Related papers: Hausdorff dimension of escaping sets of meromorphi…
A function which is transcendental and meromorphic in the plane has at least two singular values. On one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only…
We consider transcendental meromorphic function for which the set of finite singularities of its inverse is bounded. Bergweiler and Kotus gave bounds for the Hausdorff dimension of escaping sets if the function has no logarithmic…
Bergweiler and Kotus gave sharp upper bounds for the Hausdorff dimension of the escaping set of a meromorphic function in the Eremenko-Lyubich class, in terms of the order of the function and the maximal multiplicity of the poles. We show…
We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second order differential equations…
Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ \phi= \psi\circ f$, where $\phi,\psi:\C\to\C$ are affine). We show that…
In this paper we study two classes of meromorphic functions previously studied by Mayer, Kotus, and Urba\'nski. In particular we estimate a lower bound for the Julia set and the set of escaping points for non-autonomous additive and affine…
It is shown that the Hausdorff dimension of the fast escaping set of a quasiregular self-map of ${\mathbb R}^3$ can take any value in the interval $[1,3]$. The Hausdorff dimension of the Julia set of such a map is estimated under some…
The escaping set I(f) of a transcendental meromorphic function f consists of all points which tend to infinity under iteration. The Eremenko-Lyubich class B consists of all transcendental meromorphic functions for which the set of finite…
We show that for each $d\in (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.
We consider the subclass of class ${\mathcal B} $ consisting of meromorphic functions $f:{\mathbb C}\to\hat{\mathbb C}$ for which infinity is not an asymptotic value and whose all poles have orders uniformly bounded from above. This class…
The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these…
We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate…
We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an…
In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) admits divergent trajectories that exit to infinity at arbitrarily slow…
We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside the Eremenko-Lyubich class. We show that…
We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity `slowly', and which have Hausdorff dimension equal to 1. We prove these results by using the idea of…
We give a lower bound of the hyperbolic and the Hausdorff dimension of the Julia set of meromorphic functions of finite order under very general conditions.
Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff…
We consider the evolution of a connected set in Euclidean space carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order sqrt{t}…
In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of…