Related papers: Dimensions of slowly escaping sets and annular iti…

For one parameter subgroup action on a finite volume homogeneous space, we consider the set of points admitting divergent on average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension…

There are several classes of transcendental entire functions for which the Julia set consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Many authors have studied the topological properties…

The issue of whether an analytic function has wandering domains has long been of interest in complex dynamics. Sullivan proved in 1985 that rational maps do not have wandering domains. On the other hand, several transcendental entire…

We study probabilistic iterated function systems (IFS), consisting of a finite or infinite number of average-contracting bi-Lipschitz maps on R^d. If our strong open set condition is also satisfied, we show that both upper and lower bounds…

In this article we derive a formula for the Hausdorff dimension of Besicovitch-Eggleston level sets associated with non-autonomous dynamics constructed from families of countable affine iterated function systems. The formula obtained shows…

Recently Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a new technique called quasiconfomal folding. It is easy to check that his method produces an entire function of…

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…

We extend results about the dimension of the radial Julia set of certain exponential functions to quasiregular Zorich maps in higher dimensions. Our results improve on previous estimates of the dimension also in the special case of…

Let $k$ be a natural number. We consider $k$-times continuously-differentiable real-valued functions $f:E\to\mathbb{R}$, where $E$ is some interval on the line having positive length. For $0<\alpha<1$ let $I_\alpha(f)$ denote the set of…

The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$, ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to…

Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax(\bmod 1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|\lambda_1|\le|\lambda_2|\le\dots\le|\lambda_d|$. We investigate the Hausdorff dimension of the…

We prove that the infinitely generated Apollonian gasket has full Hausdorff dimension spectrum. Our proof, which is computer assisted, relies on an iterative technique introduced by the first three authors in [3] and on a flexible method…

We investigate the set $I$ of parameters $\kappa$ for which the singular value of $z\mapsto e^z+\kappa$ converges to $\infty$. The set $I$ consists of uncountably many parameter rays, plus landing points of some of these rays. We show that…

Let f be an entire function with a bounded set of singular values, and suppose furthermore that the postsingular set of f is bounded. We show that every component of the escaping set I(f) is unbounded. This provides a partial answer to a…

We prove that the set of all endpoints of the Julia set of $f(z)=\exp(z)-1$ which escape to infinity under iteration of $f$ is not homeomorphic to the rational Hilbert space $\mathfrak E$. As a corollary, we show that the set of all points…

Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other…

We determine the Hausdorff dimension of sets of irrationals in $(0,1)$ whose partial quotients in semi-regular continued fractions obey certain restrictions and growth conditions. This result substantially generalizes that of the second…

We consider Teichm\"uller geodesics in strata of translation surfaces. We prove lower and upper bounds for the Hausdorff dimension of the set of parameters generating a geodesic bounded in some compact part of the stratum. Then we compute…

We develop a general technique for realising full closed subsets of the complex plane as wandering sets of entire functions. Using this construction, we solve a number of open problems. (1) We construct a counterexample to Eremenko's…

We prove that there exist three transcendental entire functions that have infinite number of domains which lie in the wandering component of each of these functions and their composites. This result is a generalization of the result of…