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Smooth linearization is one of the central themes in the study of dynamicalsystems. The classical Belitskii's $C^1$ linearization theorem has been widelyused in the investigation of dynamical behaviors such as bifurcations, mixing,and chaotic behaviors due to its minimal requirement of partial second ordernon-resonances and low regularity of systems. In this article, we revisitBelitskii's $C^1$ linearization theorem by taking an approach based on smoothinvariant foliations and study this problem for a larger class of dynamicalsystems ({\it random dynamical systems}). We assumed that the linearized systemsatisfies the condition of Multiplicative Ergodic Theorem and the associatedLyapunov exponents satisfy Belitskii's partial second order non-resonantconditions. We first establish the existence of $C^{1,\beta}$ stable andunstable foliations without assuming the bunching condition for Lyapunovexponents, then prove a $C^{1,\beta}$ linearization theorem of Belitskii typefor random dynamical systems. As a result, we show that the classicalBelitskii's $C^1$ linearization theorem for a $C^{2}$ diffeomorphism $F$ indeedholds without assuming all eigenspaces of the linear system $DF(0)$ areinvariant under the nonlinear system $F$, a requirement previously imposed byBelitskii in his proof.

Smooth invariant foliations without a bunching condition and Belitskii's $C^{1}$ linearization for random dynamical systems