Let $K$ a cone (we can suppose that it is the cone of nonnegative vectors in $\mathbb{R}^n$), $f:K\to K$ an homogeneous (of degree 1) function. Suppose that $$\lim_{n\to \infty}\frac{f^n(x)}{\|f^n(x)\|} = u\in K$$ for every $x\in \operatorname{Int}{K}$ ($n$-th power is respect to composition of maps).
Is it true (and under which hypotesis) that for every $x\in \operatorname{Int}{K}$ it exists $\lambda(x)>0$ such that the following holds? $$\lim_{n\to \infty}f^n(x) = \lambda(x) u$$
The claim is equivalent to $\lim_n f^n(x/\lambda)=u$, by the homogeneous hypothesis. So to me, it seems forced that $\lambda$ must be equals to $\lim_n \|f^n(x)\|$.
I thought about this because the assumption is that "the projection of $f^n(x)$ on the unite sphere (intersected with the cone $V$) has limit", so when you "come back to check what $f^n(x)$ is doing", you have no informations about his behavior for big $n$, only that his direction stabilize near the direction of $u$, the norm of $f^n(x)$ can still tends to infinity..