Quasicrystals, aperiodicity and lattice systems

This thesis complements the paper Lattice gas models on self-similar aperiodic tilings in that it gives background information, motivation, new results, and proofs that had been omitted from. It can, however, be read independently from. The thesis is organised as follows. Chapter 1 discusses quasicrystals. Quasicrystals are a new form of matter; one that is ordered, yet not periodic. Their discovery has generated a lot of interest in aperiodic systems. Chapter 2 describes a class of tilings, in arbitrary dimension, that share an important property with the Penrose tilings: a certain kind of self-similarity. Chapter 3 discusses lattice systems on self-similar tilings. It introduces notation and summarizes the results of. It also discusses how lattice systems on the cubic lattice fit into the formalism. Chapter 4 is concerned with additive set functions. Such functions arise naturally when studying the existence of thermodynamic quantities, like the mean free energy. The mean free energy is the (spatial) mean of an additive set function. Chapter 5 discusses probability measures in lattice systems on the cubic lattice and relations between periodic probability measures and certain aperiodic probability measures. It shows that the mean entropy exists for 'almost-periodic' probability measures. Finally, Chapter 6 is devoted to 'unique Gibbs measures'