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V. Timorin

Publications and source records attributed to V. Timorin.

6 recordsLinked to original sources
arXiv2026

Limit cubic laminations

Let $\sigma_3:\mathbb{S}\to \mathbb{S}$ be the tripling map of the unit circle. For sequences $\{\mathcal{L}_i\}$ of $\sigma_3$-invariant dendritic laminations we study limits $(\overline{c}, \overline{d})$ of their critical portraits assuming that one such limit $\mathcal{P}=(\overline{c}_\circ, \overline{d}_\circ)$ is given. If the endpoints of $\overline{c}_\circ$ and $\overline{d}_\circ$ are non-periodic, then there is a unique lamination $\mathcal{L}$ with finite critical sets such that $\overline{c}$ and $\overline{d}$ can be any couple of critical chords compatible with $\mathcal{L}$. As the extreme opposite case we consider $\mathcal{P}=(\overline{0 \frac13}, \overline{0 \frac23})$ and describe the corresponding countable closed family of possible critical portraits $(\overline{c}, \overline{d})$ and the distinct laminations corresponding to them. These results can be useful for the construction of a model for the cubic connectedness locus.

math.DS↗
arXiv2023

Symmetric cubic polynomials

We describe a model $\mathcal{M}_3^{comb}$ for the boundary of the connectedness locus $\mathcal{M}^{sy}_3$ of the parameter space of cubic symmetric polynomials $p_c(z)=z^3-3c^2z$. We show that there exists a monotone continuous function $\pi:\partial \mathcal{M}_c^{sy}\to \mathcal{M}_3^{comb}$ which is a homeomorphism if $\mathcal{M}^{sy}_3$ is locally connected.

math.DS↗
arXiv2019

Dynamical generation of parameter laminations

Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant laminations representing Julia sets with the so-called Quadratic Minor Lamination (QML) representing a locally connected model of the Mandelbrot set. Similarly to the construction of an invariant lamination by pullbacks of certain leaves, we describe how QML can be generated by properly understood pullbacks of certain minors. In particular, we show that the minors of all non-renormalizable quadratic laminations can be obtained by taking limits of "pullbacks" of minors from the main cardioid. This is the second, amended version of the paper, to appear in Contemporary Mathematics

math.DS↗
arXiv2016

Non-degenerate locally connected models for plane continua and Julia sets

Suppose that a $X$ is an \emph{unshielded} plane continuum (i.e., $X$ coincides with the boundary of the unbounded complementary component of $X$). Then there exists a \emph{finest monotone} map $m:X\to L$, where $L$ is a locally connected continuum (i.e., $m^{-1}(y)$ is connected for each $y\in L$, and any monotone map $\varphi:X\to L'$ onto a locally connected continuum is a composition $\varphi=\varphi'\circ m$ where $\varphi':L\to L'$ is monotone). Such finest locally connected model $L$ of $X$ is easier to understand because $L$ is locally connected (in particular it can be described by a picture) and represents the finest but still understandable decomposition of $X$ into possibly complicated but pairwise disjoint \emph{fibers} (point-preimages) of $m$. However, in some cases (i.e., in case $X$ is indecomposable) $L$ is a singleton. In this paper we provide sufficient conditions for the existence of a non-degenerate model depending on the existence of certain subcontinua of $X$ and apply these results to the connected Julia sets of polynomials.

math.DS↗
arXiv2015

The combinatorial Mandelbrot set as the quotient of the space of geolaminations

We interpret the combinatorial Mandelbrot set in terms of \it{quadratic laminations} (equivalence relations $\sim$ on the unit circle invariant under $\sigma_2$). To each lamination we associate a particular {\em geolamination} (the collection $\mathcal{L}_\sim$ of points of the circle and edges of convex hulls of $\sim$-equivalence classes) so that the closure of the set of all of them is a compact metric space with the Hausdorff metric. Two such geolaminations are said to be {\em minor equivalent} if their {\em minors} (images of their longest chords) intersect. We show that the corresponding quotient space of this topological space is homeomorphic to the boundary of the combinatorial Mandelbrot set. To each equivalence class of these geolaminations we associate a unique lamination and its topological polynomial so that this interpretation can be viewed as a way to endow the space of all quadratic topological polynomials with a suitable topology.

math.DS↗