Complex dynamics

Remarks

The standard cases are rational functions $f \colon \hat{\C} \to \hat{\C}$ of degree at least $2$, polynomials of degree at least $2$, and transcendental entire functions $f \colon \C \to \C$. Example: $f(z) = z^2 + c$ with $c \in \C$ is a polynomial, hence holomorphic on $\C$, so $(\C, f)$ is a holomorphic dynamical system. Non-example: $f(z) = \bar z^2$ is not holomorphic, since $\partial f / \partial \bar z = 2 \bar z \ne 0$ for $z \ne 0$, so its iteration is not part of complex dynamics in this sense. Rational maps of degree $1$ are Möbius maps, whose dynamics is elementary, which is why the degree is usually taken to be at least $2$.