  We establish the existence of `time quasicrystals', tilings of the time axis
with two unit cells of different duration. These aperiodic tilings can be
constructed as slices through regular tilings of a space spanned by two
orthogonal time directions. We establish the result rigorously using the tools
of symbolic dynamics. We show that, of the ten physically-relevant classes of
one-dimensional quasicrystal, precisely two can appear as stable, attracting
trajectories in dynamical systems, which we term the infinite Pell and
Clapeyron words. These grow, via a generalization of the period-doubling
cascade, as a sequence of stable orbits with periods increasing as the Pell and
Clapeyron numbers, providing systematic approximations which can be
experimentally implemented. The results apply to a wide universality class of
dissipative nonlinear systems: we consider discrete-time maps, and
continuous-time dynamical systems, both autonomous and periodically driven.
This Paper proves and extends the results of a companion Letter, as well as
providing a pedagogical background.
