Hyperbolic Dimension of Julia Sets of Meromorphic Maps with Logarithmic Tracts

2012

Isr. J. Math.

Abstract. Let f be a transcendental entire function. We give conditions which imply that the Julia set and the escaping set of f have packing dimension 2. For example, this holds if there exists a positive constant c less than 1 such that the minimum modulus L(r, f ) and the maximum modulus M (r, f ) satisfy log L(r, f ) ≤ c log M (r, f ) for large r. The conditions are also satisfied if log M (2r, f ) ≥ d log M (r, f ) for some constant d greater than 1 and all large r.

Section: Introduction and Resultsmentioning

confidence: 99%

On the packing dimension of the Julia set and the escaping set of an entire function

2012

Isr. J. Math.

“…While our proof of Theorem 1.5 is based on the set of escaping points, Baranski, Karpinska and Zdunik [3] have recently shown that the set of points in the Julia set whose orbit is bounded and contained in a logarithmic tract also has Hausdorff dimension greater than 1, thereby giving an alternative proof of Theorem 1.5.…”

Section: Functions With a Logarithmic Tractmentioning

confidence: 99%

Dynamics of meromorphic functions with direct or logarithmic singularities

Proceedings of the London Mathematical Society

Rippon

Stallard

2008

144

Abstract. Let f be a transcendental meromorphic function and denote by J(f ) the Julia set and by I(f ) the escaping set. We show that if f has a direct singularity over infinity, then I(f ) has an unbounded component and I(f ) ∩ J(f ) contains continua. Moreover, under this hypothesis I(f ) ∩ J(f ) has an unbounded component if and only if f has no Baker wandering domain. If f has a logarithmic singularity over infinity, then the upper box dimension of I(f ) ∩ J(f ) is 2 and the Hausdorff dimension of J(f ) is strictly greater than 1. The above theorems are deduced from more general results concerning functions which have "direct or logarithmic tracts", but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman-Valiron theory. This method is also applied to complex differential equations.

“…Thus it actually suffices to consider the special case that P( f λ ) is bounded. The dynamics of meromorphic functions with logarithmic tracts are studied for example in [2,8].…”

Section: Remark 33mentioning

confidence: 99%

“…For further results on the Hausdorff dimension of Julia sets of entire functions we refer to [2,8] and, in particular, the survey [31]. While McMullen's result on the Hausdorff dimension of J (λe z ) thus has prompted a lot of further research, there seem to be no papers whose main intention is to extend…”

mentioning

confidence: 99%

Entire functions with Julia sets of positive measure

Aspenberg

2010

Math. Ann.

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