I was reading paper "Dynamical properties of some classes of entire functions" by A. Eremenko and M. Lyubich (http://www.numdam.org/item/?id=AIF_1992__42_4_989_0). My question is about Section 3 (more precisely, about first three paragraphs), in which authors introduced structure of a complex manifold on a parameter space of entire finite type maps.
First, authors introduce notion of topological equivalence between two entire holomorphic maps. Two entire maps $f\colon \mathbb{C} \to \mathbb{C}$ and $g \colon \mathbb{C} \to \mathbb{C}$ are called topologically equivalent if there are two homeomorphisms $\Psi\colon \mathbb{C} \to \mathbb{C}$ and $\varphi\colon \mathbb{C} \to \mathbb{C}$ such that $\Psi \circ g = f \circ \varphi$. Then they introduce set $M_g$ which is the set of all entire maps topologically equivalent to $g$. Then authors introduce additional set $M_g(\beta_1, \beta_2) \subset M_g$ (where $\beta_1, \beta_2 \in \mathbb{C} \setminus f^{-1}(S(g))$), that is subset of all maps $f$ from $M_g$ such that homeomorphisms $\Psi$ and $\varphi$ from above can be chosen such that $\varphi(\beta_1) = \beta_1$ and $\varphi_2(\beta_2) = \beta_2$.
Further, authors make a claim (p. 995, in Paragraph before Lemma 2), which I do not understand. They say: "One can easily verify that $M_g = \cup M_g(\beta_1, \beta_2)$". This claim is equivalent to saying that for any $f \in M_g$ we can choose homeomorphisms $\Psi$ and $\varphi$ from above, so $\varphi$ has two fixed points that are not from $f^{-1}(S(f))$.
Also note that in this paragraph authors work with entire maps of finite type (even though I am not sure if it is needed for the arguments mentioned earlier). Entire map $f\colon\mathbb{C} \to \mathbb{C}$ is of finite type if singular set $S(f)$ is finite. Here by singular set I mean minimal set $S(f)$ such that $f\colon \mathbb{C} \setminus f^{-1}(S(f)) \to \mathbb{C} \setminus S(f)$ is a covering. In particular, set $S(f)$ equals to closure of the set of all critical and asymptotic values of $f$.