  Quasiperiodic systems serve as fertile ground for studying localisation, due
to their propensity already in one dimension to exhibit rich phase diagrams
with mobility edges. The deterministic and strongly-correlated nature of the
quasiperiodic potential nevertheless offers challenges distinct from disordered
systems. Motivated by this, we present a theory of localisation in
quasiperiodic chains with nearest-neighbour hoppings, based on the convergence
of local propagators; exploiting the fact that the imaginary part of the
associated self-energy acts as a probabilistic order parameter for localisation
transitions and, importantly, admits a continued-fraction representation.
Analysing the convergence of these continued fractions, localisation or its
absence can be determined, yielding in turn the critical points and mobility
edges. Interestingly, we find anomalous scalings of the order parameter with
system size at the critical points, consistent with the fractal character of
critical eigenstates. The very nature of the theory implies that it goes far
beyond the leading-order self-consistent framework introduced by us recently
[Phys. Rev. B 103, L060201 (2021)]. Self-consistent theories at high orders are
in fact shown to be conceptually connected to the theory based on continued
fractions, and in practice converge to the same result. Results are exemplified
by analysing the theory for three families of quasiperiodic models covering a
range of behaviour.
