Order, randomness and orbit distributions for dynamics of birational maps over finite fields

dc.contributor.advisor Roberts, John Anthony Gerard en_US Siu, Timothy en_US 2022-03-23T11:06:28Z 2022-03-23T11:06:28Z 2019 en_US
dc.description.abstract We consider birational maps in affine space of two or more dimensions over finite fields. We see that when we look at the mappings modulo p, we lose all topology and sense of "closeness" of points that is present over the real or complex numbers. However, algebraic properties such as the presence of a reversing symmetry (reversible maps) or preserving an invariant algebraic surface (integrable maps) also reduce algebraically to the finite field. For birational maps on the finite field, we have the possibility of periodic orbits or singular orbits. We investigate how these algebraic properties manifest themselves in the orbits and show how random (i.e. probabilistic) models tailored for these properties can be used to predict various statistics of the orbits such as the number of periodic orbits, the number of periodic points and the distribution of the lengths of the orbits. Furthermore, this can be used as a diagnostic for whether a given mapping possesses such properties. We see that these properties alone seem to be the constraining property for many of the statistics of the dynamical system, and not the details of the map itself. We provide in-depth analysis and numerical studies on a few representative examples and also show the efficacy of these ideas on a menagerie of maps from the literature. en_US
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 en_US
dc.subject.other Dynamical systems en_US
dc.subject.other Orbit distribution en_US
dc.subject.other Finite fields en_US
dc.subject.other Reversibility en_US
dc.title Order, randomness and orbit distributions for dynamics of birational maps over finite fields en_US
dc.type Thesis en_US
dcterms.accessRights open access
dcterms.rightsHolder Siu, Timothy
dspace.entity.type Publication en_US
unsw.relation.faculty Science
unsw.relation.originalPublicationAffiliation Siu, Timothy, Mathematics & Statistics, Faculty of Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Roberts, John Anthony Gerard, Mathematics & Statistics, Faculty of Science, UNSW en_US School of Mathematics & Statistics *
unsw.thesis.degreetype PhD Doctorate en_US
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