eTheses Repository

# Abstract topological dynamics

Let $$\char{cmmi10}{0x54}$$ : $$\char{cmmi10}{0x58}$$ → $$\char{cmmi10}{0x58}$$ be a function from a countably infinite set $$\char{cmmi10}{0x58}$$ to itself. We consider the following problem: can we put a structure on $$\char{cmmi10}{0x58}$$ with respect to which $$\char{cmmi10}{0x54}$$ has some meaning? In this thesis, the following questions are addressed: when can we endow $$\char{cmmi10}{0x58}$$ with a topology such that $$\char{cmmi10}{0x58}$$ is homeomorphic to the rationals $$\char{msbm10}{0x51}$$ and with respect to which $$\char{cmmi10}{0x54}$$ is continuous? We characterize such functions on the rational world. The other question is: can we put an order on $$\char{cmmi10}{0x58}$$ with respect to which $$\char{cmmi10}{0x58}$$ is order-isomorphic to the rationals $$\char{msbm10}{0x51}$$, naturals $$\char{msbm10}{0x4e}$$ or integers $$\char{msbm10}{0x5a}$$ with their usual orders and with respect to which $$\char{cmmi10}{0x54}$$ is order-preserving (or order-reversing)? We give characterization of such bijections, injections and surjections on the rational world and of arbitrary maps on the naturals and integers in terms of the orbit structure of the map concerned.