The Hartman-Grobman theorem for mean hyperbolic systems

In this paper, we establish Hartman-Grobman theorem for mean hyperbolic systems on the whole axis R and half axis R⁺. Mean hyperbolic systems possess fixed average contraction and expansion rates measured at sufficiently long evolution time. The coexistence of expansion and contraction behaviors in generalized stable (unstable) subspace leads to non-hyperbolic phenomena during evolution process. For mean hyperbolic systems, the Hölder regularity and continuous dependence on perturbation for $ h(t,x) $ are discussed. Particularly, the damped wave equation with variable speed of propagation can be applied to illustrate our main result.