Escaping set

In mathematics, and particularly complex dynamics, the escaping set of an entire function ${\displaystyle f}$ consists of all points that tend to infinity under the repeated application of ${\displaystyle f}$. That is, a complex number ${\displaystyle z_{0}\in \mathbb {C} }$ belongs to the escaping set if and only if the sequence defined by ${\displaystyle z_{n+1}:=f(z_{n})}$ converges to infinity as ${\displaystyle n}$ gets large. The escaping set of ${\displaystyle f}$ is denoted by ${\displaystyle I(f)}$.

For example, for ${\displaystyle f(z)=e^{z}}$, the origin ${\displaystyle z=0}$ belongs to the escaping set, since the sequence ${\displaystyle 0,1,e,e^{e},e^{e^{e}},\dots }$ tends to infinity.