Fixed points in triangular lattice 2d Ising model

Given the following recursion relations:

$$K^\prime = 2K \Big(\frac{e^{3K}+e^{-K}}{e^{3K}+3e^{-K}}\Big)^2$$

and

$$h^\prime = 3h\Big(\frac{e^{3K}+e^{-K}}{e^{3K}+3e^{-K}}\Big),$$

where $$K = J/k_B T$$ and $$h = H/k_B T$$ with $$H$$ the external magnetic field.

I want to reconstruct Figure 9.7 in Goldenfeld Lectures on phase transitions and renormalization group. In order to do this I want to determine the fixed points but they do not coincide with the RG flow diagram sketched in the book. I thought the fixed points (defined by $$K^\prime = K$$ and $$h^\prime =h$$) were:

(1) $$K=0$$, $$h \rightarrow \pm \infty$$

(2) $$K=h=0$$

(3) $$K \rightarrow \infty$$, $$h=0$$

(4) $$K \rightarrow \infty$$, $$h \rightarrow \pm \infty$$

(5) $$K \approx 0.34$$, $$h=0$$ or $$h \rightarrow \pm \infty.$$

Where (5) describes the only non-trivial point. However in the plot I only see the points (1), (2), (3). And for (5) only the solution $$K \approx 0.34$$, $$h = 0$$. Why am I overcounting the fixed points in this problem? And how do the fixed points relate to the temperature e.g. in case (3), there we should have $$T \rightarrow 0$$ but how can we then find $$h = 0$$?