Counting roots of unity on the graphs of Laurent series over non-Archimedean local fields

We completely classify Laurent series converging on the unit circle over a non-Archimedean local field (of any characteristic) that map infinitely many roots of unity to roots of unity. For a given Laurent series $f$ over a field of positive characteristic with residue field ${\mathbb{F}}_q$, we prove effective bounds for the number of possible roots of unity in terms of the number of zeroes of the auxilliary function $f(x^q)-f(x{)}^q$ on the unit circle. In characteristic $0$ our bound is still effective but also depends on the ramification degree of the base field over ${\mathbb{Q}}_p$ as well as the size of the coefficients of $f$. This has applications to the Manin-Mumford conjecture in ${\mathbb{G}}_m^2$. In characteristic $0$, this work builds upon a pigeon-hole based method by Schmidt.