Automatic sequences are orthogonal to aperiodic multiplicative functions

Mariusz Lemańczyk; Clemens Müllner

doi:10.3934/dcds.2020260

Previous Article
Euler integral and perihelion librations
DCDS Home
This Issue
Next Article
Maximal equicontinuous generic factors and weak model sets

December 2020, 40(12): 6877-6918. doi: 10.3934/dcds.2020260

Automatic sequences are orthogonal to aperiodic multiplicative functions

Mariusz Lemańczyk ^1, and Clemens Müllner ^2,,

1.	Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopin Street 12/18, 87-100 Toruń, Poland
2.	Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria

^* Corresponding author: Clemens Müllner

Received August 2019 Revised April 2020 Published December 2020 Early access July 2020

Given a finite alphabet

$ \mathbb{A} $

and a primitive substitution

$ \theta: \mathbb{A}\to \mathbb{A}^\lambda $

(of constant length

$ \lambda $

), let

$ (X_\theta,S) $

denote the corresponding dynamical system, where

$ X_\theta $

is the closure of the orbit via the left shift

$ S $

of a fixed point of the natural extension of

$ \theta $

to a self-map of

$ \mathbb{A}^{ {\mathbb{Z}}} $

. The main result of the paper is that all continuous observables in

$ X_\theta $

are orthogonal to any bounded, aperiodic, multiplicative function

$ \boldsymbol{u}: {\mathbb{N}}\to {\mathbb{C}} $

, i.e.

$ \lim\limits_{N\to\infty}\frac1N\sum\limits_{n\leq N}f(S^nx) \boldsymbol{u}(n) = 0 $

for all

$ f\in C(X_\theta) $

and

$ x\in X_\theta $

. In particular, each primitive automatic sequence, that is, a sequence read by a primitive finite automaton, is orthogonal to any bounded, aperiodic, multiplicative function.

Keywords: Automatic sequences, multiplicative functions, Sarnak conjecture, symbolic dynamical systems.

Mathematics Subject Classification: Primary: 37A45, 11N37, 11B85, 68R15, 11A25; Secondary: 37A05.

Citation: 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

References:

