Automatic sequences as good weights for ergodic theorems

Tanja Eisner; Jakub Konieczny

doi:10.3934/dcds.2018178

Previous Article
Lower spectral radius and spectral mapping theorem for suprema preserving mappings
DCDS Home
This Issue
Next Article
The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4

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 & 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. Google Scholar
[2]	J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563. Google Scholar
[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. Google Scholar
[4]	I. Assani, Wiener Wintner Ergodic Theorems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/4538. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	S. Drappeau and C. Müllner, Exponential sums with automatic sequences, 2017, Preprint, available at https://arXiv.org/abs/1710.01091. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	T. Eisner, Linear sequences and weighted ergodic theorems, Abstr. Appl. Anal., (2013), Art. ID 815726, 5 pp. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	N. Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061–1071. doi: 10.1017/S0143385706000204. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, 2017, Preprint, available at https://arXiv.org/abs/1611.09985. Google Scholar
[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. Google Scholar
[35]	P. LaVictoire, Universally L¹-bad arithmetic sequences, J. Anal. Math., 113 (2011), 241–263. doi: 10.1007/s11854-011-0006-y. Google Scholar
[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. Google Scholar
[37]	E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[45]	C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219–3290. doi: 10.1215/00127094-2017-0024. Google Scholar
[46]	C. Müllner, Exponential Sum Estimates and Fourier Analytic Methods for Digitally Based Dynamical Systems, PhD thesis, Technische Universit at Wien, 2017. Google Scholar
[47]	K. Petersen, Ergodic Theory, vol. 2 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989, Corrected reprint of the 1983 original. Google Scholar
[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. Google Scholar
[49]	T. Tao, Poincaré's Legacies, Pages from Year two of a Mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009. Google Scholar
[50]	P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982. Google Scholar
[51]	N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415–426. doi: 10.2307/2371534. Google Scholar
[52]	M. Wierdl, Pointwise ergodic theorems along the prime numbers, Israel J. Math., 64 (1988), 315-336. doi: 10.1007/BF02882425. Google Scholar
[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. Google Scholar

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. Google Scholar
[2]	J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563. Google Scholar
[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. Google Scholar
[4]	I. Assani, Wiener Wintner Ergodic Theorems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/4538. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	S. Drappeau and C. Müllner, Exponential sums with automatic sequences, 2017, Preprint, available at https://arXiv.org/abs/1710.01091. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	T. Eisner, Linear sequences and weighted ergodic theorems, Abstr. Appl. Anal., (2013), Art. ID 815726, 5 pp. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	N. Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061–1071. doi: 10.1017/S0143385706000204. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, 2017, Preprint, available at https://arXiv.org/abs/1611.09985. Google Scholar
[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. Google Scholar
[35]	P. LaVictoire, Universally L¹-bad arithmetic sequences, J. Anal. Math., 113 (2011), 241–263. doi: 10.1007/s11854-011-0006-y. Google Scholar
[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. Google Scholar
[37]	E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[45]	C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219–3290. doi: 10.1215/00127094-2017-0024. Google Scholar
[46]	C. Müllner, Exponential Sum Estimates and Fourier Analytic Methods for Digitally Based Dynamical Systems, PhD thesis, Technische Universit at Wien, 2017. Google Scholar
[47]	K. Petersen, Ergodic Theory, vol. 2 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989, Corrected reprint of the 1983 original. Google Scholar
[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. Google Scholar
[49]	T. Tao, Poincaré's Legacies, Pages from Year two of a Mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009. Google Scholar
[50]	P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982. Google Scholar
[51]	N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415–426. doi: 10.2307/2371534. Google Scholar
[52]	M. Wierdl, Pointwise ergodic theorems along the prime numbers, Israel J. Math., 64 (1988), 315-336. doi: 10.1007/BF02882425. Google Scholar
[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. Google Scholar

[1]	Tanja Eisner, Pavel Zorin-Kranich. Uniformity in the Wiener-Wintner theorem for nilsequences. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3497-3516. doi: 10.3934/dcds.2013.33.3497
[2]	Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237
[3]	Seung Jun Chang, Jae Gil Choi. A Cameron-Storvick theorem for the analytic Feynman integral associated with Gaussian paths on a Wiener space and applications. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2225-2238. doi: 10.3934/cpaa.2018106
[4]	Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113
[5]	Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377
[6]	Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247
[7]	Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143
[8]	Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395
[9]	Kai Tao. Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021162
[10]	J. Delon, A. Desolneux, Jose-Luis Lisani, A. B. Petro. Automatic color palette. Inverse Problems & Imaging, 2007, 1 (2) : 265-287. doi: 10.3934/ipi.2007.1.265
[11]	Oliver Jenkinson. Ergodic Optimization. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197
[12]	Ferenc A. Bartha, Hans Z. Munthe-Kaas. Computing of B-series by automatic differentiation. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 903-914. doi: 10.3934/dcds.2014.34.903
[13]	L. Igual, J. Preciozzi, L. Garrido, A. Almansa, V. Caselles, B. Rougé. Automatic low baseline stereo in urban areas. Inverse Problems & Imaging, 2007, 1 (2) : 319-348. doi: 10.3934/ipi.2007.1.319
[14]	Mariusz Lemańczyk, Clemens Müllner. Automatic sequences are orthogonal to aperiodic multiplicative functions. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6877-6918. doi: 10.3934/dcds.2020260
[15]	Bun Theang Ong, Masao Fukushima. Global optimization via differential evolution with automatic termination. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 57-67. doi: 10.3934/naco.2012.2.57
[16]	Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015
[17]	Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[18]	Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349
[19]	Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271
[20]	Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325

American Institute of Mathematical Sciences

Automatic sequences as good weights for ergodic theorems

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Automatic sequences as good weights for ergodic theorems

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors