Perfect subspaces of quadratic laminations

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 figure

Density of orbits in laminations and the space of critical portraits

Thurston introduced \si_d-invariant laminations (where \si_d(z) coincides with z^d:\ucirc\to \ucirc, $d\ge 2$). He defined \emph{wandering $k$-gons} as sets \T\subset \ucirc such that \si_d^n(\T) consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets \si_d^n(\T) in the plane are pairwise disjoint. Thurston proved that \si_2 has no wandering $k$-gons and posed the problem of their existence for \si_d,\, $d\ge 3$. Call a lamination with wandering $k$-gons a \emph{WT-lamination}. Denote the set of cubic critical portraits by \A_3. A critical portrait, compatible with a WT-lamination, is called a \emph{WT-critical portrait}; let \WT_3 be the set of all of them. It was recently shown by the authors that cubic WT-laminations exist and cubic WT-critical portraits, defining polynomials with \emph{condense} orbits of vertices of order three in their dendritic Julia sets, are dense and locally uncountable in \A_3 ($D\subset X$ is \emph{condense in $X$} if $D$ intersects every subcontinuum of $X$). Here we show that \WT_3 is a dense first category subset of \A_3. We also show that (a) critical portraits, whose laminations have a condense orbit in the topological Julia set, form a residual subset of \A_3, (b) the existence of a condense orbit in the Julia set $J$ implies that $J$ is locally connected.Comment: 13 pages; accepted for publication in Discrete and Continuous Dynamical System

Finitely Suslinian models for planar compacta with applications to Julia sets

A compactum X\subset \C is unshielded if it coincides with the boundary of the unbounded component of \C\sm X. Call a compactum $X$ finitely Suslinian if every collection of pairwise disjoint subcontinua of $X$ whose diameters are bounded away from zero is finite. We show that any unshielded planar compactum $X$ admits a topologically unique monotone map $m_X:X \to X_{FS}$ onto a finitely Suslinian quotient such that any monotone map of $X$ onto a finitely Suslinian quotient factors through $m_X$. We call the pair $(X_{FS},m_X)$ (or, more loosely, $X_{FS}$) the finest finitely Suslinian model of $X$. If f:\C\to \C is a branched covering map and X \subset \C is a fully invariant compactum, then the appropriate extension $M_X$ of $m_X$ monotonically semiconjugates $f$ to a branched covering map g:\C\to \C which serves as a model for $f$. If $f$ is a polynomial and $J_f$ is its Julia set, we show that $m_X$ (or $M_X$) can be defined on each component $Z$ of $J_f$ individually as the finest monotone map of $Z$ onto a locally connected continuum.Comment: 16 pages, 3 figures; accepted for publication in Proceedings of the American Mathematical Societ

Fixed points in non-invariant plane continua

If $f:[a,b]\to \mathbb{R}$, with $a<b$, is continuous and such that $a$ and $b$ are mapped in opposite directions by $f$, then $f$ has a fixed point in $I$. Suppose that $f:\mathbb{C}\to\mathbb{C}$ is map and $X$ is a continuum. We extend the above for certain continuous maps of dendrites $X\to D, X\subset D$ and for positively oriented maps $f:X\to \mathbb{C}, X\subset \mathbb{C}$ with the continuum $X$ not necessarily invariant. Then we show that in certain cases a holomorphic map $f:\mathbb{C}\to\mathbb{C}$ must have a fixed point $a$ in a continuum $X$ so that either $a\in \mathrm{Int}(X)$ or $f$ exhibits rotation at $a$.Comment: 21 pages with corrected reference

Complementary components to the cubic Principal Hyperbolic Domain

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:1305.579

Topological polynomials with a simple core

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 figure

Laminations from the Main Cubioid

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:1106.502

Laminational models for some spaces of polynomials of any degree

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 AM

The Main Cubioid

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

The parameter space of cubic laminations with a fixed critical leaf

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 System