Sarnak’s conjecture brings together number theory, ergodic theory, and dynamical systems. Motivated by this conjecture, we started a study in ergodic theory about orders of oscillating sequences and minimally mean attractable (MMA) and minimally mean-L-stable (MMLS) flows. The Möbius function in number theory gives an example of oscillating sequences of order $d$ for all $d\gt0$. From the dynamical systems point of view, we found another class of examples of oscillating sequences of order $d$ for all $d\gt0$. All equicontinuous flows are MMA and MMLA. I will talk about two non-trivial examples of MMA and MMLS flows that are not equicontinuous. One is a Denjoy counterexample in circle homeomorphisms and the other is an infinitely renormalizable one-dimensional map. I will show that all oscillation sequences of order 1 are linearly disjoint with (or meanly orthogonal to) MMA and MMLA flows. Thus, we confirm Sarnak’s conjecture for a large class of zero topological entropy flows. For oscillating sequences of order $d\gt1$, I will show that they are linearly disjoint from all affine distal flows on the $d$-torus. One of the consequences is that Sarnak’s conjecture holds for all zero topological entropy affine flows on the $d$-torus and some nonlinear zero topological entropy flows on the $d$-torus. I will also review some current developments after our work on this topic about flows with the quasi-discrete spectrum and the Thue-Morse sequence, which has zero topological entropy and small Gowers norms and thus is a higher-order oscillating sequence.

Yunping Jiang