In the book, I read some remarks about the criticality:
Iterations of the renormalization (group) map generate a sequence of points in the space of couplings, which we call a renormalization group trajectory. Since the correlation length is reduced by a factor $b<1$ at each step, a typical renormalization-group trajectory tends to take the system away from criticality. Because the correlation length is infinite at the critical point, it takes an infinite of iterations to leave that point. In general, a system is critical not only at a given point in the coupling space but on a whole "hypersurface", which we call the critical surface, or sometimes the critical line.
I think these remarks are fine. But then the authors give a statement:
The critical surface is the set of points in coupling space whose renormalization-group trajectories end up at the fixed point: $$\lim_{n\rightarrow\infty} T^n(J)=J_c,$$ where $T$ is the renormalization-group map and $J$ represents the couplings in general.
The question is, how to properly understand this statement? For this, I particularly have two small questions. Why the critical line need to hit the fixed point? And why the critical line need to end up at the fixed point rather leave the fixed point under the renormalization group transformation?