	J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563.
[3]	I. Assani, A weighted pointwise ergodic theorem, Ann. Inst. H. Poincaré Probab. Statist., 34 (1998), 139–150. doi: 10.1016/S0246-0203(98)80021-6.
[4]	I. Assani, Wiener Wintner Ergodic Theorems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/4538.
[5]	I. Assani and R. Moore, A good universal weight for multiple recurrence averages with commuting transformations in norm, Ergodic Theory Dynam. Systems, 37 (2017), 1009–1025, available at https://arXiv.org/abs/1506.06730. doi: 10.1017/etds.2015.76.
[6]	I. Assani and K. Presser, A survey of the return times theorem, in Ergodic Theory and Dynamical Systems, De Gruyter Proc. Math., De Gruyter, Berlin, 2014, 19–58.
[7]	J. P. Bell, M. Coons and K. G. Hare, The minimal growth of a k-regular sequence, Bull. Aust. Math. Soc., 90 (2014), 195–203. doi: 10.1017/S0004972714000197.
[8]	J. P. Bell, M. Coons and K. G. Hare, Growth degree classification for finitely generated semigroups of integer matrices, Semigroup Forum, 92 (2016), 23–44. doi: 10.1007/s00233-015-9725-1.
[9]	A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Trans. Amer. Math. Soc., 288 (1985), 307–345, URL http://dx.doi.org/10.2307/2000442. doi: 10.1090/S0002-9947-1985-0773063-8.
[10]	D. Berend, M. Lin, J. Rosenblatt and A. Tempelman, Modulated and subsequential ergodic theorems in Hilbert and Banach spaces, Ergodic Theory Dynam. Systems, 22 (2002), 1653–1665. doi: 10.1017/S0143385702000846.
[11]	J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ. Math., Inst. Hautes Études Sci., 69 (1985), 5–45, URL http://www.numdam.org/item?id=PMIHES_1989__69__5_0, With an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein.
[12]	J. Bourgain, H. Furstenberg, Y. Katznelson and D. S. Ornstein, Appendix on return-time sequences, Publ. Math., Inst. Hautes Études Sci., 69 (1985), 42–45, URL http://www.numdam.org/item?id=PMIHES_1989__69__42_0.
[13]	Z. Buczolich and R. D. Mauldin, Divergent square averages, Ann. of Math. (2), 171 (2010), 1479–1530. doi: 10.4007/annals.2010.171.1479.
[14]	Q. Chu, Convergence of weighted polynomial multiple ergodic averages, Proc. Amer. Math. Soc., 137 (2009), 1363–1369. doi: 10.1090/S0002-9939-08-09614-7.
[15]	D. Cömez, M. Lin and J. Olsen, Weighted ergodic theorems for mean ergodic L₁- contractions, Trans. Amer. Math. Soc., 350 (1998), 101–117. doi: 10.1090/S0002-9947-98-01986-2.
[16]	C. Cuny and M. Weber, Ergodic theorems with arithmetical weights, Israel J. Math., 217 (2017), 139-180. doi: 10.1007/s11856-017-1441-y.
[17]	S. Drappeau and C. Müllner, Exponential sums with automatic sequences, 2017, Preprint, available at https://arXiv.org/abs/1710.01091.
[18]	M. Drmota and J. F. Morgenbesser, Generalized Thue-Morse sequences of squares, Israel J. Math., 190 (2012), 157–193. doi: 10.1007/s11856-011-0186-2.
[19]	P. Dumas, Joint spectral radius, dilation equations, and asymptotic behavior of radix-rational sequences, Linear Algebra Appl., 438 (2013), 2107–2126. doi: 10.1016/j.laa.2012.10.013.
[20]	P. Dumas, Asymptotic expansions for linear homogeneous divide-and-conquer recurrences: algebraic and analytic approaches collated, Theoret. Comput. Sci., 548 (2014), 25–53. doi: 10.1016/j.tcs.2014.06.036.
[21]	M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, vol. 259 of Graduate Texts in Mathematics, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.
[22]	T. Eisner, A polynomial version of Sarnak's conjecture, C. R. Math. Acad. Sci. Paris, 353 (2015), 569–572. doi: 10.1016/j.crma.2015.04.009.
[23]	T. Eisner, Linear sequences and weighted ergodic theorems, Abstr. Appl. Anal., (2013), Art. ID 815726, 5 pp.
[24]	T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, vol. 272 of Graduate Texts in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.
[25]	T. Eisner and B. Krause, (Uniform) convergence of twisted ergodic averages, Ergodic Theory Dynam. Systems, 36 (2016), 2172–2202. doi: 10.1017/etds.2015.6.
[26]	T. Eisner and P. Zorin-Kranich, Uniformity in the Wiener-Wintner theorem for nilsequences, Discrete Contin. Dyn. Syst., 33 (2013), 3497–3516. doi: 10.3934/dcds.2013.33.3497.
