2023/09/15 by Fornaess, John Erik, Hu, Mi · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2309.08334
In this paper, we investigate the precise behavior of orbits inside attracting basins of rational functions on \mathbb P1 and entire functions f in ℂ. Let R(z) be a rational function on \mathbb P1, \mathcal A(p) be the basin of attraction of an attracting fixed point p of R, and Ωi (i=1, 2, ⋯) be the connected components of A(p), and Ω1 contains p. Let p0∈Ω1 be close to p. If at least one Ωi is not simply connected, then there exists a constant C so that for any z0∈ Ωi, there is a point q∈ ∪k R-k(p0), k≥0 so that the Kobayashi distance dΩi(z0, q)≤ C. If all Ωi are simply connected, then the result is the same as for polynomials and is treated in an earlier paper. For entire functions f, we generally can not have similar results as for rational functions. However, if f has finitely many critical points, then similar results hold.