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Counting points of bounded height in monoid orbits

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Abstract

Given a set of endomorphisms on \(\mathbb {P}^N\), we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds.

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Acknowledgements

We thank Yuri Bilu, Andrew Bridy, Alexander Evetts, Joseph Silverman, and Umberto Zannier for discussions related to this paper. We also thank the authors of [16]; Lemma 3.2 in their paper inspired the proof of Theorem 5.1. Finally, we thank the anonymous referee for their insightful comments.

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  1. Texas State University, 601 University Dr., San Marcos, TX, 78666, USA

    Wade Hindes

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Correspondence to Wade Hindes.

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Hindes, W. Counting points of bounded height in monoid orbits. Math. Z. 301, 3395–3416 (2022). https://doi.org/10.1007/s00209-022-03021-8

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