[27]	E. H. El Abdalaoui, J. Ku laga-Przymus, M. Lemańczyk and T. de la Rue, The Chowla and the Sarnak conjectures from ergodic theory point of view, Discrete Contin. Dyn. Syst., 37 (2017), 2899–2944. doi: 10.3934/dcds.2017125.
[28]	A.-H. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory and Dynamical Systems, (2017), 1-15. doi: 10.1017/etds.2017.81.
[29]	N. Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061–1071. doi: 10.1017/S0143385706000204.
[30]	A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., 13 (1967/1968), 259-265. doi: 10.4064/aa-13-3-259-265.
[31]	B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465–540. doi: 10.4007/annals.2012.175.2.2.
[32]	B. Host and B. Kra, Uniformity seminorms on $ \ell^∞$ and applications, J. Anal. Math., 108 (2009), 219–276. doi: 10.1007/s11854-009-0024-1.
[33]	J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, 2017, Preprint, available at https://arXiv.org/abs/1611.09985.
[34]	B. Krause and P. Zorin-Kranich, A random pointwise ergodic theorem with Hardy field weights, Illinois J. Math., 59 (2015), 663–674, URL http://projecteuclid.org/euclid.ijm/1475266402.
[35]	P. LaVictoire, Universally L¹-bad arithmetic sequences, J. Anal. Math., 113 (2011), 241–263. doi: 10.1007/s11854-011-0006-y.
[36]	E. Lesigne, Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner, Ergodic Theory Dynam. Systems, 10 (1990), 513– 521. doi: 10.1017/S014338570000571X.
[37]	E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784.
[38]	E. Lesigne and C. Mauduit, Propriétés ergodiques des suites q-multiplicatives, Compositio Math., 100 (1996), 131–169, URL http://www.numdam.org/item?id=CM_1996__100_2_131_0.
[39]	E. Lesigne, C. Mauduit and B. Mossé, Le théorème ergodique le long d'une suite q-multiplicative, Compositio Math., 93 (1994), 49–79, URL http://www.numdam.org/item?id=CM_1994__93_1_49_0.
[40]	M. Lin, J. Olsen and A. Tempelman, On modulated ergodic theorems for Dunford-Schwartz operators, in Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, IL, 1997), 43 (1999), 542–567, URL http://projecteuclid.org/euclid.ijm/1255985110.
[41]	B. Martin, C. Mauduit and J. Rivat, Théorème des nombres premiers pour les fonctions digitales, Acta Arith., 165 (2014), 11–45. doi: 10.4064/aa165-1-2.
[42]	C. Mauduit, Automates finis et ensembles normaux, Ann. Inst. Fourier (Grenoble), 36 (1986), 1–25, URL http://www.numdam.org/item?id=AIF_1986__36_2_1_0. doi: 10.5802/aif.1044.
[43]	C. Mauduit, Propriétés arithmétiques des substitutions et automates infinis, Ann. Inst. Fourier (Grenoble), 56 (2006), 2525–2549, URL http://aif.cedram.org/item?id=AIF_200__56_7_2525_0, Numération, pavages, substitutions. doi: 10.5802/aif.2248.
[44]	C. Mauduit and A. Sárközy, On finite pseudorandom binary sequences. Ⅱ. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences, a further construction, J. Number Theory, 73 (1998), 256–276. doi: 10.1006/jnth.1998.2286.
[45]	C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219–3290. doi: 10.1215/00127094-2017-0024.
[46]	C. Müllner, Exponential Sum Estimates and Fourier Analytic Methods for Digitally Based Dynamical Systems, PhD thesis, Technische Universit at Wien, 2017.
[47]	K. Petersen, Ergodic Theory, vol. 2 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989, Corrected reprint of the 1983 original.
[48]	M. Queffélec, Substitution Dynamical Systems—Spectral Analysis, vol. 1294 of Lecture Notes in Mathematics, 2nd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.
[49]	T. Tao, Poincaré's Legacies, Pages from Year two of a Mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009.
[50]	P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.
[51]	N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415–426. doi: 10.2307/2371534.
[52]	M. Wierdl, Pointwise ergodic theorems along the prime numbers, Israel J. Math., 64 (1988), 315-336. doi: 10.1007/BF02882425.
[53]	P. Zorin-Kranich, A double return times theorem, Israel J. Math., 204 (2014), 85–96, available at https://arXiv.org/abs/1506.05748. doi: 10.1007/s11856-014-1112-1.

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2020 Impact Factor: 1.392

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2020 Impact Factor: 1.392

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