Reversibility and equivariance in center manifolds

We show in this chapter that for a nonautonomous differential equation (in the presence of a nonuniform exponential trichotomy; see Chapter 8), the (time) reversibility and equivariance of the associated semiflow descends respectively to the reversibility and equivariance in any center manifold. We note that time-reversal symmetries are among the fundamental symmetries in many “physical” systems, both in classical and quantum mechanics. This is due to the fact that many Hamiltonian systems are reversible (see [53] for many examples). In spite of the crucial differences between reversible and equivariant dynamical systems, the techniques that are useful in any of the two contexts usually carry over to the other one. This will be apparent along the exposition. We follow closely [14].