Showing papers on "External ray published in 2020"

A landing theorem for entire functions with bounded post-singular sets

Abstract: 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 f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

Periodic Points and Smooth Rays

Abstract: Let $P: {\mathbb C} \to {\mathbb C}$ be a polynomial map with disconnected filled Julia set $K_P$ and let $z_0$ be a repelling or parabolic periodic point of $P$. We show that if the connected component of $K_P$ containing $z_0$ is non-degenerate, then $z_0$ is the landing point of at least one {\it smooth} external ray. The statement is optimal in the sense that all but one ray landing at $z_0$ may be broken.

Rays to renormalizations.

Abstract: Let P be a non-linear polynomial, K_P the filled Julia set of P, f a renormalization of P and K_f the filled Julia set of f. We show, loosely speaking, that there is a finite-to-one function \lambda from the set of P-external rays having limit points in K_f onto the set of f-external rays to K_f such that R and \lambda(R) share the same limit set. In particular, if a point of the Julia set J_f=\partial K_f of a renormalization is accessible from C\setminus K_f then it is accessible through an external ray of P (the inverse is obvious). Another interesting corollary is that: a component of K_P\setminus K_f can meet K_f only at a single (pre-)periodic point. We study also a correspondence induced by \lambda on arguments of rays. These results are generalizations to all polynomials (covering notably the case of connected Julia set K_P) of some results of Levin-Przytycki, Blokh-Childers-Levin-Oversteegen-Schleicher and Petersen-Zakeri where the case is considered when K_P is disconnected and K_f is a periodic component of K_P.