= Contraction principle for large deviations
{title2=$J(y)=\inf_{f(x)=y}I(x)$}
If $X_N$ satisfies a <large deviation principle> in $E$ with <good rate function> $I$ and $f:E\to F$ is a <continuous map>, then $f(X_N)$ has the same <large-deviation speed> and <good rate function>
$$
J(y)=\inf\{I(x):f(x)=y\}.
$$
The preimages of open and closed sets give the two <large deviation principle> inequalities. The identity $\{J\leq M\}=f(\{I\leq M\})$ follows from compactness and <lower semicontinuity> by taking minimizing sequences in the fibres; hence $J$ is good.
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