Recurrent and periodic points in dendritic Julia sets

We relate periodic and recurrent points in dendritic Julia sets. This generalizes well-known results for interval dynamics.Comment: 13 pages; to appear in Proceedings of the American Mathematical Societ

Forcing among patterns with no block structure

Define the following order among all natural numbers except for 2 and 1: $$4\gg 6\gg 3\gg \dots \gg 4n\gg 4n+2\gg 2n+1\gg 4n+4\gg\dots$$ Let $f$ be a continuous interval map. We show that if $m\gg s$ and $f$ has a cycle with no division (no block structure) of period $m$ then $f$ has also a cycle with no division (no block structure) of period $s$. We describe possible sets of periods of cycles of $f$ with no division and no block structure.Comment: 18 pages; to appear in Topology Proceeding

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

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

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

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

Combinatorial models for spaces of cubic polynomials

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:1401.512

Quadratic-like dynamics of cubic polynomials

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 Physic

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