Related papers: Cantor Julia sets with Hausdorff dimension two

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set "cancels randomness" in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower…

We show that for large classes of entire functions the Julia set and the escaping set have packing dimension two. For example, this is the case for entire functions which are bounded on a curve tending to infinity. More generally, we show…

In this paper we prove that parabolic Julia sets of rational functions are locally computable in polynomial time.

We study the exact Hausdorff and packing dimensions of the $prime$ $Cantor$ $set$, $\Lambda_P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set…

This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a…

Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the…

Two-valued sets are local sets of the two-dimensional Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in [-a,b].…

We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy…

We prove that for any countable set $A$ of real numbers, the set of binary indefinite quadratic forms $Q$ such that the closure of $Q(\mathbb{Z}^2)$ is disjoint from $A$ has full Hausdorff dimension.

In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$ of numbers having a unique expansion in the (non-integer) base $q$, and showed that…

The long-standing problem of existence of nowhere dense rational Julia set with positive area has been solved by an example in quadratic polynomials by Buff and Ch\'eritat. Since then many efforts have been devoted to finding out new…

Recently Merenkov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpi${\'n}$ski carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles…

Multiplicative Unitaries are described in terms of a pair of commuting shifts of relative depth two. They can be generated from ambidextrous Hilbert spaces in a tensor C*-category. The algebraic analogue of the Takesaki-Tatsuuma Duality…

The control of postcritical sets of quadratic polynomials with a neutral fixed point is a main ingredient in the remarkable work of Buff and Ch\'eritat to construct quadratic Julia sets with positive area. Based on the Inou-Shishikura…

Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A.…

Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia iteration theory of rational functions extend to quasiregular self-maps of the Riemann sphere for which the degree exceeds the dilatation. We show that in this context,…

We prove that the algorithm of [13] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on…

We prove that every wandering Julia component of cubic rational maps eventually has at most two complementary components.

We prove that the meandering set for $f_a(z)=e^z+a$ is homeomorphic to the space of irrational numbers whenever $a$ belongs to the Fatou set of $f_a$. This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies that the…

We prove a quantitative distortion theorem for iterated function systems that generate sets of continued fractions. As a consequence, we obtain upper and lower bounds on the Hausdorff dimension of any set of real or complex continued…