Before revision:  Here we extend this analysisto any translation-invariant directed transition structure ( displacement and diffusion), showing that a single set of eigenvectors supports prediction via displacement-specific eigenvalues. This unifies the prediction framework with traditional models of grid cells firing driven by self-motion to perform path integration .
After revision:  Here we extend this analysisto any translation-invariant directed transition structure ( displacement and diffusion), showing that a single set of eigenvectors can support predictions over arbitrary directed actions via action-specific eigenvalues. We show how to define a "sense of direction" to combine actions to reach a target state (ignoring task-specific deviations from translation-invariance), and demonstrate that adding the Fourier representations to a deep Q network aids policy learning in continuous control tasks. We show the equivalence between the generalised prediction framework and traditional models of grid cells firing driven by self-motion to perform path integration .