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Title:
Oscillations in Ergodic Theory and Sarnak's Conjecture in Number Theory
Speaker:
Abstract:
Because of Sarnak's conjecture in number theory, I will define orders of oscillating sequences and minimally mean attractable (MMA) and minimally mean-L-stable (MMLS) dynamical systems. The Mobius function in number theory gives an example of oscillating sequences of order d for all d>0. I will show another example of oscillating sequences of order d for all d>0 from the dynamical systems point of view. All equicontinuous dynamical systems are MMA and MMLA. I will talk about two kinds of non-trivial examples of MMA and MMLS dynamical systems which are not equicontinuous. One is a Denjoy counterexample in circle homeomorphisms and the other is an infinitely renormalizable one -dimensional maps. I will show that all oscillation sequences of order 1 are linearly disjoint with (or meanly orthogonal to) MMA and MMLA dynamical systems. Thus, we confirm Sarnak's conjecture for a large class of zero topological entropy dynamical systems. For oscillating sequences of order d>1, I will show that they are linearly disjoint from all affine distal dynamical systems on the d-torus. One of the consequences is that Sanark's conjecture holds for all zero topological entropy affine dynamical systems on the d-torus as well as some nonlinear zero topological entropy dynamical systems on the d-torus. I will also review some current developments after our work on this topic and works from other people.
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