60 research outputs found

### 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

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