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## Location of Siegel capture polynomials in parameter spaces

Nonlinearity. 2021. Vol. 34. No. 4. P. 2430-2453.

A cubic polynomial with a marked fixed point 0 is called an *IS*-*capture polynomial* if it has a Siegel disk *D* around 0 and if *D* contains an eventual image of a critical point. We show that any IS-capture polynomial is on the boundary of a unique bounded hyperbolic component of the polynomial parameter space determined by the rational lamination of the map and relate IS-capture polynomials to the cubic principal hyperbolic domain and its closure.

Publication based on the results of:

Comptes Rendus Mathematique 2017 Vol. 355 No. 5 P. 590-595

W. Thurston constructed a combinatorial model of the Mandelbrot set M2M2such that there is a continuous and monotone projection of M2M2to this model. We propose the following related model for the space MD3MD3of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)∈MD3(P,c1,c2)∈MD3, then every point z in the Julia set ...

Added: May 30, 2017

Communications in Mathematical Physics 2016 Vol. 341 No. 3 P. 733-749

A small perturbation of a quadratic polynomial f with a non-repelling fixed point gives a polynomial g with an attracting fixed point and a Jordan curve Julia set, on which g acts like angle doubling. However, there are cubic polynomials with a nonrepelling fixed point, for which no perturbation results into a polynomial with Jordan ...

Added: January 11, 2016

Quadratic-like dynamics of cubic polynomials / Cornell University. Series math "arxiv.org". 2013. No. 1305.5799.

A small perturbation of a quadratic polynomial with a non-repelling fixed point gives a polynomial with an attracting fixed point and a Jordan curve Julia set, on which the perturbed polynomial acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan ...

Added: October 6, 2013

Science China Mathematics 2018 Vol. 61 No. 12 P. 2121-2138

Added: November 24, 2018

Nonlinearity 2014 Vol. 27 No. 8 P. 1879-1897

The connectedness locus in the parameter space of quadratic polynomials is called the Mandelbrot set. A good combinatorial model of this set is due to Thurston. By definition, the principal hyperbolic domain of the Mandelbrot set consists of parameter values, for which the corresponding quadratic polynomials have an attracting fixed point. The closure of the ...

Added: August 25, 2014

Transactions of the American Mathematical Society 2019 Vol. 372 No. 7 P. 4829-4849

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic. By results of Kiwi, any dendritic polynomial is semiconjugate to a topological polynomial whose topological
Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset ...

Added: April 9, 2019

Discrete and Continuous Dynamical Systems 2017 Vol. 37 No. 11 P. 5781-5795

Every plane continuum admits a finest locally connected model. The latter is a locally connected continuum onto which the original continuum projects in a monotone fashion. It may so happen that the finest locally connected model is a singleton. For example, this happens if the original continuum is indecomposable. In this paper, we provide sufficient ...

Added: August 16, 2017

The Main Cubioid / Cornell University. Series math "arxiv.org". 2013. No. 1305.5798.

We discuss different analogs of the main cardioid in the parameter space of cubic polynomials, and establish relationships between them. ...

Added: October 6, 2013

Models for spaces of dendritic polynomials / Cornell University. Series arXiv "math". 2017.

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a ...

Added: November 22, 2017

Proceedings of the American Mathematical Society 2018 Vol. 146 No. 11 P. 4649-4660

Added: August 27, 2018

Fundamenta Mathematicae 2018 Vol. 241 No. 3 P. 265-290

The paper deals with Newton maps of complex exponential functions and a surgery tool developed by P. Haissinsky. The concept of "Postcritically minimal" Newton maps of complex exponential functions are introduced, analogous to postcritically finite Newton maps of polynomials. The dynamics preserving mapping is constructed between the space of postcritically finite Newton maps of polynomials ...

Added: January 11, 2018

Plane trees, Shabat-Zapponi polynomials and Julia sets / Cornell University. Series math "arxiv.org". 2016. No. 1608.08866.

A tree, embedded into plane, is a dessin d'enfant and its Belyi function is a polynomial -- Shabat polynomial. Zapponi form of this polynomial is unique, so we can correspond to an embedded tree the Julia set of its Shabat-Zapponi polynomial. In this purely experimental work we study relations between the form of a tree ...

Added: September 5, 2016

Cutpoints of invariant subcontinua of polynomial Julia sets / Cornell University. Series arXiv "math". 2021.

We prove fixed point results for branched covering maps f of the plane. For complex polynomials P with Julia set J_P these imply that periodic cutpoints of some invariant subcontinua of J_P are also cutpoints of JP. We deduce that, under certain assumptions on invariant subcontinua Q of J_P, every Riemann ray to Q landing at a periodic repelling/parabolic point x∈Q is isotopic to a Riemann ray to J_P relative to Q. ...

Added: November 24, 2021

Laminations from the Main Cubioid / Cornell University. Series math "arxiv.org". 2013. No. 1305.5788.

According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can ...

Added: October 6, 2013

Memoirs of the American Mathematical Society 2020 Vol. 265 No. 1288 P. 1-116

The so-called “pinched disk” model of the Mandelbrot set is due to A. Douady, J. H. Hubbard and W. P. Thurston. It can be described in the language of geodesic laminations. The combinatorial model is the quotient space of the unit disk under an equivalence relation that, loosely speaking, “pinches” the disk in the plane ...

Added: May 10, 2020

The Fano variety of lines and rationality problem for a cubic hypersurface / Cornell University. Series math "arxiv.org". 2014. No. 1405.5154.

We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...

Added: May 21, 2014

Selecta Mathematica, New Series 2018 Vol. 24 No. 1 P. 21-62

We study plane partitions satisfying condition a_{n+1,m+1}=0 (this condition is called “pit”) and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal gl1 algebra, therefore our formulas can be interpreted as the characters of ...

Added: October 24, 2018

ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск: ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013

Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Russian Mathematical Surveys 2017 Vol. 71 No. 6 P. 1146-1148

In the paper a Palis problem on finding sufficient conditions on embedding of Morse-Smale diffeomorphisms in topological flow is discussed. ...

Added: May 17, 2017

Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 565-600

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties.
This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik Zamolodchikov and dynamical q-difference equations. ...

Added: October 25, 2018

Moscow University Computational Mathematics and Cybernetics 2013 Vol. 37 No. 4 P. 180-188

The article investigates a model of delays in a network of functional elements (a gate network) in an arbitrary finite complete basis B, where basis elements delays are arbitrary positive real numbers that are specified for each input and each set of boolean variables supplied on the other inputs. Asymptotic bounds of the form τ ...

Added: December 2, 2019