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Euclidean dual billiard outside a regular $N$-gon or $2N$-gon, for $N>3$ odd, has aperiodic points., Comment: 26 pages, 6 figures

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 this critical point is recurrent., Comment: 26 pages, 1 figure; accepted to a special issue of the MMJ dedicated to Yulij Ilyashenko's 80th Birthday. arXiv admin note: substantial text overlap with arXiv:2102.10325

We construct a locally connected model of the boundary of the cubic connectedness locus. The model is obtained by constructing a decomposition of the space of critical portraits and collapsing elements of the decomposition into points. This model is similar to a quotient of the quadratic combinatorial locus where all baby Mandelbrot sets are collapsed to points. All fibers of the model, possibly except one, are connected., Comment: 24 pages

The modulus of a polynomial-like (PL) map is an important invariant that controls distortion of the straightening map and, hence, geometry of the corresponding PL Julia set. Lower bounds on the modulus, called complex a priori bounds, are known in a great variety of contexts. For any rational function we complement this by an upper bound for moduli of PL maps in the satellite case that depends only on the relative period and the degree of the PL map. This rules out a priori bounds in the satellite case with unbounded relative periods. We also apply our tools to obtain lower bounds for hyperbolic lengths of geodesics in the infinitely renormalizable case, and to show that moduli of annuli must converge to 0 for a sequence of arbitrary renormalizations, under several conditions all of which are shown to be necessary., Comment: 17 pages; satellite moduli estimates simplified and extended; some less important results removed

To investigate the degree $d$ connectedness locus, Thurston studied \emph{$\sigma_d$-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 $f_c(z) = z^2 +c$. In the same spirit, we consider the space of all \emph{cubic symmetric polynomials} $f_\lambda(z)=z^3+\lambda^2 z$ in three articles. In the first one we construct the lamination $C_sCL$ together with the induced factor space $\mathbb{S}/C_sCL$ of the unit circle $\mathbb{S}$. As will be verified in the third paper, $\mathbb{S}/C_sCL$ is a monotone model of the \emph{cubic symmetric connectedness locus}, i.e., the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for constructing $C_sCL$ analogous to the Lavaurs algorithm for constructing a combinatorial model $\mathcal{M}^{comb}_2$ of the Mandelbrot set $\mathcal{M}_2$., Comment: 27 pages, 3 figures. arXiv admin note: text overlap with arXiv:2201.11434

We establish a version of the Pommerenke-Levin-Yoccoz inequality for the modulus of a polynomial-like restriction of a global polynomial and give two applications. First it is shown that if the modulus of a polynomial-like restriction of an arbitrary polynomial is bounded from below then this forces bounded combinatorics. The second application concerns parameter slices of cubic polynomials given by a non-repelling value of a fixed point multiplier. Namely, the intersection of the main cubioid and the multiplier slice lies in the closure of the principal hyperbolic domain, with only possible exception of queer components., Comment: 21 pages, 3 figures; v2: references added

To investigate the degree $d$ connectedness locus, Thur\-ston studied \emph{$\sigma_d$-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 \emph{cubic symmetric polynomials} $f_\lambda(z)=z^3+\lambda^2 z$ in a series of three articles. In the present paper, the first in the series, we construct a lamination $C_sCL$ together with the induced factor space ${\mathbb{S}}/C_sCL$ of the unit circle ${\mathbb{S}}$. As will be verified in the third paper of the series, ${\mathbb{S}}/C_sCL$ is a monotone model of the \emph{cubic symmetric connected locus}, i.e. the space of all cubic symmetric polynomials with connected Julia sets., Comment: 37 pages, 4 figures

A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be \emph{immediately renormalizable} if there exists a (connected) quadratic-like 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 this critical point is recurrent., Comment: 34 pages, 2 figures

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 give a model of the central part of a slice (the subset of the slice that can be approximated by hyperbolic polynomials with Jordan curve Julia sets), and a continuous projection from the central part to the model. The projection is defined dynamically and agrees with the dynamical-analytic parameterization of the Principal Hyperbolic Domain by Petersen and Tan Lei., Comment: 37 pages, 5 figures; to appear in Moscow Math. J. This ver: figures and explanations added

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 $x\in Q$ is isotopic to a Riemann ray to $J_P$ relative to $Q$., Comment: 17 pages

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider certain conditions guaranteeing that a polynomial which does not admit a polynomial-like connected Julia set still admits an invariant continuum on which it is topologically conjugate to a lower degree polynomial. This invariant continuum may contain extra critical points of the original polynomial that are not visible in the dynamical plane of the conjugate polynomial. Thus, we extend the notions of holomorphic renormalization and polynomial-like maps and describe a setup where new generalized versions of these notions are applicable and yield useful topological conjugacies., Comment: 31 pages, 3 figures

