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Michael Yampolsky

Publications and source records attributed to Michael Yampolsky.

At least 37 records · Page 2Linked to original sources
arXiv2019

Structural Instability of Semi-Siegel Hénon maps

We show that the dynamics of sufficiently dissipative semi-Siegel complex Hénon maps with golden-mean rotation number is not $J$-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense $G_δ$ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.

math.DS↗
arXiv2018

Caratheodory convergence and harmonic measure

We give several new characterizations of Caratheodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.

math.CV↗
arXiv2017

Rigidity, universality,and hyperbolicity of renormalization for critical circle maps with non-integer exponents

We construct a renormalization operator which acts on analytic circle maps whose critical exponent $α$ is not necessarily an odd integer $2n+1$, $n\in\mathbb N$. When $α=2n+1$, our definition generalizes cylinder renormalization of analytic critical circle maps. In the case when $α$ is close to an odd integer, we prove hyperbolicity of renormalization for maps of bounded type. We use it to prove universality and $C^{1+α}$-rigidity for such maps.

math.DS↗
arXiv2017

Renormalization for unimodal maps with non-integer exponents

We define an analytic setting for renormalization of unimodal maps with an arbitrary critical exponent. We prove the global Hyperbolicity of Renormalization conjecture for unimodal maps of bounded type with a critical exponent which is sufficiently close to an even integer.

math.DS↗
arXiv2017

Computable geometric complex analysis and complex dynamics

We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and external rays impressions.

math.CV↗
arXiv2016

Renormalization of unicritical analytic circle maps

In this paper we generalize renormalization theory for analytic critical circle maps with a cubic critical point to the case of maps with an arbitrary odd critical exponent by proving a quasiconformal rigidity statement for renormalizations of such maps.

math.DS↗
arXiv2016

Golden mean Siegel disk universality and renormalization

We provide a computer-assisted proof of one of the central open questions in one-dimensional renormalization theory -- universality of the golden-mean Siegel disks. We further show that for every function in the stable manifold of the golden-mean renormalization fixed point the boundary of the Siegel disk is a quasicircle which coincides with the closure of the critical orbit, and that the dynamics on the boundary of the Siegel disk is rigid. Furthermore, we extend the renormalization from one-dimensional analytic maps with a golden-mean Siegel disk to two-dimensional dissipative Hénon-like maps and show that the renormalization hyperbolicity result still holds in this setting.

math.DS↗
arXiv2013

Constructive geometrization of Thurston maps and decidability of Thurston equivalence

The key result in the present paper is a direct analogue of the celebrated Thurston's Theorem for marked Thurston maps with parabolic orbifolds. Combining this result with previously developed techniques, we prove that every Thurston map can be constructively geometrized in a canonical fashion. As a consequence, we give a partial resolution of the general problem of decidability of Thurston equivalence of two postcritically finite branched covers of the two-sphere.

math.DS↗
arXiv2013

Computable Caratheodory Theory

Conformal Riemann mapping of the unit disk onto a simply-connected domain $W$ is a central object of study in classical Complex Analysis. The first complete proof of the Riemann Mapping Theorem given by P. Koebe in 1912 is constructive, and theoretical aspects of computing the Riemann map have been extensively studied since. Carath{é}odory Theory describes the boundary extension of the Riemann map. In this paper we develop its constructive version with explicit complexity bounds.

math.CV↗