On effective ε-integrality in orbits of rational maps over function fields and multiplicative dependence

We prove effective bounds for the set of quasi-integral points in orbits of rational maps over function fields under some conditions, generalizing previous work of Hsia and Silverman (Pacific J Math 249(2), 321-342, 2011) for orbits over function fields of characteristic zero. We then use this to prove height bounds for algebraic functions whose orbit under a rational function has multiplicative dependent elements modulo groups of S-units, generalizing recent results over number fields.