  We present in this work an exact renormalization group (RG) treatment of a
one-dimensional $p$-wave superconductor. The model proposed by Kitaev consists
of a chain of spinless fermions with a $p$-wave gap. It is a paradigmatic model
of great actual interest since it presents a weak pairing superconducting phase
that has Majorana fermions at the ends of the chain. Those are predicted to be
useful for quantum computation. The RG allows to obtain the phase diagram of
the model and to study the quantum phase transition from the weak to the strong
pairing phase. It yields the attractors of these phases and the critical
exponents of the weak to strong pairing transition. We show that the weak
pairing phase of the model is governed by a chaotic attractor being non-trivial
from both its topological and RG properties. In the strong pairing phase the RG
flow is towards a conventional strong coupling fixed point. Finally, we propose
an alternative way for obtaining $p$-wave superconductivity in a
one-dimensional system without spin-orbit interaction.
