arXiv subjects

Athanasios Tsantaris

Publications and source records attributed to Athanasios Tsantaris.

6 recordsLinked to original sources

arXiv2023

Quasiconformal curves and quasiconformal maps in metric spaces

In this paper we study quasiconformal curves which are a special case of quasiregular curves. Namely embeddings $\Omega\rightarrow\mathbb{R}^m$ from some domain $\Omega\subset\mathbb{R}^n$ to $\mathbb{R}^m$, where $n\leq m$, which belong in a suitable Sobolev class and satisfy a certain distortion inequality for some smooth, closed and non-vanishing $n$-form in $\mathbb{R}^m$. These mappings can be seen as quasiconformal mappings between $\Omega$ and $f(\Omega)$. We prove that a quasiconformal curve always satisfies the analytic definition of quasiconformal mappings and the lower half of the modulus inequality. Moreover, we give a sufficient condition for a quasiconformal curve to satisfy the metric definition of quasiconformal mappings. We also show that a quasiconformal map from $\Omega$ to $f(\Omega)\subset \mathbb{R}^m$ is a quasiconformal $\omega$ curve for some form $\omega$ under suitable assumptions. Finally, we show the same is true when we equip the target space $f(\Omega)$ with its intrinsic metric instead of the Euclidean one.

math.CV↗

arXiv2022

Finite distortion curves: Continuity, Differentiability and Lusin's (N) property

We define finite distortion $\omega$-curves and we show that for some forms $\omega$ and when the distortion function is sufficiently exponentially integrable the map is continuous, differentiable almost everywhere and has Lusin's (N) property. This is achieved through some higher integrability results about finite distortion $\omega$-curves. It is also shown that this result is sharp both for continuity and for Lusin's (N) property. We also show that if we assume weak monotonicity for the coordinates of a finite distortion $\omega$-curve we obtain continuity.

math.CV↗

arXiv2022

Hausdorff dimension in quasiregular dynamics

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 growth condition.

math.DS↗

arXiv2021

Julia sets of Zorich maps

The Julia set of the exponential family $E_κ:z\mapstoκe^z$, $κ>0$ was shown to be the entire complex plane when $κ>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<κ\leq1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the entire $\mathbb{R}^3$ generalizing Misiurewicz's result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb{R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.

math.DS↗

arXiv2021

Explosion points and topology of Julia sets of Zorich maps

Zorich maps are higher dimensional analogues of the complex exponential map. For the exponential family $λe^z$, $λ>0$, it is known that for small values of $λ$ the Julia set is an uncountable collection of disjoint curves. The same was shown to hold for Zorich maps by Bergweiler and Nicks. In this paper we introduce a topological model for the Julia sets of certain Zorich maps, similar to the so called \textit{straight brush} of Aarts and Oversteegen. As a corollary we show that $\infty$ is an \textit{explosion point} for the set of endpoints of the Julia sets. Moreover we introduce an object called a \textit{hairy surface} which is a compactified version of the Julia set of Zorich maps and we show that those objects are not uniquely embedded in $\mathbb{R}^3$, unlike the corresponding two dimensional objects which are all ambiently homeomorphic.

math.DS↗

arXiv2021

Permutable Quasiregular Maps

Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form $f$ and $g=ϕ\circ f$, where $ϕ$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.

math.DS↗