Related papers: Julia sets with a wandering branching point

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…

Let $X$ be a variety defined over a number field and $f$ be a dominant rational self-map of $X$ of infinite order. We show that $X$ admits many algebraic points which are not preperiodic under $f$. If $f$ were regular and polarized, this…

The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in combinatorial terms, and provides a useful tool to study parameter spaces of polynomials. The theory of core…

We show that the Julia set of quadratic maps with parameters in hyperbolic components of the Mandelbrot set is given by a transseries formula, rapidly convergent at any repelling periodic point. Up to conformal transformations, we obtain…

Let $P: {\mathbb C} \to {\mathbb C}$ be a polynomial map with disconnected filled Julia set $K_P$ and let $z_0$ be a repelling or parabolic periodic point of $P$. We show that if the connected component of $K_P$ containing $z_0$ is…

We give a simple algorithm that determines whether a given post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, the algorithm produces the Hubbard tree; otherwise, the algorithm produces the…

We show that there exist real parameters $c$ for which the Julia set $J_c$ of the quadratic map $z^2+c$ has arbitrarily high computational complexity. More precisely, we show that for any given complexity threshold $T(n)$, there exist a…

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings $D$. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda)=\lambda$ for any $n \in \mathbb{N}$, where…

It has been known since Julia that polynomials commuting under composition have the same Julia set. More recently in the works of Baker and Eremenko, Fern\'andez, and Beardon, results were given on the converse question: When do two…

We give a new proof of a theorem of Hubbard-Oberste-Vorth [HOV2] for H\'enon maps that are perturbations of a hyperbolic polynomial and recover the Julia set $J^{+}$ inside a polydisk as the image of the fixed point of a contracting…

We show that Fatou components of a semi-hyperbolic rational map are John domains and that the converse does not hold. This generalizes a famous result of Carleson, Jones and Yoccoz. We show that a connected Julia set is locally connected…

Let $f \in Q(z)$ be a polynomial or rational function of degree 2. A special case of Morton and Silverman's Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of $f$ is bounded above by an…

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^\mu-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,a)$ with $\varepsilon\approx 0$ and $a$ in an open subset $A$ of…

Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a hyperbolic rational map of degree $d \geq 2$, and let $J \subset \mathbb{C}$ be its Julia set. We prove that $J$ always has positive Fourier dimension. The case where $J$ is…

We discuss the dynamics of semigroups of transcendental entire functions using Fatou-Julia theory and provide a condition for the complete invariance of escaping set and Julia set of transcendental semigroups. Results regarding limit…

An important conjecture in percolation theory is that almost surely no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1. Earlier work has established this for the…

For a sequence of complex parameters $\{c_n\}$ we consider the compositions of functions $f_{c_n} (z) = z^2 + c_n$, which is the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are…

We give a criterion to determine when two external rays land at the same point for polynomials with locally connected Julia sets. As an application, we provide an elementary proof of the monotonicity of the core entropy along arbitrary…

A. Sannami constructed an example of the differentiable Cantor set embedded in the real line whose difference set has a positive measure. In this paper, we generalize the definition of the difference sets for sets of the two dimensional…

We have introduced the notion of the bungee set and the filled Julia set of a transcendental semigroup using Fatou-Julia theory. Numerous results of the bungee set of a single transcendental entire function have been generalized to a…