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Oscillating sequences, MMA and MMLS flows and Sarnak’s conjecture

Published online by Cambridge University Press:  14 March 2017

AI-HUA FAN and
YUNPING JIANG
AI-HUA FAN
Affiliation:
School of Mathematics and Statistics, Central China Normal University, 152 Avenue Luoyu, 430077 Wuhan, Hubei, ChinaLAMFA UMR 7352, CNRS, Faculté des Sciences, Université de Picardie Jules Verne, 33, rue Saint Leu, 80039 Amiens CEDEX 1, France email ai-hua.fan@u-picardie.fr
YUNPING JIANG
Affiliation:
Department of Mathematics, Queens College of the City University of New York, Flushing, NY 11367-1597, USADepartment of Mathematics, Graduate School of the City University of New York, 365 Fifth Avenue, New York, NY 10016, USA email yunping.jiang@qc.cuny.edu
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Abstract

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We define an oscillating sequence, an important example of which is generated by the Möbius function in number theory. We also define a minimally mean attractable (MMA) flow and a minimally mean-L-stable (MMLS) flow. One of the main results is that any oscillating sequence is linearly disjoint from all MMA and MMLS flows. In particular, this confirms Sarnak’s conjecture for all MMA and MMLS flows. We provide several examples of flows that are MMA and MMLS. These examples include flows defined by all $p$ -adic polynomials of integral coefficients, all $p$ -adic rational maps with good reduction, all automorphisms of the $2$ -torus with zero topological entropy, all diagonalizable affine maps of the $2$ -torus with zero topological entropy, all Feigenbaum maps, and all orientation-preserving circle homeomorphisms.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018, pp. 1709 - 1744
Copyright
© Cambridge University Press, 2017 

