Around fixed point of renormalization group In the general formulation of renormalization group in "statistical mechanics" by P.K.Pathria, each point in parameter space is represented by a vector $\vec{K}$ and the transformed vector would be given by
$$\vec{K}'=R(\vec{K})$$
If we look at what happens when it gets very close to the fixed point $\vec{K}^*$, then we would have 
$$\vec{K}'=\vec{K}^*+\vec{k}'=R(\vec{K}^*+\vec{k})+\mathcal{O}(k_i^2) $$
where both $\vec{k}'$ and $\vec{k}$ will be very small.
Now the problem is that in this book, the author directly puts 
$$R(\vec{K}^*+\vec{k})=R(\vec{K}^*)+R(\vec{k})=\vec{K}^*+R(\vec{k})$$
and then conclude that the two deviations are related by
$$\vec{k}'=R(\vec{k})$$
The step $R(\vec{K}^*+\vec{k})=R(\vec{K}^*)+R(\vec{k})$ looks obviously problematic to me, or did I miss anything?
 A: It turns out to be a mistake in this version of the book (2nd edition), I checked the latest version (3rd edition) and it has been corrected there. Naturally, the corrected one would be
$$R(\vec{K}^*+\vec{k})=R(\vec{K}^*)+R'(\vec{K}^*)\vec{k}+...$$
which after linearization simply becomes $\vec{K}^*+R'(\vec{K}^*)\vec{k}$, and then we have a linearized relation between $\vec{k}'$ and $\vec{k}$, which is 
$$\vec{k}'=R'(\vec{K}^*)\vec{k}$$
A: You missed the fact that this operator $R$ is a linear operator, at least for sufficiently small $\vec{k}$. This is a linearization of the renormalization group transformation around the fixed point $\vec{K}^*$. Let $T$ be a renormalization group transformation. If you expand around the fixed point $\vec{K}^*$ you get:
$$
T(\vec{K}^*+\vec{k})=T(\vec{K}^*)+[T^{(1)}(\vec{K}^*)]\vec{k}+\mathcal{O}(k_i^2)
$$
Where $[T^{(1)}(\vec{K}^*)]$ is a linear operator acting on $\vec{k}$. Actually, making $R=T$ until first order in $k$ give us:
$$
R(\vec{K}^*+\vec{k})=T(\vec{K}^*)+[T^{(1)}(\vec{K}^*)]\vec{k}
$$
and
$$
R(\vec{K}^*+\vec{0})=T(\vec{K}^*)=\vec{K}^*
$$
$$
[T^{(1)}(\vec{K}^*)]\vec{k}=R(\vec{K}^*+\vec{k})-\vec{K}^*=(\vec{K}^*+\vec{k}')-\vec{K}^*=\vec{k}'
$$
Then $R$ is an affine transformation and $R(\vec{K}^*+...)-\vec{K}^*$ is a linear transformation. So, what the author realy mean is:
$$
\vec{k}'=R(\vec{K}^*+\vec{k})-\vec{K}^*=[T^{(1)}(\vec{K}^*)]\vec{k}
$$
