arXiv subjects

Dimitris Vardakis

Publications and source records attributed to Dimitris Vardakis.

5 recordsLinked to original sources

arXiv2026

Julia Sets for Sequences of Monomials

We study Julia sets arising from non-autonomous iteration of monomials on the Riemann sphere. We give a precise description of the Julia set associated to an arbitrary sequence of monomials in terms of their coefficients and degrees. Combined with the basic invariance relation for non-autonomous Julia sets, this yields a plethora of striking examples in polynomial non-autonomous dynamics: finite Julia sets of arbitrary cardinality, non-trivial Julia sets with non-empty interior, Julia sets that are perfect but not uniformly perfect, and Julia sets with empty interior but positive area. We also describe a connection between autonomous and non-autonomous Julia sets of monomial sequences through Kuratowski limits.

math.DS↗

arXiv2024

Matrix-weighted estimates beyond Calder\'on-Zygmund theory

We investigate matrix-weighted bounds for the sublinear non-kernel operators considered by F. Bernicot, D. Frey, and S. Petermichl. We extend their result to sublinear operators acting upon vector-valued functions. First, we dominate these operators by bilinear convex body sparse forms, adapting a recent general principle due to T. Hyt\"onen. Then we use this domination to derive matrix-weighted bounds, adapting arguments of F. Nazarov, S. Petermichl, S. Treil, and A. Volberg. Our requirements on the weight are formulated in terms of two-exponent matrix Muckenhoupt conditions, which surprisingly exhibit a rich structure that is absent in the scalar case. Consequently, we deduce that our matrix-weighted bounds improve the ones that were recently obtained by A. Laukkarinen. The methods we use are flexible, which allows us to complement our results with a limited range extrapolation theorem for matrix weights, extending the results of P. Auscher and J. M. Martell, as well as M. Bownik and D. Cruz-Uribe.

math.CA↗

arXiv2022

The Buffon's needle problem for random planar disk-like Cantor sets

We consider a model of randomness for self-similar Cantor sets of finite and positive $1$-Hausdorff measure. We find the sharp rate of decay of the probability that a Buffon needle lands $\delta$-close to a Cantor set of this particular randomness. Two quite different models of randomness for Cantor sets, by Peres and Solomyak, and by Shiwen Zhang, appear to have the same order of decay for the Buffon needle probability: $\frac{c}{\log\frac{1}{\delta}}$. In this note, we prove the same rate of decay for a third model of randomness, which asserts a vague feeling that any "reasonable" random Cantor set of positive and finite length will have Favard length of order $\frac{c}{\log\frac{1}{\delta}}$ for its $\delta$-neighbourhood. The estimate from below was obtained long ago by Mattila.

math.AP↗

arXiv2021

Free boundary problems via Sakai's theorem

A Schwarz function on an open domain $\Omega$ is a holomorphic function satisfying $S(\zeta)=\overline{\zeta}$ on $\Gamma$, which is part of the boundary of $\Omega$. Sakai in 1991 gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if $\Omega$ is simply connected and $\Gamma=\partial \Omega\cap D(\zeta,r)$, then $\Gamma$ has to be regular real analytic. This paper is an attempt to describe $\Gamma$ when the boundary condition is slightly relaxed. In particular, three different scenarios over a simply connected domain $\Omega$ are treated: when $f_1(\zeta)=\overline{\zeta}f_2(\zeta)$ on $\Gamma$ with $f_1,f_2$ holomorphic and continuous up to the boundary, when $\mathcal{U}/\mathcal{V}$ equals certain real analytic function on $\Gamma$ with $\mathcal{U},\mathcal{V}$ positive and harmonic on $\Omega$ and vanishing on $\Gamma$, and when $S(\zeta)=\Phi(\zeta,\overline{\zeta})$ on $\Gamma$ with $\Phi$ a holomorphic function of two variables. It turns out that the boundary piece $\Gamma$ can be, respectively, anything from $C^\infty$ to merely $C^1$, regular except finitely many points, or regular except for a measure zero set.

math.CV↗

arXiv2019

Geometry of planar curves intersecting many lines in a few points

The local Lipschitz property is shown for the graph avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand's well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is $\sigma$-finite with respect to Hausdorff length and an estimate on the Hausdorff measure of each "piece" is found.

math.AP↗