[1]	H. El Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk and T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math., 228 (2018), 707-751. doi: 10.1007/s11856-018-1784-z. Google Scholar
[2]	J.-P. Allouche and L. Goldmakher, Mock characters and the Kronecker symbol, J. Number Theory, 192 (2018), 356-372. doi: 10.1016/j.jnt.2018.04.022. Google Scholar
[3]	J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563. Google Scholar
[4]	V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk and F. K. Richter, Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics, Ergodic Theory Dynam. Systems, 39 (2019), 2332-2383. doi: 10.1017/etds.2017.130. Google Scholar
[5]	V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk and F. Richter, A structure theorem for level sets of multiplicative functions and applications, International Math. Research Notices, 5 (2020), 1300-1345. doi: 10.1093/imrn/rny040. Google Scholar
[6]	J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, From Fourier analysis and number theory to Radon transforms and geometry, Springer, New York, 2013, 67-83. doi: 10.1007/978-1-4614-4075-8_5. Google Scholar
[7]	A. Danilenko and M. Lemańczyk, Spectral multiplicities for ergodic flows, Discrete Contin. Dyn. Syst., 33 (2013), 4271-4289. doi: 10.3934/dcds.2013.33.4271. Google Scholar
[8]	F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (1977/78), 221-239. doi: 10.1007/BF00534241. Google Scholar
[9]	M. Drmota, Subsequences of automatic sequences and uniform distribution, in Uniform Distribution and Quasi-Monte Carlo Methods, Radon Series in Computational and Applied Mathematics, Vol. 15, De Gruyter, Berlin, 2014, 87-104. Google Scholar
[10]	J.-M. Deshouillers, M. Drmota and C. Müllner, Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture, Studia Math., 231 (2015), 83-95. doi: 10.4064/sm8479-2-2016. Google Scholar
[11]	T. Downarowicz and J. Serafin, Almost full entropy subshifts uncorrelated to the Möbius function, Int. Math. Res. Not. IMRN, (2019), no. 11, 3459-3472. doi: 10.1093/imrn/rnx192. Google Scholar
[12]	S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk, Sarnak's conjecture: what's new, in Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Springer, Cham, 2018,163-235. Google Scholar
[13]	S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk and C. Mauduit, Substitutions and Möbius disjointness, in Ergodic theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Amer. Math. Soc., Providence, RI, 2016,151-174. doi: 10.1090/conm/678. Google Scholar
[14]	N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002. doi: 10.1007/b13861. Google Scholar
[15]	N. Frantzikinakis and B. Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc., 30 (2017), 67-157. doi: 10.1090/jams/857. Google Scholar
[16]	N. Frantzikinaki and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math., 187 (2018), 869-931. doi: 10.4007/annals.2018.187.3.6. Google Scholar
[17]	N. Frantzikinakis and B. Host, Furstenberg systems of bounded multiplicative functions and applications, International Mathematics Research Notices, 2 (2020), 1073-7928. doi: 10.1093/imrn/rnz037. Google Scholar
[18]	H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49. doi: 10.1007/BF01692494. Google Scholar
[19]	E. Glasner, Ergodic theory via joinings, in Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. Google Scholar
[20]	A. Gomilko, D. Kwietniak and M. Lemańczyk, Sarnak's conjecture implies the Chowla conjecture along a subsequence, in Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 2213, Springer, Cham, 2018. doi: 10.1007/978-3-319-74908-2_12. Google Scholar
[21]	J. L. Herning, Spectrum and Factors of Substitution Dynamical Systems, Ph.D dissertation, George Washington University, 2013, 93 pp. Google Scholar
[22]	B. Host and F. Parreau, Homomorphismes entre systèmes dynamiques définis par substitutions, Ergodic Theory Dynam. Systems, 9 (1989), 469-477. doi: 10.1017/S0143385700005113. Google Scholar
[23]	I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225. doi: 10.1007/BF01949145. Google Scholar
[24]	O. Klurman and P. Kurlberg, A note on multiplicative automatic sequences, II, Bulletin of the London Mathematical Society, 52 (2020), 185-188. doi: 10.1112/blms.12318. Google Scholar
[25]	J. Konieczny, Möbius orthogonality for $q$-multiplicative sequences, preprint, arXiv: 1808.06196. Google Scholar
[26]	J. Konieczny, On multiplicative automatic sequences, Bulletin of the London Mathematical Society, 52 (2020), 175-184. doi: 10.1112/blms.12317. Google Scholar
[27]	J. Kułaga-Przymus and M. Lemańczyk, The Möbius function and continuous extensions of rotations, Monatsh. Math., 178 (2015), 553-582. doi: 10.1007/s00605-015-0808-6. Google Scholar
[28]	E. Lehrer, Toplogical mixing and uniquely ergodic models, Israel J. Math., 57 (1987), 239-255. doi: 10.1007/BF02772176. Google Scholar
[29]	M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math., 65 (1988), 241-263. Google Scholar
[30]	M. Lemańczyk and M. K. Mentzen, Compact subgroups in the centralizers of natural factors of an ergodic group extension of a rotation determine all factors, Ergodic Theory Dynam. Systems, 10 (1990), 763-776. doi: 10.1017/S0143385700005885. Google Scholar
[31]	K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math., 183 (2016), 1015-1056. doi: 10.4007/annals.2016.183.3.6. Google Scholar
[32]	K. Matomäki, M. Radziwiłł and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167-2196. doi: 10.2140/ant.2015.9.2167. Google Scholar
[33]	C. Mauduit and J. Rivat, Sur un problème de Gelfond : La somme des chiffres des nombres premiers, Ann. of Math., 171 (2010), 1591-1646. doi: 10.4007/annals.2010.171.1591. Google Scholar
[34]	C. Mauduit and J. Rivat, Prime numbers along Rudin-Shapiro sequences, J. Eur. Math. Soc., 17 (2015), 2595-2642. doi: 10.4171/JEMS/566. Google Scholar
[35]	M. K. Mentzen, Invariant sub-$\sigma$-algebras for substitutions of constant length, Studia Math., 92 (1989), 257-273. doi: 10.4064/sm-92-3-257-273. Google Scholar
[36]	M. K. Mentzen, Ergodic properties of group extensions of dynamical systems with discrete spectra, Studia Math., 101 (1991), 19-31. doi: 10.4064/sm-101-1-19-31. Google Scholar
[37]	C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219-3290. doi: 10.1215/00127094-2017-0024. Google Scholar
[38]	M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, second edition, Lecture Notes in Mathematics, Vol. 1294, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6. Google Scholar
[39]	P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sites/default/files/Mobius%20lectures%20Summer%202010.pdf. Google Scholar
[40]	J.-C. Schlage-Puchta, Completely multiplicative automatic functions, Integers, 11 (2011), A31, 8 pp. doi: 10.1515/INTEG.2011.055. Google Scholar
[41]	T. Tao, The logarithmically averaged and non logarithmically averaged Chowla conjectures, https://terrytao.wordpress.com/2017/10/20/the-logarithmically-averaged-and-non-logarithmically-averaged-chowla-conjectures/ Google Scholar
[42]	W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341. doi: 10.1007/BF01667386. Google Scholar
[43]	S. Yazdani, Multiplicative functions and k-automatic sequences, J. Théor. Nombres Bordeaux, 13 (2001), 651-658. doi: 10.5802/jtnb.342. Google Scholar

show all references

References:

