{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/350596"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/350596","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Effective integrality results in arithmetic dynamics","abstract":"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.","abstract_html":"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.","abstract_has_math":true,"creators":["Young, Marley"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Krieger, Holly"],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-04-01","date_published":"2023-04-01","updated_at":"2026-07-24T01:33:18Z","subjects":["arithmetic dynamics","rational semigroups","unicritical polynomials","heights","equidistribution","non-archimedean potential theory"],"languages":["eng"],"rights":[],"rights_urls":["https://www.repository.cam.ac.uk/bitstreams/4b15a73a-28a9-43a0-a95f-ab674809e9c5/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.97034","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Krieger, Holly"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["Cambridge Australia Scholarships Cambridge Trust"]},{"key":"dc:creator","label":"Author","values":["Young, Marley"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2023-04-01"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/350596"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["arithmetic dynamics","rational semigroups","unicritical polynomials","heights","equidistribution","non-archimedean potential theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.repository.cam.ac.uk/bitstreams/4b15a73a-28a9-43a0-a95f-ab674809e9c5/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.17863/CAM.97034"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/209af78b-1c3c-4dc7-94aa-90eb00d1f1d2/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["21e039036dba65fab74d6493120eb2b1","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Effective integrality results in arithmetic dynamics"]}]}],"canonical_facts":{"dc:contributor.advisor":["Krieger, Holly"],"dc:contributor.sponsor":["Cambridge Australia Scholarships Cambridge Trust"],"dc:creator":["Young, Marley"],"dc:date.issued":["2023-04-01"],"dc:description.abstract":["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."],"dc:format.checksum.md5":["21e039036dba65fab74d6493120eb2b1","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["https://doi.org/10.17863/CAM.97034"],"dc:identifier.uri":["https://www.repository.cam.ac.uk/bitstreams/209af78b-1c3c-4dc7-94aa-90eb00d1f1d2/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/350596"],"dc:rights":["https://www.repository.cam.ac.uk/bitstreams/4b15a73a-28a9-43a0-a95f-ab674809e9c5/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["arithmetic dynamics","rational semigroups","unicritical polynomials","heights","equidistribution","non-archimedean potential theory"],"dc:title":["Effective integrality results in arithmetic dynamics"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:33:18Z"}