  Revisiting the Lorenz '63 equations in the regime of large of Rayleigh
number, we study the occurrence of periodic solutions and quantify
corresponding time averages of selected quantities. Perturbing from the
integrable limit of infinite $\rho$, we provide a full proof of existence and
stability of symmetric periodic orbits, which confirms previous partial
results. Based on this, we expand time averages in terms of elliptic integrals
with focus on the much studied average `transport', which is the mode reduced
excess heat transport of the convection problem that gave rise to the Lorenz
equations. We find a hysteresis loop between the periodic attractors and the
non-zero equilibria of the Lorenz equations. These have been proven to maximize
transport, and we show that the transport takes arbitrarily small values in the
family of periodic attractors. In particular, when the non-zero equilibria are
unstable, we quantify the difference between maximal and typically realized
values of transport. We illustrate these results by numerical simulations and
show how they transfer to various extended Lorenz models.
