Let
M
2
M_2
be the space of quadratic rational maps
f
:
P
1
→
P
1
f:\textbf {P}^1\to \textbf {P}^1
, modulo the action by conjugation of the group of Möbius transformations. In this paper a compactification
X
X
of
M
2
M_2
is defined, as a modification of Milnor’s
M
¯
2
≃
CP
2
\overline {M}_2\simeq \textbf {CP}^2
, by choosing representatives of a conjugacy class
[
f
]
∈
M
2
[f]\in M_2
such that the measure of maximal entropy of
f
f
has conformal barycenter at the origin in
R
3
\textbf {R}^3
and taking the closure in the space of probability measures. It is shown that
X
X
is the smallest compactification of
M
2
M_2
such that all iterate maps
[
f
]
↦
[
f
n
]
∈
M
2
n
[f]\mapsto [f^n]\in M_{2^n}
extend continuously to
X
→
M
¯
2
n
X \to \overline {M}_{2^n}
, where
M
¯
d
\overline {M}_d
is the natural compactification of
M
d
M_d
coming from geometric invariant theory.