Non-recurrent meromorphic functions

Theorem 1. Let $f: \mathbb{C} \to \overline{\mathbb{C}}$ be a transcendental meromorphic function in the class $\mathcal{B}$ such that $J(f) = \overline{\mathbb{C}}$ . If additionally f satisfies the Misiurewicz condition then every compact and forward invariant set K, disjoint from the set Crit(f) of critical points, is hyperbolic. For rational mappings Theorem 1 is known to follow from the work of Mañé, although it has not been stated in the literature. In the meromorphic case, a somewhat different method has to be used. The proof of Theorem 1 comes in the next section. In Section 3 we will give another theorem, under some additional hypotheses, about non-existence of invariant line-fields.

Lemma 1. For every R > 0 and every $x \in \mathbb{C}$ there is $\varepsilon > 0$ such that every connected component $D_i$ of $f^{-1}(B(x,\varepsilon))$ with $f^{-1}(B(x,\varepsilon)) \cap \overline{B(0,R)}$ $\neq \emptyset$ is a topological disk containing exactly one preimage of x and $f: D_i \to B(x,\varepsilon)$ is a proper map. Proof. We fix R > 0 and $x \in \mathbb{C}$ . If $R_1 > R$ , for $\varepsilon$ small enough all connected components of $f^{-1}(B(x,\varepsilon))$ which intersect B(0,R) are contained in $B(0,R_1)$ , otherwise we would end up with a preimage of x on every circle with radius between R and $R_1$ . By continuity of f, the boundary of each such component $D_i$ is mapped into $C(x,\varepsilon)$ . Hence, $f_{|D_i|}$ is proper onto $B(x,\varepsilon)$ , in particular it contains a preimage $x_i$ of x. So, there are only finitely many components $D_i$ and, possibly for smaller $\varepsilon$ , each contains exactly one preimage of x. From the local form of an analytic function, for $\varepsilon$ small enough each $D_i$ is a topological disk. $\blacksquare$

Lemma 2. For every $\varepsilon > 0$ , M > 0, and R > 0 there is $\delta > 0$ such that for every $n \geq 1$ if $U, K \subset \mathbb{C}$ , $K \subset U$ , U is a topological disk while K is compact and simply connected, $\operatorname{mod}(U \setminus K) \geq M$ , $\operatorname{diam} K < \delta$ , and g is an analytic branch of $f^{-n}$ defined on U such that $\operatorname{dist}(g(K), 0) < R$ , then $\operatorname{diam} g(K) < \varepsilon$ . Proof. Let $M, R, \varepsilon > 0$ be given and suppose that the assertion does not hold. Then choose a sequence of triples $(U_p, K_p, g_p)$ with diam $K_p \to 0$ such that $g_p : U_p \to \mathbb{C}$ is an inverse branch of $f^{n_p}$ and diam $g_p(K_p) \geq \varepsilon > 0$ . Also, make sure that $\mathrm{dist}(0, g_p(K_p)) < R$ and $\mathrm{mod}(U_p \setminus K_p) \geq 2M$ for every p. By decreasing each $U_p$ and using the Koebe distortion theorem, we may assume without loss of generality that the distortion of $g_p$ on $U_p$ is bounded uniformly in p, $g_p(U_p)$ are quasi-disks of uniform distortion and $\mathrm{mod}(U_p \setminus K_p) \geq M$ . Since the diameters of $K_p$ go to 0, but the diameters of $g_p(K_p)$ stay bounded away from 0, $\mathrm{inf}\{|g_p'(z)|: z \in U_p\} \to \infty$ as $p \to \infty$ . There is $\eta > 0$ , depending on $\varepsilon$ and M, such that for every p the set $g_p(U_p)$ contains a ball of radius $\eta$ which intersects B(0,R). By taking an accumulation point of the centers of such balls for various p, we find a ball of radius $\eta/2$ which is contained in $g_p(U_p)$ for infinitely many p, without loss of generality for all p. On this ball, we get a sequence of iterates $f^{n_p}$ with derivatives tending uniformly toward 0. This means that $n_p$ tends to $\infty$ with p. Then, however, we get a contradiction with the density of repelling periodic orbits in the Julia set, which in our case is the whole plane. $\blacksquare$ The proof of Lemma 2 is one of few places where the hypothesis that the Julia set is the whole sphere gets used. One might expect by analogy with the rational case that an assumption of that type is superfluous. Lemma 2, however, obviously fails even in the rational case if the Fatou set contains attracting or parabolic basins, so we have to leave that issue open for now.

