I have a deterministic map $M:[0,1] \times \mathbb{R} \to [0,1]\times \mathbb{R}$ such that $M(x_1,0)=(x_1,0)$ for all $0 \leq x_1 \leq 1$ and the Jacobian $$ \nabla M(x_1,0)=\begin{bmatrix} 1 & 0 \\ 0 & \alpha \end{bmatrix}, \quad \alpha<1, $$ for $0\leq x_1 \leq x_{\mathrm{threshold}}$ and $1-x_{\mathrm{threshold}} \leq x_1 \leq 1$ for some $x_{\mathrm{threshold}} \in [0,1]$.
Since the points approaching from the x-axis always stay as they are and those approaching from the y-direction are attracted for all such $(x_1,0)$, can I conclude that these points are stable fixed points? Strictly speaking they don't attract all the points initialized in a small neighborhood so I m not sure what's the nature of these fixed points.