A sharp growth condition for a fast escaping spider's web

We show that the fast escaping set $A(f)$ of a transcendental entire function $f$ has a structure known as a spider's web whenever the maximum modulus of $f$ grows below a certain rate. We give examples of entire functions for which the fast escaping set is not a spider's web which show that this growth rate is best possible. By our earlier results, these are the first examples for which the escaping set has a spider's web structure but the fast escaping set does not. These results give new insight into a conjecture of Baker and a conjecture of Eremenko

Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Tsuji related to the Denjoy-Carleman-Ahlfors theorem.Comment: 17 page

Baker domains of meromorphic functions

Let $f$ be a transcendental meromorphic function and $U$ be an invariant Baker domain of $f$. We obtain a new estimate for the growth of the iterates of $f$ in $U$, and we use this estimate to improve an earlier result relating the geometric properties of $U$ and the proximity of $f$ in $U$ to the identity function. We illustrate the latter result by considering transcendental meromorphic functions $f$ of the form f(z) = az + bz^ke^{-z}(1+o(1)) \; \mbox{ as } \Re (z) \rightarrow \infty, where $k \in \bf N$, $a > 1$ and $b > 0$, and we show that these functions have Baker domains which contain an unbounded set of critical points and an unbounded set of critical values

Dimensions of Julia sets of meromorphic functions with finitely many poles

Let $f$ be a transcendental meromorphic function with finitely many poles such that the finite singularities of $f^{-1}$ lie in a bounded set. We show that the Julia set of $f$ has Hausdorff dimension strictly greater than one and packing dimension equal to two. The proof for Hausdorff dimension simplifies the earlier argument given for transcendental entire functions

Dimensions of Julia sets of meromorphic functions

We show that for any meromorphic function $f$ the Julia set $J(f)$ has constant local upper and lower box dimensions, $\overline{d}(J(f))$ and $\underline{d}(J(f))$, respectively, near all points of $J(f)$ with at most two exceptions. Further, the packing dimension of the Julia set is equal to $\overline{d}(J(f))$. Using this result we show that, for any transcendental entire function $f$ in the class $B$ (that is, the class of functions such that the singularities of the inverse function are bounded), both the local upper box dimension and packing dimension of $J(f)$ are equal to 2. Our approach is to show that the subset of the Julia set containing those points that escape to infinity as quickly as possible has local upper box dimension equal to 2

Microcomputers and Mathematics

Body cremated. George A. Mason - husbandhttps://stars.library.ucf.edu/cfm-ch-register-vol25/1382/thumbnail.jp