Lemma 3. For every $R, \eta > 0$ there is $\delta > 0$ such that for every $x, y \in B(0,R)$ if x = f(y) and $\operatorname{dist}(y,\operatorname{Crit}(f)) \geq \eta$ , then there is an inverse branch g of f defined on $B(x,\delta)$ such that g(x) = y.

Lemma 4. Let U be a topological disk in the plane, and $K \subset U$ be compact and simply connected such that $\operatorname{mod}(U \setminus K) \geq M$ . Suppose that $K_i$ , $i = 0, \ldots, m$ , are simply connected sets which satisfy $K_0 = K$ , $f(K_i) = K_{i-1}$ for $i = 1, \ldots, m$ . Analogously, define $U_i$ to be the connected component of $f^{-i}(U)$ which contains $K_i$ . Assume also $K_m \cap P(f) \neq \emptyset$ and that there is an inverse branch $\gamma: U \to U_{m-1}$ of $f^{m-1}$ for which $\gamma(K) = K_{m-1}$ . For every M>0 there is $\varepsilon>0$ such that if $\operatorname{diam} U<\varepsilon$ and W is a topological disk with $K\subset W\subset \overline{W}\subset U$ and $\operatorname{mod}(U\setminus \overline{W})\geq M/2$ , then there is an inverse branch $g:W\to U_m$ of $f^m$ such that $g(K)=K_m$ .

Lemma 5. Let U be a topological disk in the complex plane, and $K \subset U$ be compact and simply connected. Let $(P_i)_{i=1}^N$ be a finite collection of open connected sets with union $U \setminus K$ . Consider another topological disk $K \subset W \subset U$ and assume that there is an inverse branch $g: W \to \mathbb{C}$ of f. Suppose also that $P_i \cap W$ is connected for all i. Let $S_i$ be the connected component of $f^{-1}(P_i)$ which contains $g(P_i)$ . Then the following holds: - If $f: S_i \to P_i$ is univalent for each i = 1, ..., N, then g has an analytic continuation defined on U. - If $f: S_i \to P_i$ is not univalent and $P_i$ is n-regular for some n (see Definition 1), then $S_i$ does not contain any point of the form $f^k(c)$ , where $c \in \text{Sing}(f)$ and k < n.

Proposition 1. Let 0 < r < R and $z_0 \in \mathbb{C}$ . Suppose that an inverse branch g of $f^n$ , for some $n \geq 1$ , is defined on a neighborhood of $\overline{B(z_0, r)}$ , and that $g(B(z_0, r)) \cap P(f) \neq \emptyset$ . Suppose also that the annulus $B(z_0, R) \setminus \overline{B(z_0, r)}$ is the union of finitely many n-regular sets $P_i$ , i = 1, ..., N, each of which is a "rectangle" bounded by arcs of $C(z_0, r)$ , $C(z_0, R)$ , and two radii of $B(z_0, R)$ . For every $\eta > 1$ and N there are $\delta > 0$ and $\varepsilon > 0$ , independent of n, such that whenever $R/r \geq 1 + \eta$ and $R < \varepsilon$ , then g has an analytic continuation to $B(z_0, (1+\delta)r)$ .

Proposition 2. There is $\varepsilon > 0$ such that for each $z_0 \in \mathbb{C}$ , the ball $B(z_0, \varepsilon)$ is regular.

