Let $F$ be a meromorphic function on $\mathbb{C}$.

I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ does not percolate for $\varepsilon$ sufficiently small, i.e. it has only bounded connected components. My function $F$ is not periodic nor a polynomial and has infinitely many zeros and poles.

Any idea of a helpful tool?

More specifically, I actually want to prove that "gradient lines" do not percolate in $E_\varepsilon$, where a gradient line is just a function $y(t)\in\mathbb{C},t\geq 0$ such that $y'(t)=F(y(t))$.

So to sum up: I want to find conditions under which there exists $\varepsilon>0$ such that $F$ does not have unbounded gradient lines lying in $E_\varepsilon$.

Thanks!

Let $F$ be a meromorphic function on $\mathbb{C}$.

I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ does not percolate for $\varepsilon$ sufficiently small, i.e. it has only bounded connected components. My function $F$ is not periodic nor a polynomial and has infinitely many zeros and poles.

Any idea of a helpful tool?

More specifically, I actually want to prove that "gradient lines" do not percolate in $E_\varepsilon$, where a gradient line is just a function $y(t)\in\mathbb{C},t\geq 0$ such that $y'(t)=F(y(t))$.

So for those who want a binary question: Are there non-periodic non-polynomial meromorphic functions such that there exists $\varepsilon>0$ such that $F$ does not have unbounded gradient lines lying in $E_\varepsilon$?

Thanks!

Let $F$ be a meromorphic function on $\mathbb{C}$.

I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ does not percolate for $\varepsilon$ sufficiently small, i.e. it has only bounded connected components. My function $F$ is actually random and stationary, which means that it is rather generic (not periodic, not a polynomial, zeros and poles of order 1 and all over the plane, etc...).

Any idea of a helpful tool?

More specifically, I actually want to prove that "gradient lines" do not percolate in $E_\varepsilon$, where a gradient line is just a function $y(t)\in\mathbb{C},t\geq 0$ such that $y'(t)=F(y(t))$.

So to sum up: I want to find conditions under which there exists $\varepsilon>0$ such that $F$ does not have unbounded gradient lines lying in $E_\varepsilon$.

Thanks!

Let $F$ be a meromorphic function on $\mathbb{C}$.

I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ does not percolate for $\varepsilon$ sufficiently small, i.e. it has only bounded connected components. My function $F$ is not periodic nor a polynomial and has infinitely many zeros and poles.

Any idea of a helpful tool?

More specifically, I actually want to prove that "gradient lines" do not percolate in $E_\varepsilon$, where a gradient line is just a function $y(t)\in\mathbb{C},t\geq 0$ such that $y'(t)=F(y(t))$.

So to sum up: I want to find conditions under which there exists $\varepsilon>0$ such that $F$ does not have unbounded gradient lines lying in $E_\varepsilon$.

Thanks!