Bauer, MarkNguyen, Dang KhoaGunn, Keira2024-05-162024-05-162024-05-13Gunn, K. (2024). Some problems on the dynamics of positive characteristic tori (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.https://hdl.handle.net/1880/118760https://doi.org/10.11575/PRISM/46357The positive characteristic tori T_F are a set of counterparts to the real torus T=R/Z. In positive characteristic we define the ``integers'' as polynomials with coefficients from a finite field F (Z_F:=F[t]) and the ``reals'' as the field of Laurent series with coefficients in F (R_F:=F((t))) so that the positive characteristic torus over F is similarly defined: T_F:=R_F/Z_F. While T and T_F have some structural and operational similarities, they behave fundamentally differently, particularly with regards to dynamics. In this dissertation we seek analogues to known results in R and T. We find that both sets have similar ergodicity results but that orbits of affine maps and other areas of dynamics show significant differences. In particular, we construct an Artin-Mazur zeta function that looks significantly different to its counterpart in T, demonstrate that Furstenberg's orbital density theorem falls apart in positive characteristic, and establish that the intersection of orbits of affine maps rely on sets that depend on powers of the characteristic of F rather than arithmetic progressions. At first glance, the simplicity of working in T_F and its similarities to T suggest that we should be able to find many of the same simple results; however in reality the structure of T_F consists of infinitely defined sub-structures constructed by shifts of Frobenius maps into itself and these sub-structures present themselves frequently in a manner that does not occur in TenUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.Number TheoryToriDynamicsPositive CharacteristicMathematicsdoctoral thesis