Lemma 6. Consider an inverse orbit $(x_{-n})_{n=0}^{\infty}$ , $x_{-n} = f(x_{-n-1})$ , and assume that the set $\{x_{-n} : n = 0, 1, \ldots\}$ is bounded in a disk B(0, R) and its distance from Crit(f) is $\eta > 0$ . For every $R, \eta$ there is $\varepsilon > 0$ such that for every n there is an inverse branch of $f^n$ defined on $B(x_0, \varepsilon)$ which sends $x_0$ to $x_{-n}$ .

Proposition 3. Let K be a forward invariant compact set which does not intersect Crit(f). Then K is hyperbolic.

Proposition 4. Suppose that f is a meromorphic map whose Julia set is the whole plane, which satisfies the Misiurewicz condition and additionally $\infty \notin P(f)$ . Also assume that the set of points whose orbits converge to $\infty$ has zero Lebesgue measure. Then, for all $z \in \mathbb{C}$ except a set of zero Lebesgue measure, there exist $\varepsilon > 0$ and a sequence $n_k \to \infty$ and a compact set $K \subset \mathbb{C}$ such that, for every k, $f^{n_k}(z) \subset K$ and there is a neighborhood of z which is mapped univalently by $f^{n_k}$ onto $B(f^{n_k}(z), \varepsilon)$ .

Theorem 2. Suppose that f is a meromorphic function whose Julia set is the whole plane, f satisfies the Misiurewicz condition and $\infty \notin P(f)$ . Suppose that the set of points whose forward orbits converge to $\infty$ has measure 0. In addition, assume that f has no more than 2 inherently univalently omitted values in the sphere, and if one such value is $\infty$ , then the other one is not a pole. Then there is no line-field on the plane invariant under $f^*$ . Following the pattern set by [7], our proof will be given in two steps. First, we will show that a potential invariant line-field is analytic on some open set, and then we will prove that a holomorphic line-field cannot exist. More precisely:

Proposition 5. Suppose that f is a meromorphic function whose Julia set is the whole plane and satisfies the Misiurewicz condition, $\infty \notin P(f)$ and $\nu$ is a line-field on the plane invariant under $f^*$ . If the orbit of almost every point does not converge to $\infty$ , then there is an open non-empty subset $U \subset \mathbb{C}$ such that $\nu$ is univalent on U. Notice that the hypothesis of Proposition 5 does not contain any restrictions on univalently omitted values. That some additional assumptions in Theorem 2 are necessary is shown by rational examples of Lattés (see [5, p. 30]). The proof of Proposition 5 is standard. Take a point z which is a Lebesgue point of $\nu$ and which also has a sequence $n_k$ of passages to the large scale as specified in the claim of Proposition 4. Assume that $f^{n_k}(z) \to w$ and consider $U := B(w, \varepsilon/2)$ , where $\varepsilon$ comes from Proposition 4. Define univalent mappings $g_k(u) = A_k(f^{-n_k}(u)-z)+z$ , where $A_k$ are positive numbers chosen so that $|g'_k(w)| = 1$ for all k. Then $g_k$ converge uniformly on U to a univalent function G with G(w) = z and |G'(w)| = 1. Consider a constant line-field $\nu_0 := \nu(z)d\overline{u}/du$ and $\nu_k(u) = \nu(z + A_k^{-1}(u-z))$ . Since z is a Lebesgue point, $\nu_k(u)$ converge to the constant $\nu(z)$ in $L^1(G(U))$ . For every k we have $\nu = (g_k^{-1})^(\nu_k d\overline{u}/du)$ . But clearly the coefficient of the line-field on the right-hand side converges to the coefficient of $(G^{-1})^(\nu_0)$ in $L^1$ , and so $\nu$ is univalent as claimed. 3.1. Restrictions on analytic line-fields. In this section we will prove Theorem 2. The proof will be based on two lemmas concerning analytic line-fields. If $\nu = h^*(d\overline{w}/dw)$ on some open set and h is holomorphic, we will refer to h as a local linearizing coordinate.

