### ?

## Cutpoints of Invariant Subcontinua of Polynomial Julia Sets

Arnold Mathematical Journal. 2022. Vol. 8. P. 271-284.

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 J_P. 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.

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., Timorin V., Transactions of the American Mathematical Society 2022 Vol. 375 No. 8 P. 5313-5359

In this paper, we study slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the main cubioid in this parameter space. The main cubioid is the set of affine conjugacy classes of complex cubic polynomials that ...

Added: June 15, 2022

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

Blokh A., Haïssinsky P., Oversteegen L. et al., Advances in Mathematics 2023 Vol. 428 Article 109135

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider invariant continua that are not polynomial-like Julia sets because of extra critical points. However, under certain assumptions, these invariant continua can be identified with Julia sets of lower degree polynomials up to a topological conjugacy. Thus we extend the concept of renormalization. ...

Added: August 16, 2023

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., 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., Selinger N. et al., Conformal Geometry and Dynamics 2023 Vol. 27 No. 8 P. 264-293

To investigate the degree $d$ connectedness locus, Thurston studied $\sigma_d$-invariant laminations, where $\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials $f(z) = z^2 +c$. In the spirit of Thurston's work, we consider the space of all cubic symmetric polynomials $f_\lambda(z)=z^3+\lambda^2 z$ in a series of three articles. In ...

Added: August 16, 2023

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

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

Blokh A., Oversteegen L., Ptacek R. et al., / Cornell University. Series math "arxiv.org". 2015.

We interpret the combinatorial Mandelbrot set in terms of quadratic laminations (equivalence relations ∼ on the unit circle invariant under σ2). To each lamination we associate a particular geolamination (the collection L∼ of points of the circle and edges of convex hulls of ∼-equivalence classes) so that the closure of the set of all of ...

Added: November 19, 2015

Blokh A., Oversteegen L., Timorin V., , in : Contemporary Mathematics 744 Dynamics: Topology and Numbers (2020). : United States of America : American Mathematical Society, 2020. Ch. 13. P. 205-229.

Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant
laminations representing Julia sets with the so-called Quadratic Minor Lamination (QML) representing a locally connected model of the Mandelbrot set. Similarly to the construction of ...

Added: November 7, 2020

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

Springer Nature Switzerland AG, 2019

Gathering the proceedings of the 11th CHAOS2018 International Conference, this book highlights recent developments in nonlinear, dynamical and complex systems. The conference was intended to provide an essential forum for Scientists and Engineers to exchange ideas, methods, and techniques in the field of Nonlinear Dynamics, Chaos, Fractals and their applications in General Science and the ...

Added: October 29, 2021

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., 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

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

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

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

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., Shepelevtseva A., Arnold Mathematical Journal 2019 Vol. 5 P. 435-481

We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching for invariant spanning trees. This procedure uses bisets over the fundamental group of a punctured sphere. We also introduce a new combinatorial invariant of Thurston classes—the ivy ...

Added: May 10, 2020

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

Blokh A., Oversteegen L., Ptacek R. M. et al., Discrete and Continuous Dynamical Systems 2016 Vol. 36 No. 9 P. 4665-4702

Polynomials from the closure of the principal hyperbolic domain of the cubic connectedness locus have some specific properties, which were studied in a recent paper by the authors. The family of (affine conjugacy classes of) all polynomials with these properties is called the Main Cubioid. In this paper, we describe a combinatorial counterpart of the ...

Added: July 6, 2016

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

Sirotin V., Arkhipova M., Dubrova T. A. et al., Bielsko-Biala : University of Bielsko-Biala Press, 2016

The main attributes of modern enterprises should be the flexibility and the ability of forecasting the future. Constant adaptation to the changing environment and the rapidity of undertaking certain actions which are conditioned by specific situations determine the rules for the future position of market competition. Effective and efficient adjustment of the company in line ...

Added: November 2, 2016