Given a complex number $c_1$, is there a theorem, or any other way that guarantees that $c_1$ belongs to at least one Julia set? Let $c_1\in \mathbb{C}$. Let $f_{c_2}(z)=z^2+c_2$ be a quadratic polynomial with $c_2\in\mathbb{C}$, and let be the Julia set defined as 
$$J(f_{c_2})=\{ z\in\mathbb{C}:\forall n\in\mathbb{N}, |f_{c_2}^n(0)|\leq \max(c_2,2)\},$$
where $f_{c_2}^n$ is the $n$th iterate  of $f_{c_2}$. My question is: is there a theorem, or some other way that guarantees that it is always possible to find a $c_2$ such that $c_1\in J(f_{c_2})$? or is it something trivial, since $c_2$ can be any number in the complex plane, and therefore in principle it is always possible to find such parameter that would permit that $c_1\in J(f_{c_2})$?
 A: For aesthetic reasons (to get rid of subscripts) I will change the notation slightly: The question is whether for any given $w \in \mathbb{C}$ there exists $c \in \mathbb{C}$ such that $w \in J(f_c)$.
If $w=0$, you can use $c=-2$, since $J(f_{-2}) = [-2,2]$ contains $0$. If $|w| > 1/2$, you can define $c = w-w^2$, then $f_c(w) = w^2+c = w$ and $|f'(w)| = |2w| > 1$, so that $w$ is a repelling fixed point of $f_c$, and thus $w \in J(f_c)$.
For $0<|w|\le 1/2$ there is an argument using some slightly more advanced complex dynamics. Let $C$ denote the open main cardioid of the Mandelbrot set. Pick any $c_0 \in \partial C$ for which $f_{c_0}$ has a Cremer point, i.e., a non-linearizable irrationally indifferent fixed point. In particular, $J(f_{c_0})$ has no interior, so either $w \in J(f_{c_0})$, or $w \in A_\infty(f_{c_0})$, where $A_\infty$ denotes the basin of infinity. In the first case we are done, picking $c=c_0$. In the second case, it is easy to see that $w \in A_\infty(f_c)$ for $|c-c_0|$ sufficiently small, since the condition that $|f_c^n(w)| > R$ for some escape radius $R$ is an open condition on $c$, for fixed $n$. This now means that there exists $c_1 \in C$ for which $w \in A_\infty(f_{c_1})$. Also, we know that $w \in K(f_{0})^\circ$ (where the circle denotes interior of the filled-in Julia set.) Let $A$ and $B$ be the sets of those $c \in C$ for which $w$ is in $K(f_c)^\circ$ and $A_\infty(f_c)$, respectively. Both of these are open (for $A$ this requires another argument that the condition of converging to an attracting fixed point is an open condition, which you can find in complex dynamics texts) and non-empty. If $w \notin J(f_c)$ for all $c \in C$, then $C = A \cup B$ would be a partition of $C$ into open, disjoint, non-empty sets, contradicting the fact that $C$ is connected.
I don't know whether there is a more elementary argument for the case $0<|w| \le 1/2$, that would still be an interesting question.
