arXiv subjects

Oleg Ivrii

Publications and source records attributed to Oleg Ivrii.

22 records · Page 2Linked to original sources

arXiv2016

The geometry of the Weil-Petersson metric in complex dynamics

In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. Via the Bers embedding, one may view the Weil-Petersson metric as a metric on the main cardioid of the Mandelbrot set. We prove that the metric completion attaches the geometrically finite parameters from the Euclidean boundary of the main cardioid and conjecture that this is the entire completion. For the upper bound, we estimate the intersection of a circle $S_r = \{z : |z| = r\}$, $r \approx 1$, with an invariant subset $\mathcal G \subset \mathbb{D}$ called a half-flower garden, defined in this work. For the lower bound, we use gradients of multipliers of repelling periodic orbits on the unit circle. Finally, utilizing the convergence of Blaschke products to vector fields, we compute the rate at which the Weil-Petersson metric decays along radial degenerations.

math.DS↗

arXiv2016

On Makarov's principle in conformal mapping

We examine several characteristics of conformal maps that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin. While these quantities need not be equal in general, they agree for domains whose boundaries are regular fractals such as Julia sets or limit sets of quasi-Fuchsian groups. We give a new proof of these dynamical equalities. We also show that these characteristics have the same universal bounds and prove a central limit theorem for extremals. Our method is based on analyzing the local variance of dyadic martingales associated to Bloch functions.

math.CV↗

arXiv2016

Quasicircles of dimension 1+k^2 do not exist

A well-known theorem of S. Smirnov states that the Hausdorff dimension of a $k$-quasicircle is at most $1+k^2$. Here, we show that the precise upper bound $D(k) = 1+Σ^2 k^2 + \mathcal O(k^{8/3-\varepsilon})$ where $Σ^2$ is the maximal asymptotic variance of the Beurling transform, taken over the unit ball of $L^\infty$. The quantity $Σ^2$ was introduced in a joint work with K. Astala, A. Perälä and I. Prause where it was proved that $0.879 < Σ^2 \le 1$, while recently, H. Hedenmalm discovered that surprisingly $Σ^2 <1$. We deduce the asymptotic expansion of $D(k)$ from a more general statement relating the universal bounds for the integral means spectrum and the asymptotic variance of conformal maps. Our proof combines fractal approximation techniques with the classical argument of J. Becker and Ch. Pommerenke for estimating integral means.

math.DS↗

arXiv2015

Asymptotic variance of the Beurling transform

We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a $k$-quasicircle is at most $1+k^2$, it is natural to expect that the maximum asymptotic variance $Σ^2 = 1$. In this paper, we prove $0.87913 \le Σ^2 \le 1$. For the lower bound, we give examples of polynomial Julia sets which are $k$-quasicircles with dimensions $1+ 0.87913 \, k^2$ for $k$ small, thereby showing that $Σ^2 \ge 0.87913$. The key ingredient in this construction is a good estimate for the distortion $k$, which is better than the one given by a straightforward use of the $λ$-lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating $Σ^2$ in terms of nearly circular polynomial Julia sets.

math.CV↗