[1]	H. El Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk and T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math., 228 (2018), 707-751. doi: 10.1007/s11856-018-1784-z. Google Scholar
[2]	J.-P. Allouche and L. Goldmakher, Mock characters and the Kronecker symbol, J. Number Theory, 192 (2018), 356-372. doi: 10.1016/j.jnt.2018.04.022. Google Scholar
[3]	J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563. Google Scholar
[4]	V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk and F. K. Richter, Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics, Ergodic Theory Dynam. Systems, 39 (2019), 2332-2383. doi: 10.1017/etds.2017.130. Google Scholar
[5]	V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk and F. Richter, A structure theorem for level sets of multiplicative functions and applications, International Math. Research Notices, 5 (2020), 1300-1345. doi: 10.1093/imrn/rny040. Google Scholar
[6]	J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, From Fourier analysis and number theory to Radon transforms and geometry, Springer, New York, 2013, 67-83. doi: 10.1007/978-1-4614-4075-8_5. Google Scholar
[7]	A. Danilenko and M. Lemańczyk, Spectral multiplicities for ergodic flows, Discrete Contin. Dyn. Syst., 33 (2013), 4271-4289. doi: 10.3934/dcds.2013.33.4271. Google Scholar
[8]	F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (1977/78), 221-239. doi: 10.1007/BF00534241. Google Scholar
[9]	M. Drmota, Subsequences of automatic sequences and uniform distribution, in Uniform Distribution and Quasi-Monte Carlo Methods, Radon Series in Computational and Applied Mathematics, Vol. 15, De Gruyter, Berlin, 2014, 87-104. Google Scholar
[10]	J.-M. Deshouillers, M. Drmota and C. Müllner, Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture, Studia Math., 231 (2015), 83-95. doi: 10.4064/sm8479-2-2016. Google Scholar
[11]	T. Downarowicz and J. Serafin, Almost full entropy subshifts uncorrelated to the Möbius function, Int. Math. Res. Not. IMRN, (2019), no. 11, 3459-3472. doi: 10.1093/imrn/rnx192. Google Scholar
[12]	S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk, Sarnak's conjecture: what's new, in Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Springer, Cham, 2018,163-235. Google Scholar
[13]	S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk and C. Mauduit, Substitutions and Möbius disjointness, in Ergodic theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Amer. Math. Soc., Providence, RI, 2016,151-174. doi: 10.1090/conm/678. Google Scholar
[14]	N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002. doi: 10.1007/b13861. Google Scholar
[15]	N. Frantzikinakis and B. Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc., 30 (2017), 67-157. doi: 10.1090/jams/857. Google Scholar
[16]	N. Frantzikinaki and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math., 187 (2018), 869-931. doi: 10.4007/annals.2018.187.3.6. Google Scholar
[17]	N. Frantzikinakis and B. Host, Furstenberg systems of bounded multiplicative functions and applications, International Mathematics Research Notices, 2 (2020), 1073-7928. doi: 10.1093/imrn/rnz037. Google Scholar
[18]	H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49. doi: 10.1007/BF01692494. Google Scholar
[19]	E. Glasner, Ergodic theory via joinings, in Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. Google Scholar
[20]	A. Gomilko, D. Kwietniak and M. Lemańczyk, Sarnak's conjecture implies the Chowla conjecture along a subsequence, in Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 2213, Springer, Cham, 2018. doi: 10.1007/978-3-319-74908-2_12. Google Scholar
[21]	J. L. Herning, Spectrum and Factors of Substitution Dynamical Systems, Ph.D dissertation, George Washington University, 2013, 93 pp. Google Scholar
[22]	B. Host and F. Parreau, Homomorphismes entre systèmes dynamiques définis par substitutions, Ergodic Theory Dynam. Systems, 9 (1989), 469-477. doi: 10.1017/S0143385700005113. Google Scholar
[23]	I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225. doi: 10.1007/BF01949145. Google Scholar
[24]	O. Klurman and P. Kurlberg, A note on multiplicative automatic sequences, II, Bulletin of the London Mathematical Society, 52 (2020), 185-188. doi: 10.1112/blms.12318. Google Scholar
[25]	J. Konieczny, Möbius orthogonality for $q$-multiplicative sequences, preprint, arXiv: 1808.06196. Google Scholar
[26]	J. Konieczny, On multiplicative automatic sequences, Bulletin of the London Mathematical Society, 52 (2020), 175-184. doi: 10.1112/blms.12317. Google Scholar
[27]	J. Kułaga-Przymus and M. Lemańczyk, The Möbius function and continuous extensions of rotations, Monatsh. Math., 178 (2015), 553-582. doi: 10.1007/s00605-015-0808-6. Google Scholar
[28]	E. Lehrer, Toplogical mixing and uniquely ergodic models, Israel J. Math., 57 (1987), 239-255. doi: 10.1007/BF02772176. Google Scholar
[29]	M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math., 65 (1988), 241-263. Google Scholar
[30]	M. Lemańczyk and M. K. Mentzen, Compact subgroups in the centralizers of natural factors of an ergodic group extension of a rotation determine all factors, Ergodic Theory Dynam. Systems, 10 (1990), 763-776. doi: 10.1017/S0143385700005885. Google Scholar
[31]	K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math., 183 (2016), 1015-1056. doi: 10.4007/annals.2016.183.3.6. Google Scholar
[32]	K. Matomäki, M. Radziwiłł and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167-2196. doi: 10.2140/ant.2015.9.2167. Google Scholar
[33]	C. Mauduit and J. Rivat, Sur un problème de Gelfond : La somme des chiffres des nombres premiers, Ann. of Math., 171 (2010), 1591-1646. doi: 10.4007/annals.2010.171.1591. Google Scholar
[34]	C. Mauduit and J. Rivat, Prime numbers along Rudin-Shapiro sequences, J. Eur. Math. Soc., 17 (2015), 2595-2642. doi: 10.4171/JEMS/566. Google Scholar
[35]	M. K. Mentzen, Invariant sub-$\sigma$-algebras for substitutions of constant length, Studia Math., 92 (1989), 257-273. doi: 10.4064/sm-92-3-257-273. Google Scholar
[36]	M. K. Mentzen, Ergodic properties of group extensions of dynamical systems with discrete spectra, Studia Math., 101 (1991), 19-31. doi: 10.4064/sm-101-1-19-31. Google Scholar
[37]	C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219-3290. doi: 10.1215/00127094-2017-0024. Google Scholar
[38]	M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, second edition, Lecture Notes in Mathematics, Vol. 1294, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6. Google Scholar
[39]	P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sites/default/files/Mobius%20lectures%20Summer%202010.pdf. Google Scholar
[40]	J.-C. Schlage-Puchta, Completely multiplicative automatic functions, Integers, 11 (2011), A31, 8 pp. doi: 10.1515/INTEG.2011.055. Google Scholar
[41]	T. Tao, The logarithmically averaged and non logarithmically averaged Chowla conjectures, https://terrytao.wordpress.com/2017/10/20/the-logarithmically-averaged-and-non-logarithmically-averaged-chowla-conjectures/ Google Scholar
[42]	W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341. doi: 10.1007/BF01667386. Google Scholar
[43]	S. Yazdani, Multiplicative functions and k-automatic sequences, J. Théor. Nombres Bordeaux, 13 (2001), 651-658. doi: 10.5802/jtnb.342. Google Scholar

