Let \(f\) be an entire function. The set of singularities of the inverse of \(f\) consists of critical and asymptotic values in \(\mathbb{C}\) of \(f\), and is denoted by \(\mathrm{Sing}(f^{-1})\). We put, for some integer \(k,\) \[ \mathcal{P}( f ) := \overline{\bigcup_{n\ge0} f^n(\mathrm{Sing}( f^{-1}))} {\qquad \mathrm{and} \qquad}\mathcal{P}^k ( f ) :=\overline{\bigcup_{ n\ge k} f^n(\mathrm{Sing}( f^{-1}))}. \] As usual, let \(\mathcal{F}( f )\) and \(\mathcal{J} ( f )\) denote the Fatou and Julia sets of \(f\) respectively.
Definition. A transcendental entire function \(f\) satisfies the Misiurewicz condition if it is non-hyperbolic and satisfies the following conditions:
(i) \(\mathcal{P}( f ) \bigcap \mathcal{F}( f )\) is compact;
(ii ) There exists \(k \in \mathbb{N}\) such that \(\mathcal{P}^k ( f )\bigcap\mathcal{J} ( f )\) is hyperbolic.
It is clear by the definition above that Misiurewicz functions are post-singularly bounded and thus belong to the Eremenko-Lyubich class \[ \mathcal{B} := \{ f : \mathbb{C}\rightarrow\mathbb{C} \mbox{ transcendental and entire,\;} \mathrm{Sing}( f^{-1}) \mbox{\; is bounded} \}. \] We consider the Speiser class \[ \mathcal{S} := \{ f : \mathbb{C}\rightarrow\mathbb{C} \mbox{ transcendental and entire,\;} \mathrm{Sing}( f^{-1}) \mbox{\;is finite} \}. \] Functions in this class are often called Speiser functions. Let \(f, g \in \mathcal{S}\) be entire. We say that \(f\) and \(g\) are quasiconformally equivalent if there exist two quasiconformal maps \(\varphi,\psi:\mathbb{C}\to \mathbb{C}\) such that \(\varphi\circ f = g\circ\psi\). The parameter space \(\mathcal{M}_f\) in which \(f\in S\) lies is defined as the set of all functions \(g\) which are quasiconformally equivalent to \(f\) . The parameter space \(\mathcal{M}_f\) turns out to be a complex manifold of dimension \(\#\mathrm{Sing}( f^{-1}) + 2.\)
The authors prove the following results.
Theorem 1. Let \(f \in \mathcal{S}\) be Misiurewicz. Then \(f\) can be approximated by hyperbolic maps in \(\mathcal{M}_f\).
Theorem 2. Let \(f \in \mathcal{S}\) be Misiurewicz. Assume that \(\mathcal{J} ( f )\) has zero Lebesgue measure. Then \(f\) is a Lebesgue density point of hyperbolic maps in\(\mathcal{M}_f \).
Theorem 3. Let \(f\) be a Speiser function. Then the set of Misiurewicz parameters in \(\mathcal{M}_f\) has Lebesgue measure zero.