"Estimating mutual information" [A Kraskov, H Stögbauer, P Grassberger - Physical Review E, 2004] states that
Mutual information is invariant under reparametrization of the marginal variables. If $X ′ = F(X)$ and $Y′ =G(Y)$ are homeomorphisms [ie. smooth uniquely invertible maps], then $$I(X,Y ) = I(X′,Y′)$$
This paper is also used on Wikipedia to justify the same claim.
But if that is true, doesn't that imply that if $F=G$ and $X=Y$ we get $$H(X) = I(X,X) = I(X',X') = I(F(X),F(X)) = H(F(X))$$ where $H$ is the differential entropy?
Wouldn't this be "proof" that differential entropy is also invariant wrt. such transformations (which it obviously isn't because e.g. for a constant $a$, $H(aX) = H(X) + \log|a| \neq H(X)$)?
In particular I'm wondering if $I(X,F(X))=H(X)=H(F(X))$ for all homeomorphisms $F$.
Can someone help me reduce my uncertainty?
PS: I'm talking about the continous case, ie. differential entropy and differential mutual information