Abstract
If f is a parabolic holomorphic self-map of the unit disc \(\mathbb {D}\subset \mathbb {C}\), then either for every point z one has \(\lim _{n\to \infty } k_{\mathbb {D}}(f^{n+1}(z),f^n(z))>0\), or the Poincaré distance of any two f-orbits converges to zero. It is an open question whether such a dichotomy holds in the unit ball \(\mathbb {B}^q\subset \mathbb {C}^q\). We show how this question is related to the theory of canonical Kobayashi hyperbolic semi-models, to commuting holomorphic self-maps of the ball and to a purely geometric problem about biholomorphisms of the ball.
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Acknowledgements
This work was supported by the SIR grant “NEWHOLITE—New methods in holomorphic iteration” n. RBSI14CFME.
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Arosio, L. (2017). On Parabolic Dichotomy. In: Bracci, F. (eds) Geometric Function Theory in Higher Dimension. Springer INdAM Series, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-73126-1_3
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