Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-02-06T11:28:51.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Degree gaps for multipliers and the dynamical André–Oort conjecture

Part of: Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  13 November 2020

Patrick Ingram
Patrick Ingram*
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ONM3J 1P3, Canada
*
e-mail:pingram@yorku.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We demonstrate how recent work of Favre and Gauthier, together with a modification of a result of the author, shows that a family of polynomials with infinitely many post-critically finite specializations cannot have any periodic cycles with multiplier of very low degree, except those that vanish, generalizing results of Baker and DeMarco, and Favre and Gauthier.

Keywords

Arithmetic dynamicscritical heightunlikely intersections

MSC classification

Primary: 37P45: Families and moduli spaces
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 867 - 876
Check for updates
Copyright
© Canadian Mathematical Society 2020

References

Baker, M. and DeMarco, L., Special curves and postcritically finite polynomials . Forum Math. Pi 1(2013), e3. http://dx.doi.org/10.1017/fmp.2013.2 CrossRefGoogle Scholar
DeMarco, L., Wang, X., and Ye, H., Bifurcation measures and quadratic rational maps . Proc. Lond. Math. Soc. 111(2015), 149–180. http://dx.doi.org/10.1112/plms/pdv024 CrossRefGoogle Scholar
Favre, C. and Gauthier, T., Classification of special curves in the space of cubic polynomials . Int. Math. Res. Not. IMRN (2018), no. 2, 362–411. http://dx.doi.org/10.1093/imrn/rnw245 Google Scholar
Favre, C. and Gauthier, T., The arithmetic of polynomial dynamical pairs. Preprint, 2020. https://arxiv.org/abs/2004.13801 Google Scholar
Ghioca, D., Hsia, L.-C., and Tucker, T. J., Preperiodic points for families of rational maps . Proc. Lond. Math. Soc. 110(2015), 395–427. http://dx.doi.org/10.1112/plms/pdu051 CrossRefGoogle Scholar
Ghioca, D., Krieger, H., and Nguyen, K. D., A case of the dynamical André–Oort conjecture . Int. Math. Res. Not. (2016), no. 3, 738–758. http://dx.doi.org/10.1093/imrn/rnv143 CrossRefGoogle Scholar
Ghioca, D., Krieger, H., Nguyen, K. D., and Ye, H., The dynamical André–Oort conjecture: unicritical polynomials . Duke Math. J. 166(2017), 1–25. http://dx.doi.org/10.1215/00127094-3673996 Google Scholar
Ghioca, D. and Ye, H., A dynamical variant of the André–Oort conjecture. Int. Math. Res. Not. (2018), 2447–2480. http://dx.doi.org/10.1093/imrn/rnw314 Google Scholar
Ingram, P, A finiteness result for post-critically finite polynomials. Int. Math. Res. Not. (2012) no. 3, 524–543. http://dx.doi.org/10.1093/imrn/rnr030CrossRefGoogle Scholar
Ingram, P., Variation of the canonical height for a family of polynomials . J. Reine Angew. Math. 685(2013), 73–97. http://dx.doi.org/10.1515/crelle-2012-0017 Google Scholar
Ingram, P., The critical height is a moduli height . Duke Math. J. 167(2018), 1311–1346. http://dx.doi.org/10.1215/00127094-2017-0053 Google Scholar
Ingram, P., Critical orbits of polynomials with a periodic point of specified multiplier . Math. Zeit. 291(2019), 1245–1262. http://dx.doi.org/10.1007/s00209-018-2118-x CrossRefGoogle Scholar
Silverman, J. H., Moduli spaces and arithmetic dynamics . CRM Monograph Series, 30, American Mathematical Society, Providence, RI, 2012. http://dx.doi.org/10.1090/crmn/030 Google Scholar