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Dynamics of generic endomorphisms of Oka–Stein manifolds

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Abstract

We study the dynamics of a generic endomorphism f of an Oka–Stein manifold X. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several descriptions of the Fatou set and the Julia set of f. In particular, we show that the Julia set is the derived set of the set of attracting periodic points of f and that it is also the closure of the set of repelling periodic points of f. Among other results, we prove that f is chaotic on the Julia set and that every periodic point of f is hyperbolic. We also give an explicit description of the “Conley decomposition” of X induced by f into chain-recurrence classes and basins of attractors. For \(X={\mathbb {C}}\), we prove that every Fatou component is a disc and that every point in the Fatou set is attracted to an attracting cycle or lies in a dynamically bounded wandering domain (whether such domains exist is an open question).

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Notes

  1. By an Oka–Stein manifold we simply mean a complex manifold that is both Oka and Stein.

  2. This lemma states that if p is a chain-recurrent point of \(f\in {{\text {End}}}\,X\), V is a neighbourhood of p in X, and W is a neighbourhood of f in \({{\text {End}}}\,X\), then there is an endomorphism in W with a periodic point in V.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133, Rome, Italy

    Leandro Arosio

  2. School of Mathematical Sciences, University of Adelaide, Adelaide, SA, 5005, Australia

    Finnur Lárusson

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  1. Leandro Arosio
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  2. Finnur Lárusson
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Correspondence to Finnur Lárusson.

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L. Arosio was supported by SIR grant “NEWHOLITE—New methods in holomorphic iteration”, no. RBSI14CFME, and partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. F. Lárusson was partially supported by Australian Research Council grant DP150103442. Part of this work was done when F. Lárusson visited Rome in February 2020. He thanks the University of Rome Tor Vergata for financial support and hospitality.

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Arosio, L., Lárusson, F. Dynamics of generic endomorphisms of Oka–Stein manifolds. Math. Z. 300, 2467–2484 (2022). https://doi.org/10.1007/s00209-021-02874-9

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