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

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Copyright: Siu, Timothy
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.
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Siu, Timothy
Roberts, John Anthony Gerard
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PhD Doctorate
UNSW Faculty
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