We construct a model of the cubic connectedness locus., Comment: 17 pages

A cubic polynomial $f$ with a periodic Siegel disk containing an eventual image of a critical point is said to be a \emph{Siegel capture polynomial}. If the Siegel disk is invariant, we call $f$ a \emph{IS-capture polynomial} (or just an IS-capture; IS stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all cubic polynomials and show that any IS-capture is on the boundary of a unique hyperbolic component determined by the rational lamination of the map. We also relate IS-captures to the cubic Principal Hyperbolic Domain and its closure (by definition, the \emph{cubic Principal Hyperbolic Domain} consists of cubic hyperbolic polynomials with Jordan curve Julia sets) and prove that, in the slice of cubic polynomials given by a fixed multiplier at one of the fixed points, the closure of the cubic principal hyperbolic domain might possibly only have bounded complementary domains $U$ such that (1) critical points of $f\in U$ are distinct and belong to $J(f)$, and (2) $J(f)$ has positive Lebesgue measure and carries an invariant line field., Comment: 34 pages; in the new (third) version we corrected some typos and added Section 6 on the existence of IS-capture polynomials

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 graph., Comment: A small but essential correction in Theorem A compared to the published version; 55 pages, 8 figures

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady and, in different terms, by Thurston. Thurston used quadratic invariant laminations as a major tool. As has been previously shown by the authors, the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant laminations. The topology in the space of laminations is defined by the Hausdorff distance. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that taken for the Mandelbrot set. The result (the quotient space) is obtained from the Mandelbrot set by "unpinching" the transitions between adjacent hyperbolic components. In the second case, we do not identify non-renormalizable laminations while identifying renormalizable laminations according to which hyperbolic lamination they can be "unrenormalised" to., Comment: 29 pages, 4 figures

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{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 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., Comment: 23 pages, 1 figure; in this version the proof of a key lemma was shortened; to appear in Trans. Amer. Math. Soc

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 have certain dynamical properties generalizing those of polynomials $z^2+c$ for $c$ in the filled main cardioid., Comment: 60 pages, 3 figures

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors., Comment: 40 pages; 2 figures; to appear in Ergodic Theory and Dynamical Systems

We study the closure of the cubic Principal Hyperbolic Domain and its intersection $\mathcal{P}_\lambda$ with the slice $\mathcal{F}_\lambda$ of the space of all cubic polynomials with fixed point $0$ defined by the multiplier $\lambda$ at $0$. We show that any bounded domain $\mathcal{W}$ of $\mathcal{F}_\lambda\setminus\mathcal{P}_\lambda$ consists of $J$-stable polynomials $f$ with connected Julia sets $J(f)$ and is either of \emph{Siegel capture} type (then $f\in \mathcal{W}$ has an invariant Siegel domain $U$ around $0$ and another Fatou domain $V$ such that $f|_V$ is two-to-one and $f^k(V)=U$ for some $k>0$) or of \emph{queer} type (then at least one critical point of $f\in \mathcal{W}$ belongs to $J(f)$, the set $J(f)$ has positive Lebesgue measure, and carries an invariant line field)., Comment: 12 pages; one figure; to appear in Proc. Amer. Math. Soc. arXiv admin note: substantial text overlap with arXiv:1305.5799

We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves., Comment: 18 pages, 6 figures

A model for the Mandelbrot set is due to Thurston and is stated in the language of geodesic laminations. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials, even conjectural models are missing, one possible reason being that the higher degree analog of the MLC conjecture is known to be false. We provide a combinatorial model for an essential part of the parameter space of complex cubic polynomials, namely, for the space of all cubic polynomials with connected Julia sets all of whose cycles are repelling (we call such polynomials \emph{dendritic}). The description of the model turns out to be very similar to that of Thurston., Comment: 52 pages, 12 figures (in the new version a few typos have been corrected and some proofs have been expanded). arXiv admin note: substantial text overlap with arXiv:1401.5123

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 (whence the name of the model). The significance of the model lies in particular in the fact that this quotient is planar and therefore can be easily visualized. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials no combinatorial model is known. One possible reason may be that the higher degree analog of the MLC conjecture is known to be false. We investigate to which extent a geodesic lamination is determined by the location of its critical sets and when different choices of critical sets lead to essentially the same lamination. This yields models of various parameter spaces of laminations similar to the "pinched disk" model of the Mandelbrot set., Comment: 98 pages, 18 figures; to appear in the Memoirs of the AMS

Section 235 of Gauss' fundamental treatise "Disquisitiones Arithmeticae" establishes basic properties that compositions of binary quadratic forms must satisfy. Although this section is very technical, it contains truly important results. We review section 235 using a more invariant language and simplifying the arguments. We also make the statements slightly stronger by removing unnecessary assumptions., Comment: 12 pages; v3: minor correction in Proposition 4.3, a reference added

If the complement of a closed convex set in a closed convex cone is bounded, then this complement minus the apex of the cone is called a coconvex set. Coconvex sets appear in singularity theory (they are closely related to Newton diagrams) and in commutative algebra. Such invariants of coconvex sets as volumes, mixed volumes, number of integer points, etc., play an important role. This paper aims at extending various results from the theory of convex bodies to the coconvex setting. These include the Aleksandrov-Fenchel inequality and the Ehrhart duality., Comment: 17 pages, 1 figure

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., Comment: Now 23 pages. In the new version we strengthen some of the results using new arguments. We also expand some proofs and add some references. A preprint "Complementary components to the cubic Principal Hyperbolic Domain" with related results is being posted to arxiv too. The paper is to appear at Communications in Mathematical Physics

