  An important question is to describe topological conjugacy classes of
dynamical systems. Here we show that within the space of real analytic
one-dimensional maps with critical points of prescribed order, the conjugacy
class of a map is a real analytic manifold. This extends results of
Avila-Lyubich-de Melo \cite{ALM} for the quasi-quadratic unimodal case and of
Clark \cite{C} for the more general unimodal case. Their methods fail in the
case where there are several critical points, and for this reason we introduce
the new notions of {\em pruned Julia set} of a real analytic map, and associate
to a real analytic map an {\em external map} of the circle {\em with
discontinuities} and {\em a pruned polynomial-like} complex extension of the
real analytic map. Using this we are also able to show that topological
conjugacy classes are connected (something which was not even known in the
general unimodal setting). Even more, this space is contractible.
  In a companion paper, further applications of this paper will be given. It
will be shown that within any real analytic family of real analytic
one-dimensional maps, hyperbolic parameters form an open and dense subset.
