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

TL;DR

This paper extends Belitskii's $C^1$ linearization theorem to random dynamical systems using smooth invariant foliations, removing the bunching condition and relaxing invariance requirements on eigenspaces.

Contribution

It introduces a new approach based on smooth invariant foliations to prove $C^{1,eta}$ linearization for a broader class of random dynamical systems without bunching conditions.

Findings

Established existence of $C^{1,eta}$ stable and unstable foliations without bunching.

Proved a $C^{1,eta}$ linearization theorem for random dynamical systems.

Showed classical Belitskii's $C^1$ linearization holds without eigenspace invariance.

Abstract

Smooth linearization is one of the central themes in the study of dynamical systems. The classical Belitskii's $C^1$ linearization theorem has been widely used in the investigation of dynamical behaviors such as bifurcations, mixing, and chaotic behaviors due to its minimal requirement of partial second order non-resonances and low regularity of systems. In this article, we revisit Belitskii's $C^1$ linearization theorem by taking an approach based on smooth invariant foliations and study this problem for a larger class of dynamical systems ({\it random dynamical systems}). We assumed that the linearized system satisfies the condition of Multiplicative Ergodic Theorem and the associated Lyapunov exponents satisfy Belitskii's partial second order non-resonant conditions. We first establish the existence of $C^{1,\beta}$ stable and unstable foliations without assuming the bunching condition for Lyapunov exponents, then prove a $C^{1,\beta}$ linearization theorem of Belitskii type for random dynamical systems. As a result, we show that the classical Belitskii's $C^1$ linearization theorem for a $C^{2}$ diffeomorphism $F$ indeed holds without assuming all eigenspaces of the linear system $DF(0)$ are invariant under the nonlinear system $F$, a requirement previously imposed by Belitskii in his proof.