We discuss different analogs of the main cardioid in the parameter space of cubic polynomials, and establish relationships between them., Comment: 25 pages

According to a recent paper \cite{bopt13}, polynomials from the closure $\bar{\rm PHD}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\mathrm{CU}$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\mathrm{CU}^c$ of laminations which can be associated to polynomials from $\mathrm{CU}$., Comment: 38 pages, 5 figures (in the new version a few typos have been corrected and a few proofs have been expanded). To appear in Discrete and Continuous Dynamical Systems. arXiv admin note: text overlap with arXiv:1106.5022

We prove a version of the Aleksandrov-Fenchel inequality for mixed volumes of coconvex bodies. This version is motivated by an inequality from commutative algebra relating intersection multiplicities of ideals., Comment: 5 pages

We discuss the problem of counting vertices in Gelfand-Zetlin polytopes. Namely, we deduce a partial differential equation with constant coefficients on the exponential generating function for these numbers. For some particular classes of Gelfand-Zetlin polytopes, the number of vertices can be given by explicit formulas., Comment: 11 pages, 3 figures

In parameter slices of quadratic rational functions, we identify arcs represented by matings of quadratic polynomials. These arcs are on the boundaries of hyperbolic components., Comment: 25 pages

We define a very general class of rational functions f:CP^1 --> CP^1 such that for every function f of this class, there exists a countable family of smooth curves \gamma_i and a critically finite hyperbolic function R such that the dynamical system obtained from f by cutting along the curves \gamma_i is topologically semi-conjugate to R., Comment: 4 pages

Aiming at a generalization of a classical theorem of Moebius, we study maps that take line intervals to plane curves, and also maps that take line intervals to conics from certain linear systems., Comment: 9 pages

We define the (dynamical) core of a topological polynomial (and the associated lamination). This notion extends that of the core of a unimodal interval map. Two explicit descriptions of the core are given: one related to periodic objects and one related to critical objects. We describe all laminations associated with quadratic and cubic topological polynomials with a simple core (in the quadratic case, these correspond precisely to points on the Main Cardioid of the Mandelbrot set)., Comment: 47 pages, 8 figures

We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope., Comment: 33 pages, 4 figures, introduction rewritten, Section 4 restructured, typos corrected

We establish a general result on the existence of partially defined semiconjugacies between rational functions acting on the Riemann sphere. The semiconjugacies are defined on the complements to at most one-dimensional sets. They are holomorphic in a certain sense., Comment: 20 pages

In this (mostly expository) paper, we review a proof of the following old theorem of R.L. Moore: for a closed equivalence relation on the 2-sphere such that all equivalence classes are connected and non-separating, and not all points are equivalent, the quotient space is homeomorphic to the 2-sphere. The proof uses a general topological theory close to but simpler than an original theory of Moore. The exposition is organized so that to make applications of Moore's theory (not only Moore's theorem) in complex dynamics easier, although no dynamical applications are mentioned here., Comment: 19 pages

Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston--Teichm\"uller theory. We will discuss a topological theory of regluing, and trace a direction in which a holomorphic theory can develop., Comment: 38 pages

Consider a quadratic rational self-map of the Riemann sphere such that one critical point is periodic of period 2, and the other critical point lies on the boundary of its immediate basin of attraction. We will give explicit topological models for all such maps. We also discuss the corresponding parameter picture.

The classical Tait-Kneser theorem states that the osculating circles of a smooth plane curve, free from curvature extrema, are pairwise disjoint. We prove a number of analogs of this theorem, e.g., for ovals of osculating cubics, osculating polynomials and trigonometric polynomials; in each case, we will obtain a non-differentiable foliation with smooth leaves., Comment: 14 pages, 6 figures, v2: references added

A quadratic form f is said to have semigroup property if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with semigroup property. If there is an integer bilinear map s such that f(s(x,y))=f(x)f(y) for all vectors x and y from the integer 2-dimensional lattice, then the form f has semigroup property. We give an explicit description of all pairs (f,s) with the property stated above. We do not know any other examples of forms with semigroup property., Comment: v3: minor changes, referenced added; 28 pages, 1 figure

We list all analytic diffeomorphisms between an open subset of the 4-dimensional projective space and an open subset of the 4-dimensional sphere that take all line segments to arcs of round circles. These are the following: restrictions of the quaternionic Hopf fibrations and projections from a hyperplane to a sphere from some point. We prove this by finding the exact solutions of the corresponding system of partial differential equations., Comment: 10 pages

Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has rank at least 2, is equivalent to a fractional quadratic rounding. A fractional quadratic map is just the ratio of a quadratic map and a quadratic polynomial. We also show that any rounding gives rise to a quadratic map between spheres. The known results on quadratic maps between spheres have some interesting implications concerning roundings., Comment: 14 pages, v3: minor changes to improve readability

Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this map takes all lines passing though the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra does. We also describe a connection between our result and the Hurwitz--Radon theorem about sums of squares., Comment: 12 pages; to appear in Funct. Anal. and its Appl.; v4: minor corrections