I am reading Douady and Hubbards "On the dynamics of polynomial-like mappings". I am relatively new dynamics of complex maps, and I would appreciate some help with aspects of the following.
Definition: Let $f:U'\rightarrow U$ and $g: V'\rightarrow V$ be two polynomial-like maps. They are termed quasi-conformally equivalent if there exists a quasi-conformal map $\phi$ sending a neighbourhood of $K(f)$ to a neighbourhood of $K(g)$ such that $\phi\circ f = g\circ\phi$. They are termed hybrid equivalent if additionally $\phi$ can be chosen such that $\bar\partial\phi=0$ on $K(f)$.
What does the symbol "$\bar\partial\phi$" represent in this context?
It represents $\frac{\partial \phi}{\partial \bar z}$, the Wirtinger derivative defined by $$ \frac{\partial \phi}{\partial \bar z} := \frac 12 \left(\frac{\partial \phi}{\partial x} + i\frac{\partial \phi}{\partial y}\right) $$