References

Auslander, J.. Mean-L-stable systems. Illinois J. Math. 3 (1959), 566–579.CrossRefGoogle Scholar
Bourgain, J.. On the correlation of the Möbius function with random rank one systems. Preprint, 2011,arXiv:1112.1031.Google Scholar
Bourgain, J., Sarnak, P. and Ziegler, T.. Disjointness of Möbius from horocycle flows. From Fourier Analysis and Number Theory to Radon Transforms and Geometry (Developments in Mathematics, 28) . Springer, New York, 2013, pp. 67–83.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . Eds. Dold, A. and Eckmann, B.. Springer, Heidelberg, 1975, 108 pp.CrossRefGoogle Scholar
Bruckner, A. M. and Hu, T.. Equicontinuity of iterates of an interval map. Tamkang J. Math. 21(3) (1990), 287–294.CrossRefGoogle Scholar
Cho, S., Min, K. and Yang, S.. Equicontinuity of iterates of a map on the circle. Bull. Korean Math. Soc. 30(2) (1993), 239–244.Google Scholar
Collet, P. and Eckmann, J.-P.. Iterated Maps on the Interval as Dynamical Systems (Progress in Physics, 1) . Birkhäuser, Boston, 1980.Google Scholar
Coullet, P. and Tresser, C.. Itération dèndomorphismes et groupe de renormalisation. C. R. Acad. Sci. Paris Ser. A–B 287 (1978), A577–A580 (J. Phys. Coll. 39 (C5) (1978), 25–28; supplement au 39:8).Google Scholar
Daboussi, H. and Delange, H.. On multiplicative arithmetical functions whose modulus does not exceed one. J. Lond. Math. Soc. (2) 26(2) (1982), 245–264.CrossRefGoogle Scholar
Davenport, H.. On some infinite series involving arithmetical functions (II). Q. J. Math. Oxford 8 (1937), 313–320.CrossRefGoogle Scholar
Delange, H.. Sur les fonctions multiplicatives complexes de module. Ann. Inst. Fourier 44(5) (1994), 1323–1349.CrossRefGoogle Scholar
Denjoy, A.. Sur les courbes definies par les équations différentielles à la surface du tore. J. Math. Pures Appl. (9) 11 (1932), 333–375.Google Scholar
El Abdalaoui, H., Kulaga-Przymus, J., Lemanczyk, M. and de La Rue, Th.. The Chowla and the Sarnak conjectures from ergodic theory point of view. Preprint, 2014, arXiv:1410.1673.Google Scholar
Fan, A. H., Fan, S. L., Liao, L. M. and Wang, Y. F.. On minimal decomposition of $p$ -adic rational functions with good reduction. Preprint, 2015.Google Scholar
Fan, A. H. and Liao, L. M.. On minimal decomposition of p-adic polynomial dynamical systems. Adv. Math. 228 (2011), 2116–2144.CrossRefGoogle Scholar
Fan, A. H. and Schmeling, J.. Everywhere divergence of one-sided ergodic Hilbert transform. Preprint.Google Scholar
Fan, A. H. and Schneider, D.. On an inequality of Littlewood–Salem. Ann. Inst. Henri Poincaré 39(2) (2003), 193–216.CrossRefGoogle Scholar
Feigenbaum, M.. Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19 (1978), 25–52.CrossRefGoogle Scholar
Feigenbaum, M.. The universal metric properties of non-linear transformations. J. Stat. Phys. 21 (1979), 669–706.CrossRefGoogle Scholar
Fomin, S.. On dynamical systems with a purely point spectrum. Dokl. Akad. Nauk SSSR 77 (1951), 29–32 (in Russian).Google Scholar
Frantzikinakis, N. and Host, B.. Higher order Fourier analysis of multiplicative functions and applications. Preprint, 2014, arXiv:1403.0945.Google Scholar
Ge, L. M.. Topology of natural numbers and entropy of arithmetic functions. Preprint, 2015.CrossRefGoogle Scholar
Gottschalk, W. H.. Orbit-closure decompositions and almost periodic properties. Bull. Amer. Math. Soc. (N.S.) 50 (1944), 915–919.CrossRefGoogle Scholar
Green, B. and Tao, T.. Quadratic uniformity of the Möbius function. Preprint, 2006, arXiv:math/0606087v2 [math.NT], 22 September 2007.Google Scholar
Green, B. and Tao, T.. The Möbius function is strongly orthogonal to nilsequences. Ann. of Math. (2) 175 (2012), 541–566.CrossRefGoogle Scholar
Hua, L. G.. Additive Theory of Prime Numbers (Translations of Mathematical Monographs, 13) . American Mathematical Society, Providence, RI, 1966.Google Scholar
Jiang, Y.. Renormalization and Geometry in One-Dimensional and Complex Dynamics (Advanced Series in Nonlinear Dynamics, 10) . World Scientific, River Edge, NJ, 1996.CrossRefGoogle Scholar
Jiang, Y., Morita, T. and Sullivan, D.. Expanding direction of period doubling operator. Comm. Math. Phys. 144(3) (1992), 509–520.CrossRefGoogle Scholar
Kahane, J. P.. Some Random Series of Functions. Cambridge University Press, Cambridge, 1985.Google Scholar
Kahane, J. P. and Saias, E.. Fonction complètement multiplicatives de somme nulle. Preprint, 2015,https://hal.archives-ouvertes.fr/hal-01177065.Google Scholar
Karagulyan, D.. On Möbius orthogonality for interval maps of zero entropy and orientation-preserving circle homeomorphisms. Ark. Mat., to appear, doi:10.1007/s11512-014-0208-5.CrossRefGoogle Scholar
Krengel, U.. Ergodic Theorems. Walter de Gruyter, New York, 1982.Google Scholar
Lanford, O. E. III. A computer-assisted proof of Feigenbaum conjectures. Bull. Amer. Math. Soc. (N.S.) 6 (1982), 427–434.CrossRefGoogle Scholar
Li, J., Tu, S. and Ye, X. D.. Mean-equicontinuity and mean sensitivity. Ergod. Th. & Dynam. Sys. 35 (2015), 2587–2612.CrossRefGoogle Scholar
Liu, J. Y. and Sarnak, P.. The Möbius function and distal flows. Duke Math. J. 164(7) (2015), 1353–1399.CrossRefGoogle Scholar
Liu, J. Y. and Zhan, T.. Exponential sums involving the Möbius function. Indag. Math. (N.S.) 7(2) (1996), 271–278.Google Scholar
Lyubich, M.. Feigenbaum–Coullet–Tresser universality and Milnor’s hairiness conjecture. Ann. of Math. (2) 149 (1999), 319–420.CrossRefGoogle Scholar
Mauduit, C. and Rivat, J.. Sur un problème de Gelfond: la somme des chiffres des nombres premiers. Ann. of Math. (2) 171 (2010), 1591–1646.CrossRefGoogle Scholar
McMullen, C.. Complex Dynamics and Renormalization (Annals of Mathematics Studies, 135) . Princeton University Press, Princeton, NJ, 1994.Google Scholar
McMullen, C.. Renormalization and 3-Manifolds Which Fiber Over the Circle (Annals of Mathematics Studies, 135) . Princeton University Press, Princeton, NJ, 1996.CrossRefGoogle Scholar
de Melo, W. and van Strien, S.. One-Dimensional Dynamics. Springer, Heidelberg, 1993.CrossRefGoogle Scholar
Milnor, J. and Thurston, W.. On Iterated Maps of the Interval: I and II (Lecture Notes in Mathematics, 1342) . Springer, New York, 1988, pp. 465–563.Google Scholar
Oxtoby, J. C.. Ergodic sets. Bull. Amer. Math. Soc. 58 (1952), 116–136.CrossRefGoogle Scholar
Parry, W.. Topics in Ergodic Theory, Vol. 75. Cambridge University Press, Cambridge, 1981.Google Scholar
Petersen, K.. Ergodic Theory (Cambridge Studies in Advanced Mathematics, 2) . Cambridge University Press, Cambridge, 1983.CrossRefGoogle Scholar
Sarnak, P.. Three Lectures on the Möbius Function, Randomness and Dynamics (IAS Lecture Notes) . 2009, http://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf.Google Scholar
Sarnak, P.. Möbius randomness and dynamics. Not. S. Afr. Math. Soc. 43 (2012), 89–97.Google Scholar
Silverman, J.. The Arithmetic of Dynamical Systems (Graduate Texts in Mathematics, 241) . Springer, New York, 2007.CrossRefGoogle Scholar
Sinai, Ya.. Introduction to Ergodic Theory. Princeton University Press, Princeton, NJ, 1976.Google Scholar
Sullivan, D.. Bounds, quadratic differentials, and renormalization conjectures. Mathematics into the Twenty-First Century (American Mathematical Society Centennial Publications, 2) . American Mathematical Society, Providence, RI, 1992, pp. 417–466.Google Scholar
Valaristos, A.. Equicontinuity of iterates of circle maps. Int. J. Math. Math. Sci. 21(3) (1998), 453–458.CrossRefGoogle Scholar
Zygmund, A.. Trigonometric Series. Cambridge University Press, Cambridge, 1959.Google Scholar