  In this paper we prove complex bounds, also referred to as a priori bounds,
for real analytic (and even C3) interval maps. This means that we associate to
such a map a complex box mapping (which provides a kind of Markov structure),
together with universal geometric bounds on the shape of the domains. Such
bounds show that the first return maps to these domains are well-controlled,
and consequently form one of the corner stones in many recent results on
one-dimensional dynamics: renormalisation theory, rigidity results, density of
hyperbolicity, local connectivity.

Dynamical Systems