Related papers: Square Sierpi\'nski carpets and Latt\`es maps

We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpi\'nski carpet $S_3$ is a Euclidean isometry. For carpets in a more general family, the standard $1/p$-Sierpi\'nski carpets $S_p$, $p\ge 3$ odd, we show that the…

We prove that if $\xi$ is a quasisymmetric homeomorphism between Sierpi\'nski carpets that are the Julia sets of postcritically-finite rational maps, then $\xi$ is the restriction of a M\"obius transformation to the Julia set. This implies…

In this paper, the main focus is on the Sierpinski carpet Julia sets of the rational maps with non-recurrent critical points. We study the uniform quasicircle property of the peripheral circles, the relatively separated property of the…

We prove that if $f$ and $g$ are postcritically finite rational maps whose Julia sets $\mathcal{J}(f), \mathcal{J}(g)$, respectively, are Sierpi\'nski carpets, and if $\xi$ is a quasiregular map of the Riemann sphere $\widehat{\mathbb{C}}$…

Let $S_i$, $i\in I$, be a countable collection of Jordan curves in the extended complex plane $\Sph$ that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated,…

We study a new class of square Sierpi\'nski carpets $F_{n,p}$ ($5\leq n, 1\leq p<\frac{n}{2}-1$) on $\mathbb{S}^2$, which are not quasisymmetrically equivalent to the standard Sierpi\'{n}ski carpets. We prove that the group of…

We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_\lambda(z)=z^m+\lambda/z^\ell$, where $\ell$, $m\geq 2$ are integers satisfying $1/\ell+1/m<1$ and $\lambda\in\mathbb{C}\setminus\{0\}$. If the free critical points…

We prove that all Sierpi\'nski carpets in the plane are non-removable for (quasi)conformal maps. More precisely, we show that for any two Sierpi\'nski carpets $S,S'\subset \hat{\mathbb{C}}$ there exists a homeomorphism $f\colon…

In this paper, we prove that a postcritically finite rational map with non-empty Fatou set is Thurstion equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpinski carpet

We prove the existence of rational maps whose Julia sets are Sierpi\'{n}ski carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely…

We introduce Schottky maps-conformal maps between relative Schottky sets, and study their local rigidity properties. This continues the investigations of relative Schottky sets initiated in [S. Merenkov, "Planar relative Schottky sets and…

We establish periodic quasiconformal extension theorems for periodic orientation-preserving quasisymmetric self homeomorphisms of quasicircles or quasi-round carpets. As applications, we prove that, if $f$ is a periodic…

In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpi\'nski carpet, then there is an integer $n_0$, such that, for any $n\ge…

We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpi\'nski carpet into the plane. In the case of so called dyadic slit carpets, several characterizations are obtained. One characterization is in terms of a…

Motivated by questions in geometric group theory we define a quasisymmetric co-Hopfian property for metric spaces and provide an example of a metric Sierpi\'nski carpet with this property. As an application we obtain a quasi-isometrically…

We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable…

We initiate a study of the quasisymmetric uniformization of naturally arising random fractals and show that many of them fall outside the realm of quasisymmetric uniformization to simple canonical spaces. We begin with the trace, the graph…

We prove that all Sierpi\'nski spaces in ${\mathbb{S}}^n$, $n\geq 2$, are non-removable for (quasi)conformal maps, generalizing the result of the first named author arXiv:1809.05605. More precisely, we show that for any Sierpi\'nski space…

The complement of the union of a collection of disjoint open disks in the $2$-sphere is called a Schottky set. We prove that a subset $S$ of the $2$-sphere is quasiconformally equivalent to a Schottky set if and only if every pair of…

We study the family of renormalization transformations of the generalized $d$--dimensional diamond hierarchical Potts model in statistical mechanic and prove that their Julia sets and non-escaping loci are always connected, where $d\geq 2$.…