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Denjoy–Wolff Points on the Bidisk via Models

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Abstract

Let \(F=(\phi , \psi ):\mathbb {D}^2\rightarrow \mathbb {D}^2\) denote a holomorphic self-map of the bidisk without interior fixed points. It is well-known that, unlike the case with self-maps of the disk, the sequence of iterates

$$\begin{aligned} \{F^n:=F\circ F\circ \cdots \circ F\} \end{aligned}$$

needn’t converge. The cluster set of \(\{F^n\}\) was described in a classical 1954 paper of Hervé. Motivated by Hervé’s work and the Hilbert space perspective of Agler, McCarthy and Young on boundary regularity, we propose a new approach to boundary points of Denjoy–Wolff type for the coordinate maps \(\phi , \psi .\) We establish several equivalent descriptions of our Denjoy–Wolff points, some of which only involve checking specific directional derivatives and are particularly convenient for applications. Using these tools, we are able to refine Hervé’s theorem and show that, under the extra assumption of \(\phi \) and \(\psi \) possessing Denjoy–Wolff points with certain regularity properties, one can draw much stronger conclusions regarding the behavior of \(\{F^n\}.\)

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Acknowledgements

The second author would like to thank John McCarthy and Greg Knese for helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, Gainesville, FL, USA

    Michael T. Jury

  2. Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO, USA

    Georgios Tsikalas

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Correspondence to Georgios Tsikalas.

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Jury partially supported by National Science Foundation Grant DMS 2154494. Tsikalas partially supported by National Science Foundation Grant DMS 2054199 and by Onassis Foundation—Scholarship ID: F ZR 061-1/2022-2023.

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Jury, M.T., Tsikalas, G. Denjoy–Wolff Points on the Bidisk via Models. Integr. Equ. Oper. Theory 95, 30 (2023). https://doi.org/10.1007/s00020-023-02751-6

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