Backward orbits in the unit ball

2018/07/31 by Arosio, Leandro, Guerini, Lorenzo
#32H50 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.11767

Abstract

We show that, if f\colon \mathbbB^q→ \mathbbB^q is a holomorphic self-map of the unit ball in ℂ^q and ζ∈ ∂ \mathbbB^q is a boundary repelling fixed point with dilation λ>1, then there exists a backward orbit converging to ζ with step log λ. Morever, any two backward orbits converging to the same boundary repelling fixed point stay at finite distance. As a consequence there exists a unique canonical pre-model (\mathbbB^k,ℓ, τ) associated with ζ where 1≤ k≤ q, τ is a hyperbolic automorphism of \mathbbB^k, and whose image ℓ(\mathbbB^k) is precisely the set of starting points of backward orbits with bounded step converging to ζ. This answers questions in [8] and [3,4].

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