One Note on N-Sensitivity in Mean Forms

In 2022, Li, Ye and Yu introduced multivariate sensitivity version of notions of mean m-Sensitivity and m-sensitivity in the mean for m ≥ 2. In this manuscript, we mainly focus on the investigation of multivariate sensitivity in mean forms. First, we prove that for a linear system, the equivalence between mean m-sensitivity and m-sensitivity in the mean holds without any more conditions. Subsequently, we introduce the notion of m-equicontinuity in the mean, and obtain an Auslander–Yorke type dichotomy between m-equicontinuity in the mean and m-sensitivity in the mean for minimal systems. As a consequence, we demonstrate that the equivalence between mean m-sensitivity and m-sensitivity in the mean is valid under the condition of minimality for a general system, thereby affirmatively resolving the conjecture proposed in [Li, J., Ye, X. D., Yu, T.: Equicontinuity and sensitivity in mean forms. J. Dynam. Differential Equations, 34, 133–154 (2022)].