BerksonPortaFormula

The Berkson-Porta formula, named after Edward Berkson and Alberto Porta, provides an expression for the infinitesimal generator of a one-parameter semigroup of holomorphic maps of the unit disc into itself. Given a one-parameter semigroup ${ \phi_t }$ of holomorphic self-maps of the unit disc $D$, the infinitesimal generator is a linear operator $A$ acting on the space of holomorphic functions on $D$.

The Berkson-Porta formula expresses the infinitesimal generator $A$ as:

$$A(f)(z) = \lim_{t \to 0} \frac{f(\phi_t(z)) - f(z)}{t},$$

where $A(f)$ is the result of applying the operator $A$ to the function $f$, and $\phi_t(z)$ is the value of the holomorphic map at time $t$ for a given point $z$ in the unit disc.

The formula is particularly useful in the study of holomorphic dynamics and complex analysis. The infinitesimal generator $A$ can be used to analyze the local behavior of the semigroup near fixed points, as well as its global properties, such as ergodicity and mixing.