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.
In this article we give an expository account of the holomorphic motion theorem based on work of Mãne-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have |ǫ log ǫ| moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarz's lemma and integration over the holomorphic variable to produce Hölder continuity on the mappings. We also prove, by using holomorphic motions, that Kobayashi's and Teichmüller's metrics on the Teichmüller space of a Riemann surface coincide. Finally, we present an application of holomorphic motions to complex dynamics, that is, we prove the Fatou linearization theorem for parabolic germs by involving holomorphic motions.
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