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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, China LAMFA 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, USA Department 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

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.

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

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