Related papers: Rational maps whose Julia sets are Cantor circles

Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then…

Jacobian conjectures (that nonsingular implies a global inverse) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The birational…

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.

We prove that if $\xi$ is a quasisymmetric homeomorphism between Sierpi\'nski carpets that are the Julia sets of postcritically-finite rational maps, then $\xi$ is the restriction of a M\"obius transformation to the Julia set. This implies…

It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has…

Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in…

We give a topological description of the space of quadratic rational maps with superattractive two-cycles: its "non-escape locus" M2 (the analog of the Mandelbrot set M) is locally connected, it is the continuous image of M under a…

It is well-known that the Julia set J(f) of a rational map is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this article we prove that an analogous result is…

We introduce and study a non uniform hyperbolicity condition for complex rational maps, that does not involve a growth condition. We call this condition Backward Contraction. We show this condition is weaker than the Collet-Eckmann…

The dynamical classification of rational maps is a central concern of holomorphic dynamics. Much progress has been made, especially on the classification of polynomials and some approachable one-parameter families of rational maps; the goal…

In this paper we study varieties covered by rational or elliptic curves. First, we show that images of Calabi-Yau or irreducible symplectic varieties under rational maps are almost always rationally connected. Second, we investigate…

We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.

We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical…

A normal subgroup of the (extended) mapping class group of a surface is said to be geometric if its automorphism group is the mapping class group. We prove that in the case of the Cantor tree surface, every normal subgroup is geometric. We…

For a continuous map on a topological graph containing a unique loop S, it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and…

We show that a topological Cantor set in the line has at most countably many real-analytic, onto self-maps.

The moduli space ${\rm M}_{d}$, of complex rational maps of degree $d \geq 2$, is a connected complex orbifold which carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps…

In this paper we prove that parabolic Julia sets of rational functions are locally computable in polynomial time.

We show that Misiurewicz maps for which the Julia set is not the whole sphere are Lebesgue density points of hyperbolic maps.

There are two natural definitions of the Julia set for complex H\'enon maps: the sets $J$ and $J^\star$. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts…