Automatic sequences are orthogonal to aperiodic multiplicative functions

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 and 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.
[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.
[3]	J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002. doi: 10.1007/b13861.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	J. L. Herning, Spectrum and Factors of Substitution Dynamical Systems, Ph.D dissertation, George Washington University, 2013, 93 pp.
[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.
[23]	I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225. doi: 10.1007/BF01949145.
[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.
[25]	J. Konieczny, Möbius orthogonality for $q$-multiplicative sequences, preprint, arXiv: 1808.06196.
[26]	J. Konieczny, On multiplicative automatic sequences, Bulletin of the London Mathematical Society, 52 (2020), 175-184. doi: 10.1112/blms.12317.
[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.
[28]	E. Lehrer, Toplogical mixing and uniquely ergodic models, Israel J. Math., 57 (1987), 239-255. doi: 10.1007/BF02772176.
[29]	M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math., 65 (1988), 241-263.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[37]	C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219-3290. doi: 10.1215/00127094-2017-0024.
[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.
[39]	P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sites/default/files/Mobius%20lectures%20Summer%202010.pdf.
[40]	J.-C. Schlage-Puchta, Completely multiplicative automatic functions, Integers, 11 (2011), A31, 8 pp. doi: 10.1515/INTEG.2011.055.
[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/
[42]	W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341. doi: 10.1007/BF01667386.
[43]	S. Yazdani, Multiplicative functions and k-automatic sequences, J. Théor. Nombres Bordeaux, 13 (2001), 651-658. doi: 10.5802/jtnb.342.

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.
[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.
[3]	J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002. doi: 10.1007/b13861.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	J. L. Herning, Spectrum and Factors of Substitution Dynamical Systems, Ph.D dissertation, George Washington University, 2013, 93 pp.
[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.
[23]	I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225. doi: 10.1007/BF01949145.
[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.
[25]	J. Konieczny, Möbius orthogonality for $q$-multiplicative sequences, preprint, arXiv: 1808.06196.
[26]	J. Konieczny, On multiplicative automatic sequences, Bulletin of the London Mathematical Society, 52 (2020), 175-184. doi: 10.1112/blms.12317.
[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.
[28]	E. Lehrer, Toplogical mixing and uniquely ergodic models, Israel J. Math., 57 (1987), 239-255. doi: 10.1007/BF02772176.
[29]	M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math., 65 (1988), 241-263.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[37]	C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219-3290. doi: 10.1215/00127094-2017-0024.
[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.
[39]	P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sites/default/files/Mobius%20lectures%20Summer%202010.pdf.
[40]	J.-C. Schlage-Puchta, Completely multiplicative automatic functions, Integers, 11 (2011), A31, 8 pp. doi: 10.1515/INTEG.2011.055.
[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/
[42]	W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341. doi: 10.1007/BF01667386.
[43]	S. Yazdani, Multiplicative functions and k-automatic sequences, J. Théor. Nombres Bordeaux, 13 (2001), 651-658. doi: 10.5802/jtnb.342.

[1]	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
[2]	Wen-Guei Hu, Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705
[3]	H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549
[4]	David Cheban. Belitskii--Lyubich conjecture for $C$-analytic dynamical systems. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Mariusz Lemańczyk Clemens Müllner
on Google Scholar
Mariusz Lemańczyk Clemens Müllner

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Mariusz Lemańczyk Clemens Müllner
on Google Scholar
Mariusz Lemańczyk Clemens Müllner

[Back to Top]