Non-recurrent meromorphic functions

Theorem 1. Let f : C →C be a transcendental meromorphic function in the class B such that J(f) = 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 Ma˜n´e, 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 nex

Lemma 1. For every R > 0 and every x ∈C there is ε > 0 such that every connected component Di of f −1(B(x, ε)) with f −1(B(x, ε)) ∩B(0, R) ̸= ∅is a topological disk containing exactly one preimage of x and f : Di →B(x, ε) is a proper map.

Lemma 2. For every ε > 0, M > 0, and R > 0 there is δ > 0 such that for every n ≥1 if U, K ⊂C, K ⊂U, U is a topological disk while K is

Lemma 3. For every R, η > 0 there is δ > 0 such that for every x, y ∈ B(0, R) if x = f(y) and dist(y, Crit(f)) ≥η, then there is an inverse branch g of f defined on B(x, δ) such that g(x) = y.

Lemma 4. Let U be a topological disk in the plane, and K ⊂U be compact and simply connected such that mod(U \K) ≥M. Suppose that Ki, i = 0, . . . , m, are simply connected sets which satisfy K0 = K, f(Ki) = Ki−1 for i = 1, . . . , m. Analogously, define Ui to be the connected component of f −i(U) which contains Ki. Assume also Km ∩P(f) ̸= ∅and that there is an inverse branch γ : U →Um−1 of f m−1 for which γ(K) = Km−1. For every M > 0 there is ε > 0 such that if diam U < ε and W is a topological d

Lemma 5. Let U be a topological disk in the complex plane, and K ⊂U be compact and simply connected. Let (Pi)N i=1 be a finite collection of open connected sets with union U \ K. Consider another topological disk K ⊂ W ⊂U and assume that there is an inverse branch g : W →C of f. Suppose also that Pi ∩W is connected for all i. Let Si be the connected component of f −1(Pi) which contains g(Pi). Then the following holds: • If f : Si →Pi is univalent for each i = 1, . . . , N, then g has an analytic

Proposition 1. Let 0 < r < R and z0 ∈C. Suppose that an inverse branch g of f n, for some n ≥1, is defined on a neighborhood of B(z0, r), and that g(B(z0, r))∩P(f) ̸= ∅. Suppose also that the annulus B(z0, R)\B(z0, r) is the union of finitely many n-regular sets Pi, i = 1, . . . , N, each of which is a “rectangle” bounded by arcs of C(z0, r), C(z0, R), and two radii of B(z0, R). For every η > 1 and N there are δ > 0 and ε > 0, independent of n, such that whenever R/r ≥1 + η and R < ε, then g has a

Proposition 2. There is ε > 0 such that for each z0 ∈C, the ball B(z0, ε) is regular.

Proposition 2 now follows by induction. Hyperbolicity

Lemma 6. Consider an inverse orbit (x−n)∞ n=0, x−n = f(x−n−1), and assume that the set {x−n : n = 0, 1, . . .} is bounded in a disk B(0, R) and its distance from Crit(f) is η > 0. For every R, η there is ε > 0 such that for every n there is an inverse branch of f n defined on B(x0, ε) which sends x0 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 ∞̸∈P(f). Also assume that the set of points whose orbits converge to ∞ has zero Lebesgue measure. Then, for all z ∈C except a set of zero Lebesgue measure, there exist ε > 0 and a sequence nk →∞and a compact set K ⊂C such that, for every k, f nk(z) ⊂K and there is a neighborhood of z which is mapped univalently by f nk onto B(f nk(z), ε).

Theorem 2. Suppose that f is a meromorphic function whose Julia set is the whole plane, f satisfies the Misiurewicz condition and ∞̸∈P(f). Suppose that the set of points whose forward orbits converge to ∞has mea- sure 0. In addition, assume that f has no more than 2 inherently univa- lently omitted values in the sphere, and if one such value is ∞, 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 give

Theorem 2. The proof will be based on two lemmas concerning analytic line-fields. If ν = h∗(dw/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 z0. Assume also that there is a line-field ν, holomorphic and invariant in a punctured neighborhood of z0. Then there is a function h meromorphic in a neighborhood of z0 such that ν(z) = ( p h(z))∗dz dz on that neighborhood.

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

Theorem 2. References [1] I. N. Baker, J. Kotus and Y. L¨u, Iterates of meromorphic functions I , Ergodic Theory Dynam. Systems 11 (1991), 241–248. [2] —, —, —, Iterates of meromorphic functions IV , Results Math. 22 (1992), 651–656. [3] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151–188. [4] W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), 355–373. [5] L. Car

Definition 1. A set K ⊂C is called n-regular if for every c ∈Sing(f) and every k ∈N such that k ≤n and f k(c) ∈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 ⊂C is called regular if it is n-regular for every n ∈N. A set K ⊂C is called hyperbolic if there are N ∈N and k > 1 such that |(f N)′(x)| > k for every x ∈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

Definition 3. Let f be a meromorphic function on the complex plane. Then z ∈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 ν is holomorphic at z0 if there is a holomor- phic non-constant function h from a neighborhood of z0 into C such that ν = h∗(dw/dw) a.e. on this neighborhood. If h′(z0) ̸= 0, then ν is univalent. Proposition 5. Suppose that f is a meromorphic function whose Julia set is the whole plane and satisfies the Misiurewicz condition, ∞̸∈P(f) and ν is a line-field on the plane invariant under f ∗. If the orbit of almost every point does not converge to ∞, then there is an open non