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

Abstract

We define anoscillating sequence, an important example of which is generated by the Möbius function in number theory. We also define aminimally mean attractable (MMA) flowand aminimally 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.