Year: 2020
Type: article
Abstract: TheDouady-Hubbard landing theoremfor periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomialfwith 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 functionfwith bounded postsingular set. Iffhas finite order of growth, then it is known that the escaping setI(f) contains certain curves calledperiodic 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 functionfof infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets ofI(f), calleddreadlocks. 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.
Authors: Anna Miriam Benini, Lasse Rempe
Institutions: University of Parma, University of Liverpool
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,750
OpenAlex ID: W2772365706