  Dynamical systems often admit geometric properties that must be taken into
account when studying their behaviour. We show that many such properties can be
encoded by means of quiver representations. These properties include classical
symmetry, hidden symmetry and feedforward structure, as well as subnetwork and
quotient relations in network dynamical systems. A quiver equivariant dynamical
system consists of a collection of dynamical systems with maps between them
that send solutions to solutions. We prove that such quiver structures are
preserved under Lyapunov-Schmidt reduction, center manifold reduction, and
normal form reduction.
