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In this article, we present a landing theorem for periodic dynamic rays for transcendental entire maps which have bounded post-singular sets, by using standard hyperbolic geometry results.

In this article, we prove that for several one-dimensional holomorphic families of holomorphic maps, in the parameter plane, there exists a local piece of a curve that lands at a given parabolic parameter, in the spirit of well-known…

We provide a short proof of a conic version of the colorful Carath\'eodory theorem for oriented matroids. Holmsen's extension of the colorful Carath\'eodory theorem to oriented matroids (Advances in Mathematics, 2016) already encompasses…

We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one,…

We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof…

We answer a question of Schleicher by showing that, for an exponential map with nonescaping singular value, every periodic ray lands. This is an analog of a theorem of Douady and Hubbard concerning polynomials. We also prove a partial…

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present…

For a hyperbolic subgroup H of a hyperbolic group G, we describe sufficient criteria to guarantee the following. 1) Geodesic rays in H starting at the identity land at a unique point of the boundary of G. 2)The inclusion of H into G does…

We consider the convergence of pointed multiply connected domains in the Caratheodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the…

B\'ar\'any's colorful generalization of Carath\'eodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized B\'ar\'any's theorem by replacing color classes with matroid…

The main objective of this paper is to show that balls under invariant metrics on hyperbolic planar domains are finitely-connected. As applications, we give new and transparent proofs of classical results on conformal mappings of planar…

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic…

We prove global existence for quasilinear wave equations outside of a wide class of obstacles. The obstacles may contain trapped hyperbolic rays as long as there is local exponential energy decay for the associated linear wave equation.…

We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively…

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial with bounded postcritical set, every periodic external ray lands at a…

We generalize a combinatorial formula of Douady from the main cardioid to other hyperbolic components $H$ of the Mandelbrot set, constructing an explicit piecewise linear map which sends the set of angles of external rays landing on $H$ to…

This is a survey on Valiron's Theorem about the convergence properties of orbits of analytic self-maps of the disk of hyperbolic type and related questions in one and several variables.

We prove two colorful Carath\'eodory theorems for strongly convex hulls, generalizing the colorful Carat\'eodory theorem for ordinary convexity by Imre B\'ar\'any, the non-colorful Carath\'eodory theorem for strongly convex hulls by the…

In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the center point theorem, and Tverberg's theorem on partitions of a point set.

Let $G$ be a relatively hyperbolic group that admits a decomposition into a finite graph of relatively hyperbolic groups structure with quasi-isometrically (qi) embedded condition. We prove that the set of conjugates of all the vertex and…