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Quadratic-like dynamics of cubic polynomials
Cornell University
,
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 curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.
Blokh A., Oversteegen L., Ptacek R. et al., 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
Blokh A., Oversteegen L., Ptacek R. M. et al., 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
Blokh A., Cheritat A., Oversteegen L. et al., 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 ...
Added: April 26, 2021
Timorin V., Oversteegen L., Blokh A. et al., / 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
Timorin V., Blokh A., Oversteegen L. et al., / 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
Ptacek R., Blokh A., Oversteegen L. et al., 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
Mamayusupov K., 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
Blokh A., Oversteegen L., Shepelevtseva A. et al., Moscow Mathematical Journal 2022 Vol. 22 No. 2 P. 265-294
The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such slices as parameter spaces were studied by S. Zakeri, so we call them Zakeri slices. We ...
Added: May 27, 2022
Blokh A., Oversteegen L., Ptacek R. et al., Proceedings of the American Mathematical Society 2018 Vol. 146 No. 11 P. 4649-4660
Added: August 27, 2018
Galkin S., Shinder E., / 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
Timorin V., Ross P., Lex O. et al., / 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
Blokh A., Oversteegen L., Timorin V., 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
Blokh A., Oversteegen L., Timorin V., Science China Mathematics 2018 Vol. 61 No. 12 P. 2121-2138
Added: November 24, 2018
Kochetkov Y., / Cornell Univercity. Series arXiv.org e-print archive "arXiv.math". 2024. No. 2401.11208.
A cubic Galois polynomial is a cubic polynomial with rational coefficients that defines a cubic Galois field. Its
discriminant is a full square and its roots $x_1,x_2,x_3$ (enumerated in some order) are real. There exists (and only one) quadratic polynomial $q$ with rational coefficients such that $q(x_1)=x_2, q(x_2)=x_3, q(x_3)=x_1$. The polynomial $r=q(q) \text{ mod } p$ cyclically permutes roots of $p$ ...
Added: February 5, 2024
Timorin V., Oversteegen L., Blokh A., / 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
Alexander B., Lex O., Ptacek R. M. et al., 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
Blokh A., Oversteegen L., Ptacek R. et al., 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
Kotelnikova M. V., Aistov A., 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
Borzykh D., 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
В. Л. Попов, 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
Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18
Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...
Added: November 16, 2020
Grines V., Gurevich E., Pochinka O., 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
Okounkov A., Aganagic M., 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