Multiplicatively dependent points on curves and applications to algebraic dynamical systems

Bombieri, Masser and Zannier (1999) proved that the intersection of a curve defined over a number
field  with the union of all proper algebraic subgroups of the multiplicative group $\mathbb{G}_m^n$
is a set of bounded height (unless this is false for an obvious reason). It is important to note that this
set is still infinite as the degree of the points is not bounded.
In this talk we present recent results on multiplicative relations of points on algebraic curves, when
restricted to certain proper subfields of the algebraic closure of $\mathbb{Q}$, complementing those of
Bombieri, Masser and Zannier (1999). Some of our initial motivation comes from studying multiplicative
relations in orbits of algebraic dynamical systems, for which we present several results.