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We give effective bounds for the set quasi-integral points in orbits ofnon-isotrivial rational maps over function fields under some conditions,generalizing previous work of Hsia and Silverman (2011) for orbits overfunction fields of characteristic zero. We then use this to prove finitenessresults for algebraic functions whose orbit under a rational function hasmultiplicative dependent elements modulo rings of $S$-integers, generalizingrecent results over number fields.

On effective $\epsilon$-integrality in orbits of rational maps over function fields and multiplicative dependence