On prime endomorphisms of the free group of rank two

We study prime endomorphisms of the free group $F_2$. The main results provide lower and upper bounds for the logarithmic density ${\delta }_{\mathrm{n}\mathrm{p}}$ of non-prime endomorphisms: $$\frac{3}{4}\le {\delta }_{\mathrm{n}\mathrm{p}}\le \frac{27}{28}\mathrm{.}$$ Equivalently, we prove corresponding lower and upper bounds for the exponential decay of the probability that a random endomorphism is non-prime, aligning with a heuristic argument of Ian Agol.