Related papers: No invariant line fields on Cantor Julia sets

In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their…

Let $f:\hat{\mathbb C}\to\hat{\mathbb C}$ be a hyperbolic rational map of degree $d\ge2$ on the Riemann sphere. We give several conditions which are equivalent to the condition for the Julia set $J_f$ to be a Cantor set. It has been known…

We prove that for meromorphic maps with logarithmic tracts (e.g. entire or meromorphic maps with a finite number of poles from class $\mathcal B$), the Julia set contains a compact invariant hyperbolic Cantor set of Hausdorff dimension…

It has been shown that Cantor bubble Julia sets can appear in the dynamics of polynomials and their singular perturbations. In this paper, we present a criterion that guarantees the existence of Cantor bubble Julia sets for certain rational…

This article discusses some topological properties of the dynamical plane ($z$-plane) of the holomorphic family of meromorphic maps $\lambda + \tan z^2$ for $ \lambda \in \mathbb C$. In the dynamical plane, we prove that there is no Herman…

Adopting the approach of [7] we study rational function carrying invariant line fields on the Julia set. In particular, we show that under certain weak conditions all possible measurable invariant line fields of a rational function on its…

We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable…

It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is…

The no invariant line fields conjecture is one of the main outstanding problems in traditional complex dynamics. In this paper we consider non-autonomous iteration where one works with compositions of sequences of polynomials with suitable…

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of…

We consider the structure of substantially dissipative complex H\'enon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points…

This article discusses some topological properties of the dynamical plane ($z$-plane) of the holomorphic family of meromorphic maps $\lambda \tan z^2$ for $ \lambda \in \mathbb C^*$. In the dynamical plane, I prove that there is no Herman…

Let $f$ be a rational map with an infinitely-connected fixed parabolic Fatou domain $U$. We prove that there exists a rational map $g$ with a completely invariant parabolic Fatou domain $V$, such that $(f,U)$ and $(g,V)$ are conformally…

In this note, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational as well as transcendental meromorphic functions…

Non-renormalizable Newton maps are rigid. More precisely, we prove that their Julia set carries no invariant line fields and that the topological conjugacy is equivalent to quasi-conformal conjugacy in this case.

We give explicit examples of pairs of Julia sets of hyperbolic rational maps which are homeomorphic but not quasisymmetrically homeomorphic.

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…

For a hyperbolic polynomial automorphism of C^2 with a disconnected Julia set, and under a mild dissipativity condition, we give a topological description of the components of the Julia set. Namely, there are finitely many "quasi-solenoids"…

Let f be an infinitely-renormalizable quadratic polynomial and J_\infty be the intersection of forward orbits of "small" Julia sets of simple renormalizations of f. We prove that J_\infty contains no hyperbolic sets.

We prove that every wandering exposed Julia component of a rational map is to a singleton, provided that each wandering Julia component containing critical points is non-recurrent. Moreover, we show that the Julia set contains only finitely…