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Rel leaves of the Arnoux–Yoccoz surfaces

A Correction to this article was published on 11 August 2021

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We analyze the rel leaves of the Arnoux–Yoccoz translation surfaces. We show that for any genus \(\mathbf {g}\geqslant 3\), the leaf is dense in the connected component of the stratum \({\mathcal {H}}(\mathbf {g}-1 ,\mathbf {g}-1)\) to which it belongs, and the one-sided imaginary-rel trajectory of the surface is divergent. For one surface on this trajectory, namely the Arnoux–Yoccoz surface itself, the horizontal foliation is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but for all other surfaces, the horizontal foliation is completely periodic. The appendix proves a field theoretic result needed for denseness of the leaf: for any \(n \geqslant 3\), the field extension of \({\mathbb {Q}}\) obtained by adjoining a root of \(X^n-X^{n-1}-\cdots -X-1\) has no totally real subfields other than \({\mathbb {Q}}\).

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This work was stimulated by insightful comments of Michael Boshernitzan, who conjectured Corollary 1.6. We thank Alex Wright for directing our attention to the case \(\mathbf {g}=2\) and for his proofs of Theorems 1.9 and 5.3. Theorem 1.4, which is a crucial step in our proof of Theorem 1.8, was proved in response to our queries by Lior Bary-Soroker, Mark Shusterman, and Umberto Zannier. We thank them for agreeing to include their results in Appendix A of this paper. We thank Ivan Dynnikov, Pascal Hubert and Sasha Skripchenko for pointing out the connections to their prior work and other insightful remarks. We are also grateful to David Aulicino, Josh Bowman, Duc-Manh Nguyen and John Smillie for useful discussions. We also are grateful to the anonymous referee for useful comments which helped to improve the paper. This collaboration was supported by BSF Grant 2010428. The first author’s work is supported by NSF Grant DMS-1500965 as well as a PSC-CUNY Award (funded by The Professional Staff Congress and The City University of New York). The second author’s work was supported by ERC starter Grant DLGAPS 279893. Appendix acknowledgements We are grateful to Barak Weiss for telling us about the problem of finding the maximal totally real subfields of number fields arising in dynamics. Special thanks go to Patrick Hooper whose computer verification of Corollary A.8 for all \(n \leqslant 1000\) greatly stimulated our work. We would also like to thank Moshe Jarden for his comments on drafts of this work. The first and second appendix authors were partially supported by the Israel Science Foundation Grant No. 952/14. The third appendix author was partially supported by the ERC-Advanced Grant “Diophantine problems” (Grant Agreement No. 267273).

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Correspondence to W. Patrick Hooper.

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With an Appendix by Lior Bary-Soroker (Tel Aviv University,, Mark Shusterman (Tel Aviv University,, and Umberto Zannier (Scuola Normale Superiore,

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Hooper, W.P., Weiss, B. Rel leaves of the Arnoux–Yoccoz surfaces. Sel. Math. New Ser. 24, 875–934 (2018).

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Mathematics Subject Classification

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