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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Backward orbits in the unit ball
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by Leandro Arosio and Lorenzo Guerini PDF
Proc. Amer. Math. Soc. 147 (2019), 3947-3954 Request permission


We show that if $f\colon \mathbb {B}^q\to \mathbb {B}^q$ is a holomorphic self-map of the unit ball in $\mathbb {C}^q$ and $\zeta \in \partial \mathbb {B}^q$ is a boundary repelling fixed point with dilation $\lambda >1$, then there exists a backward orbit converging to $\zeta$ with step $\log \lambda$. 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 premodel $(\mathbb {B}^k,\ell , \tau )$ associated with $\zeta$ where $1\leq k\leq q$, $\tau$ is a hyperbolic automorphism of $\mathbb {B}^k$, and whose image $\ell (\mathbb {B}^k)$ is precisely the set of starting points of backward orbits with bounded step converging to $\zeta$. This answers questions of Ostapyuk (2011) and the first author (2015, 2017).
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Additional Information
  • Leandro Arosio
  • Affiliation: Dipartimento Di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133 Roma, Italy
  • MR Author ID: 937673
  • Email:
  • Lorenzo Guerini
  • Affiliation: Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, 1090GE Amsterdam, the Netherlands
  • MR Author ID: 1277958
  • Email:
  • Received by editor(s): July 31, 2018
  • Received by editor(s) in revised form: January 11, 2019
  • Published electronically: May 1, 2019
  • Additional Notes: The first author was supported by SIR grant NEWHOLITE – “New methods in holomorphic iteration”, no. RBSI14CFME
  • Communicated by: Filippo Bracci
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 3947-3954
  • MSC (2010): Primary 32H50; Secondary 32A40, 37F99
  • DOI:
  • MathSciNet review: 3993787