Linearisability and Integrability of Discrete Dynamical Systems from Cluster and LP Algebras

From the bipartite belt of a cluster algebra one may obtain generalisations of frieze patterns. It has been proven that linear relations exist within these frieze patterns if the associated quiver is, up to mutation equivalence, Dynkin or affine. The second chapter of this work is devoted to reproving this fact, for affine D and E types, using alternative methods to the known proof, allowing much more detail. We prove the existence of periodic quantities for affine ADE friezes with periods that mirror the widths of the tubes of their Auslander-Reiten quivers. Furthermore we interpret these friezes as discrete dynamical systems, given by a generalised cluster map. We prove the integrability of a reduction of this cluster map for each affine E type and for affine D with an even number of vertices. In our third chapter we consider recurrences that lie beyond cluster algebras, in LP algebras, named because mutation in these algebras has the Laurent property, like cluster algebras. We examine two particular examples of these recurrences and show that they can be linearised. We also show that they can be obtained by reductions of lattice equations. Finally we consider a 2-dimensional version ofthe Laurent property and give large sets of initial values such that these lattice equations possess this generalised Laurent property.