On Parabolic Dichotomy

Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 26)

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.

Keywords

Holomorphic dynamics Models Iteration 

Mathematics Subject Classification

32H50 39B12 26A18 

Notes

Acknowledgements

This work was supported by the SIR grant “NEWHOLITE—New methods in holomorphic iteration” n. RBSI14CFME.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento Di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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