Effective integrality results in arithmetic dynamics

Given a rational function f defined over a number field K, S. Ih conjectured the finiteness of f-preperiodic points which are S-integral relative to a given non-preperiodic point β. This conjecture remains open, but certain special cases have been proved. We formulate a generalisation of Ih's conjecture, considering a semigroup $\mathcal{G}$ generated by rational functions (along with an appropriate notion of preperiodic points) defined over K, instead of a single map, and prove some of the known cases in this context. We moreover make our results effective. Given an arbitrary, finitely generated rational semigroup $\mathcal{G}$, we prove our generalisation of Ih's conjecture under certain local conditions on the non-preperiodic point β, generalising a result of Petsche. As an application, we obtain bounds on the number of S-units in certain doubly-indexed dynamical sequences. In the case of a single, unicritical polynomial f_c(z)=z^d+c, with β set to be the critical point 0, for parameters c outside a small region, we give an explicit bound which depends only on the number of places of bad reduction for f_c. As part of the proof, we obtain novel lower bounds for the v-adically smallest preperiodic point of f_c for each place v of K. Finally, when $\mathcal{G}$ is a finitely generated semigroup of monomial maps, we prove the conjecture without any assumptions on β, and moreover give a bound which is uniform as β varies over number fields of bounded degree. This generalises results of Baker, Ih and Rumely, which were made uniform by Yap.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2023

Author dc:creator

• Young, Marley

Advisor dc:contributor.advisor

• Krieger, Holly

• arithmetic dynamics
• rational semigroups
• unicritical polynomials
• heights
• equidistribution
• non-archimedean potential theory

Licence

• https://www.repository.cam.ac.uk/bitstreams/4b15a73a-28a9-43a0-a95f-ab674809e9c5/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.97034

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/350596