Related papers: Backward Julia sets for a class of p-adic H\'enon …

Let $f$ and $g$ be two H\'enon-Sibony maps of $\mathbb{C}^k$. We show that if they have the same forward Julia set, then they share a common iterate, thereby extending Lamy's results from dimension 2.

This note initiates the study of the Fatou\,--\,Julia sets of a complex harmonic mapping. Along with some fundamental properties of the Fatou and the Julia sets, we observe some contrasting behaviour of these sets as those with in case of a…

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets go to 2. We prove that the Lebesgue measure of these Julia sets tend to…

We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.

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…

In this paper we study, for the first time, Julia limiting directions of quasiregular mappings in $\mathbb{R}^n$ of transcendental-type. First, we give conditions under which every direction is a Julia limiting direction. Along the way, our…

Let $f$ be a generically finite polynomial map $f: \mathbb{C}^n\to \mathbb{C}^m$ of algebraic degree $d$. Motivated by the study of the Jacobian Conjecture, we prove that the set $S_f$ of non-properness of $f$ is covered by parametric…

For a post-critically finite hyperbolic rational map $f$, we show that its Julia set $\mathcal{J}_f$ has Ahlfors-regular conformal dimension one if and only if $f$ is a crochet map, i.e., there is an $f$-invariant connected graph $G$…

This paper is a continuation of authors work: Fatou and Julia like sets,Ukranian J. Math., to appear/arXiv:2006.08308[math.CV](see [4]). Here, we introduce escaping like set and generalized escaping like set for a family of holomorphic…

The Julia set of the exponential family $E_{\kappa}:z\mapsto\kappa e^z$, $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<\kappa\leq1/e$ the Julia…

For a H\'enon map of the form $H(x, y) = (y, p(y) - ax)$, where $p$ is a polynomial of degree at least two and $a \not= 0$, it is known that the sub-level sets of the Green's function $G^+_H$ associated with $H$ are Short $\mathbb C^2$'s.…

We study the dynamics of a generic automorphism $f$ of a Stein manifold with the density property. Such manifolds include all linear algebraic groups. Even in the special case of $\mathbb C^n$, $n\geq 2$, most of our results are new. We…

For rational numbers $c$, we present a trichotomy of the set of totally real (totally $p$-adic, respectively) preperiodic points for maps in the quadratic unicritical family $f_c(x)=x^2+c$. As a consequence, we classify quadratic…

We study how the orbits of the singularities of the inverse of a meromorphic function prescribe the dynamics on its Julia set, at least up to a set of (Lebesgue) measure zero. We concentrate on a family of entire transcendental functions…

The problem of backward dynamics over the ring of p-adic integers is studied. It is shown that Inverse Limit Theory provides the right framework. Backward iterations of a polynomial with p-adic integer coefficients are constructed by…

We prove that several dynamically defined fractals in $\mathbb{C}$ and $\mathbb{C}^2$ which arise from different type of polynomial dynamical systems can not be the same objects. One of our main results is that the closure of Misiurewicz…

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with…

Let f be a polynomial endomorphism of degree d>1 of C^k (k>1) which extends to a holomorphic endomorphism of P^k. Assume that the maximal order Julia set of f is laminated by real hypersurfaces in some open set. We show that f is homogenous…

By a symmetry of the Julia set of a polynomial, also referred as polynomial Julia set, we mean an Euclidean isometry preserving the Julia set. Each such symmetry is in fact a rotation about the centroid of the polynomial. In this article, a…

We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod $k\geq 1$, is an irreducible algebraic curve. We also establish an analogous result for quadratic rational maps. We then study a…