Publication:
Algebraic aspects of integrability and reversibility in maps

dc.contributor.advisor Roberts, John en_US
dc.contributor.author Jogia, Danesh Michael en_US
dc.date.accessioned 2022-03-21T16:40:52Z
dc.date.available 2022-03-21T16:40:52Z
dc.date.issued 2008 en_US
dc.description.abstract We study the cause of the signature over finite fields of integrability in two dimensional discrete dynamical systems by using theory from algebraic geometry. In particular the theory of elliptic curves is used to prove the major result of the thesis: that all birational maps that preserve an elliptic curve necessarily act on that elliptic curve as addition under the associated group law. Our result generalises special cases previously given in the literature. We apply this theorem to the specific cases when the ground fields are finite fields of prime order and the function field $mathbb{C}(t)$. In the former case, this yields an explanation of the aforementioned signature over finite fields of integrability. In the latter case we arrive at an analogue of the Arnol'd-Liouville theorem. Other results that are related to this approach to integrability are also proven and their consequences considered in examples. Of particular importance are two separate items: (i) we define a generalization of integrability called mixing and examine its relation to integrability; and (ii) we use the concept of rotation number to study differences and similarities between birational integrable maps that preserve the same foliation. The final chapter is given over to considering the existence of the signature of reversibility in higher (three and four) dimensional maps. A conjecture regarding the distribution of periodic orbits generated by such maps when considered over finite fields is given along with numerical evidence to support the conjecture. en_US
dc.identifier.uri http://hdl.handle.net/1959.4/40947
dc.language English
dc.language.iso EN en_US
dc.publisher UNSW, Sydney en_US
dc.rights CC BY-NC-ND 3.0 en_US
dc.rights.uri https://creativecommons.org/licenses/by-nc-nd/3.0/au/ en_US
dc.subject.other maps over finite fields en_US
dc.subject.other integrability en_US
dc.subject.other reversibility en_US
dc.subject.other algebraic dynamics en_US
dc.subject.other Geometry, Algebraic en_US
dc.subject.other Finite fields (Algebra) en_US
dc.subject.other Maps en_US
dc.title Algebraic aspects of integrability and reversibility in maps en_US
dc.type Thesis en_US
dcterms.accessRights open access
dcterms.rightsHolder Jogia, Danesh Michael
dspace.entity.type Publication en_US
unsw.accessRights.uri https://purl.org/coar/access_right/c_abf2
unsw.identifier.doi https://doi.org/10.26190/unsworks/17873
unsw.relation.faculty Science
unsw.relation.originalPublicationAffiliation Jogia, Danesh Michael, Mathematics & Statistics, Faculty of Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Roberts, John, Mathematics & Statistics, Faculty of Science, UNSW en_US
unsw.relation.school School of Mathematics & Statistics *
unsw.thesis.degreetype PhD Doctorate en_US
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