  For a positive self-similar Markov process, X, we construct a local time for
the random set, $\Theta$, of times where the process reaches its past supremum.
Using this local time we describe an exit system for the excursions of X out of
its past supremum. Next, we define and study the ladder process (R,H)
associated to a positive self-similar Markov process X, namely a bivariate
Markov process with a scaling property whose coordinates are the right inverse
of the local time of the random set $\Theta$ and the process X sampled on the
local time scale. The process (R,H) is described in terms of a ladder process
linked to the L\'{e}vy process associated to X via Lamperti's transformation.
In the case where X never hits 0, and the upward ladder height process is not
arithmetic and has finite mean, we prove the finite-dimensional convergence of
(R,H) as the starting point of X tends to 0. Finally, we use these results to
provide an alternative proof to the weak convergence of X as the starting point
tends to 0. Our approach allows us to address two issues that remained open in
Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to
remove a redundant hypothesis and how to provide a formula for the entrance law
of X in the case where the underlying L\'{e}vy process oscillates.
