  Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic
dynamical systems, as they allow one to distill their dynamical structure. We
consider here the Lorenz 1963 model with the classic parameters' value. We
investigate how a chaotic trajectory can be approximated using a complete set
of UPOs up to symbolic dynamics' period 14. At each instant, we rank the UPOs
according to their proximity to the position of the orbit in the phase space.
We study this process from two different perspectives. First, we find that
longer period UPOs overwhelmingly provide the best local approximation to the
trajectory. Second, we construct a finite-state Markov chain by studying the
scattering of the orbit between the neighbourhood of the various UPOs. Each UPO
and its neighbourhood are taken as a possible state of the system. Through the
analysis of the subdominant eigenvectors of the corresponding stochastic matrix
we provide a different interpretation of the mixing processes occurring in the
system by taking advantage of the concept of quasi-invariant sets.
