  Inertial manifold theory, saddle point property and exponential dichotomy
have been treated as different topics in the literature with different proofs.
As a common feature, they all have the purpose of `splitting' the space to
understand the dynamics. We present a unified proof for the inertial manifold
theorem, which as a local consequence yields the saddle-point property with a
fine structure of invariant manifolds and the roughness of exponential
dichotomy. In particular, we use these tools in order to establish the
hyperbolicity of certain global solutions for non-autonomous parabolic partial
differential equations.
