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Models for spaces of dendritic polynomials
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 of the closed bidisk. This construction generalizes the
“pinched disk” model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.
Publication based on the results of:
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., 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
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
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
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
Кузютин Д. В., Смирнова Е. Л., Razgulyaeva L. et al., Изд-во МБИ, 2010
The present textbook is intended for students preparing to study mathematics at a higher education institution, to prepare to pass the exam. ...
Added: November 4, 2014
Kochetkov Y., / 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
Baranov A., Zarouf R., Bulletin des Sciences Mathematiques 2013 No. 4 P. 541-556
Given n ≥ 1 and r ∈ [0, 1), we consider the set Rn, r of rational functions having at most n poles all outside of 1/rD, were D is the unit disc of the complex plane. We give an asymptotically sharp Bernstein-type inequality for functions in Rn, r in weighted Bergman spaces with “polynomially” ...
Added: January 17, 2014
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., Oversteegen L., Ptacek R. et al., Proceedings of the American Mathematical Society 2018 Vol. 146 No. 11 P. 4649-4660
Added: August 27, 2018
Blokh A., Oversteegen L., Timorin V., Moscow Mathematical Journal 2023 Vol. 23 No. 4 P. 441-461
A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be immediately renormalizable if there exists a (connected) QL invariant filled Julia set $K^*$ such that $b\in K^*$. In that case, exactly one critical point of $P$ does not belong to $K^*$. We show that if, in addition, the Julia set of $P$ has no (pre)periodic cutpoints, then ...
Added: November 29, 2023
Tran-Van-Minh A., Cazé R. D., Abrahamsson T. et al., Frontiers in Cellular Neuroscience 2015 Vol. 9 No. March, Article number 67
Nonlinear dendritic integration is thought to increase the computational ability of neurons. Most studies focus on how supralinear summation of excitatory synaptic responses arising from clustered inputs within single dendrites result in the enhancement of neuronal firing, enabling simple computations such as feature detection. Recent reports have shown that sublinear summation is also a prominent ...
Added: February 24, 2015
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., Timorin V., Nonlinearity 2024 Vol. 37 No. 3 Article 035003
We establish a version of the Pommerenke–Levin–Yoccoz inequality for the modulus of a polynomial-like (PL) restriction of a polynomial and give two applications. First we show that if the modulus of a PL restriction of a polynomial is bounded from below then this restricts the combinatorics of the polynomial. The second application concerns parameter slices ...
Added: February 16, 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
Timorin V., Blokh A., Oversteegen L. et al., / 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
Vyugin I. V., Макарычев С. В., / Cornell University. Series arXiv "math". 2015.
We present an upeer bound of the number of solutions (x,y) of a polynomial equation P(x,y)=0 over a field F_p in the case where x,y from G, G is a subgroup of F_p^*. ...
Added: June 22, 2015
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