Related papers: Bowen's formula for meromorphic functions

We describe the two sets of meromorphic univalent functions in the class $\Sigma$, for which the sequence of Faber polynomials $\{F_j\}_{j=1}^\infty $ have the roots with following properties respectively:…

Let $g(z)=\int_0^zp(t)\exp(q(t))\,dt+c$ where $p,q$ are polynomials and $c\in\mathbb{C}$, and let $f$ be the function from Newton's method for $g$. We show that under suitable assumptions the Julia set of $f$ has Lebesgue measure zero.…

Let $f$ be a transcendental entire function of finite order which has an attracting periodic point $z_0$ of period at least $2$. Suppose that the set of singularities of the inverse of $f$ is finite and contained in the component $U$ of the…

A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the…

We show the existence of transcendental entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ with Hausdorff-dimension $1$ Julia sets, such that every Fatou component of $f$ has infinite inner connectivity. We also show that there exist…

We prove that the meandering set for $f_a(z)=e^z+a$ is homeomorphic to the space of irrational numbers whenever $a$ belongs to the Fatou set of $f_a$. This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies that the…

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ of H\"older $ {\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper…

In this paper, we continue our investigation on sub-additive pressures for $C^1$-smooth partially hyperbolic diffeomorphisms. Under the assumption of unstable almost product property, we show that the unstable Bowen topological pressure on…

We show that any uniformly escaping and wandering dynamics of a holomorphic function on a compact subset of the plane can be realised by a transcendental meromorphic function on $\mathbb{C}$. More precisely, let $\varphi$ be a holomorphic…

If $f$ is a meromorphic function from the complex plane ${\mathbb C}$ to the extended complex plane $\overline{ {\mathbb C} }$, for $r > 0$ let $n(r)$ be the maximum number of solutions in $\{z\colon |z| \leq r \}$ of $f(z) = a$ for $a \in…

Let $g$ be a polynomial automorphism of $\C^2$. We study the Hausdorff dimension and topological dimension of the Julia set of $g$. We show that when $g$ is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly…

In this paper, for a transcendental meromorphic function $f$ and $a\in \mathbb{C}$, we have exhaustively studied the nature and form of solutions of a new type of non-linear differential equation of the following form which has never been…

Any finite union of disjoint, mutually exterior Jordan curves in the complex plane can be approximated arbitrarily well in the Hausdorff topology by polynomial Julia sets. Furthermore, the proof is constructive.

Let $M$ be the class of all transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C} \bigcup\{\infty\}$ with at least two poles or one pole that is not an omitted value, and $M_o =\{f \in M:f{has at least one omitted value}\}$.…

In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these…

This paper studies the thermodynamic formalism in the context of complex dynamics. We establish the thermodynamics formalism for the class of hyperbolic transcendental meromorphic functions of B-class, where the poles have bounded…

The holomorphy conjecture for suspensions of plane curve singularities and the holomorphy and monodromy conjectures for L\^e-Yomdin singularities of surfaces are proved. The first part of this paper provides formul{\ae} for the motivic and…

The classical Julia-Wolff-Caratheodory Theorem is one of the main tools to study the boundary behavior of holomorphic self-maps of the unit disc of $\C$. In this paper we prove a Julia-Wolff-Caratheodory's type theorem in the case of the…

Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$, and $\varphi(\not\equiv 0,\infty)$ be a small function of $f$. In this paper, We give a quantitative estimation of the characteristic function $T(r,…

We show that the escaping sets and the Julia sets of bounded type transcendental entire functions of order $\rho$ become 'smaller' as $\rho\to\infty$. More precisely, their Hausdorff measures are infinite with respect to the gauge function…