Operator ergodic theorems with Möbius "weights"

2026/07/17 by El Houcein El Abdalaoui, Michael Lin
#math.DS

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Abstract

Motivated by Sarnak's conjecture in topological dynamics for the Möbius function μ, we study, for a power-bounded T on a Banach space E, the weak convergence (*) \frac1N∑_n=1^N μ(n)Tⁿv → 0 weakly ∀ v∈ E. For that, we introduce a notion of dynamical entropy for operators, which we denote h^*_top(T), and show that if Sarnak's conjecture is true, then h^*_top(T)=0 implies the desired convergence (*). We conclude an equivalent operator formulation of Sarnak's conjecture. For several classes of operators we prove that (*) holds, and that h^*_top(T)=0.

Citations

• Topological entropy for noncompact sets

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