[1]	Wen-Guei Hu, Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705
[2]	H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549
[3]	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
[4]	David Cheban. Belitskii--Lyubich conjecture for $C$-analytic dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 945-959. doi: 10.3934/dcdsb.2015.20.945
[5]	Alexander Gomilko, Mariusz Lemańczyk, Thierry de la Rue. On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao. Journal of Modern Dynamics, 2021, 17: 529-555. doi: 10.3934/jmd.2021018
[6]	Daniel Glasscock, Andreas Koutsogiannis, Florian Karl Richter. Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5891-5921. doi: 10.3934/dcds.2019258
[7]	David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253
[8]	David Cheban. I. U. Bronshtein's conjecture for monotone nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1095-1113. doi: 10.3934/dcdsb.2019008
[9]	Elena K. Kostousova. On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty. Conference Publications, 2011, 2011 (Special) : 864-873. doi: 10.3934/proc.2011.2011.864
[10]	João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465
[11]	Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331
[12]	Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172
[13]	Felipe García-Ramos, Brian Marcus. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 729-746. doi: 10.3934/dcds.2019030
[14]	David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 287-300. doi: 10.3934/dcdss.2009.2.287
[15]	Simone Fiori. Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2785-2808. doi: 10.3934/dcdsb.2014.19.2785
[16]	Yong Li, Hongren Wang, Xue Yang. Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2607-2623. doi: 10.3934/dcdsb.2018123
[17]	Chunlei Xie, Yujuan Sun. Construction and assignment of orthogonal sequences and zero correlation zone sequences for applications in CDMA systems. Advances in Mathematics of Communications, 2020, 14 (1) : 1-9. doi: 10.3934/amc.2020001
[18]	Moises Delgado, Heeralal Janwa. Some new results on the conjecture on exceptional APN functions and absolutely irreducible polynomials: The gold case. Advances in Mathematics of Communications, 2017, 11 (2) : 389-396. doi: 10.3934/amc.2017033
[19]	El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449
[20]	Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

American Institute of Mathematical Sciences

Automatic sequences are orthogonal to aperiodic multiplicative functions

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Automatic sequences are orthogonal to aperiodic multiplicative functions

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors