Oscillating sequences, MMA and MMLS flows and Sarnak's conjecture
Résumé
We define an oscillating sequence, an important example of which is generated by the Mains 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.