In my previous research, during my PhD and several years as a postdoc, I
studied endomorphisms of the projective line. More specifically, I
focused on rational functions that can be represented as the quotient of
two single-variable polynomials, working primarily over number fields
or function fields (often defined over finite fields).
The main objects of interest were the periodic and preperiodic points of
these rational functions. Periodic points are those whose forward
orbits form cycles, while preperiodic points have finite forward orbits
but are not necessarily part of a cycle. As we will see, many concepts
arising in this context can be naturally interpreted within the theory
of elliptic curves.
In addition, I investigated particular families of rational functions
characterized by a natural notion of good reduction, which enables us to
reduce the problem to a setting over finite fields. Throughout this
work, moduli spaces play a central role in understanding the structure
and classification of such dynamical systems.
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