Let $P, Q\in \mathbb{Z}[x, y]$ be polynomials with zero constant terms. Assume the induced map $\phi_{P, Q}:\mathbb{Z}^2\to\mathbb{Z}^2$ is injective. An example is $P=x+x^3 y^2, Q=y+x^2 y^3$.
Can the map $f_{P, Q}:\mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}$ defined by $$f_{P, Q}(n)=\sup_{|\phi_{P, Q}(v)|\leq n}|v|$$ grow faster than $2^{2^n}$?
The norm above is the $L^1$ norm.