In dynamics, they talk about Abelian differentials on surfaces, are they the same as holomorphic differentials?
Quadratic differentials are multiple valued and can change sign as you move around a zero.
In dynamics, they talk about Abelian differentials on surfaces, are they the same as holomorphic differentials?
Quadratic differentials are multiple valued and can change sign as you move around a zero.
An Abelian differential is just a traditional name for a holomorphic or meromorphic differential on a compact Riemann surface.
A quadratic differential is just an element of $S^2(\Omega^1)$, the symmetric square of the sheaf of differentials. I do not really see what you mean when you say «[they] are multiple valued and can change sign as you move around a zero».