Lemma 7. Suppose that g is holomorphic in a neighborhood of a repelling fixed point $z_0$ . Assume also that there is a line-field $\nu$ , holomorphic and invariant in a punctured neighborhood of $z_0$ . Then there is a function h meromorphic in a neighborhood of $z_0$ such that $$\nu(z) = (\sqrt{h(z)})^* \frac{d\overline{z}}{dz}$$ on that neighborhood.

Lemma 8. Suppose that f is a meromorphic function on the complex plane, f satisfies the Misiurewicz condition, $\infty \notin P(f)$ , and the Julia set of f is the whole plane. Assume that f fixes a line-field $\mu$ on the plane, which is holomorphic in some open set U. Then $\mu$ is holomorphic at every point of the Riemann sphere except for the inherently univalently omitted values of f.

Definition 1. A set $K \subset \mathbb{C}$ is called n-regular if for every $c \in \text{Sing}(f)$ and every $k \in \mathbb{N}$ such that $k \leq n$ and $f^k(c) \in K$ , there exists an inverse branch g of $f^k$ defined on K such that $g(f^k(c)) = c$ .

Definition 2. A set $K \subset \mathbb{C}$ is called regular if it is n-regular for every $n \in \mathbb{N}$ . A set $K \subset \mathbb{C}$ is called hyperbolic if there are $N \in \mathbb{N}$ and k > 1 such that $|(f^N)'(x)| > k$ for every $x \in K$ , provided that $f^N$ is defined at x. In this paper diam K denotes the Euclidean diameter of the set K, and mod(L) is the modulus of an annulus L. Let $S := \{f : \mathbb{C} \to \overline{\mathbb{C}} : \operatorname{Sing}(f) \text{ is finite} \}$ and $\mathcal{B} := \{f : \mathbb{C} \to \overline{\mathbb{C}} : \operatorname{Sing}(f) \text{ is bounded} \}$ . The class $\mathcal{B}$ has been studied by many people. However, the dynamics of maps in the class $\mathcal{B}$ is richer and consequently less understood than for S. For example in S the classification of periodic components is the same as for rational maps, which is not the case for S (see [3], [2] and [8]). $B(x,\varepsilon)$ and $C(x,\varepsilon)$ will refer to the open Euclidean disk and circle of center and radius given, respectively. Misiurewicz condition. The following will be referred to as the Misiurewicz condition: $P(f) \subset B(0,R) \cup \{\infty\}$ for some R > 0 and $\overline{P(f)} \cap \operatorname{Crit}(f) = \emptyset$ . The Misiurewicz condition comes from interval dynamics, where it simply says that P(f) is disjoint from $\operatorname{Crit}(f)$ . It was later used in the study of dynamics of rational maps and is widely viewed as the first generalization going beyond the realm of post-critically finite maps. In the case of meromorphic dynamics which we are dealing with here, $\infty$ can be viewed as a critical point of sorts with asymptotic values akin to critical values arising from it. Then our condition is tantamount to the classical Misiurewicz condition, with one exception that falling into $\infty$ is allowed. We shall prove the following.

Definition 3. Let f be a meromorphic function on the complex plane. Then $z \in \overline{\mathbb{C}}$ will be called a univalently omitted value for f if every preimage of z is either a critical point of f or belongs to P(f). We will say that z is inherently univalently omitted if z is univalently omitted together with all its forward images. Obviously, omitted values are univalently omitted as well.

Definition 4. A line-field $\nu$ is holomorphic at $z_0$ if there is a holomorphic non-constant function h from a neighborhood of $z_0$ into $\mathbb{C}$ such that $\nu = h^*(d\overline{w}/dw)$ a.e. on this neighborhood. If $h'(z_0) \neq 0$ , then $\nu$ is univalent.