THERMOGAMAS - Quasicrystals, a Meeting Place for Mathematical Physics, Ergodic Theory, and Topology (Jacek Miȩkisz) (3/3)

Talk given during the THERMOGAMAS Workshop.

Abstract: We will address a fundamental question how global order arises from local rules. This is a story of quasicrystals which are non-periodic structures but nevertheless are uniform in certain senses. We present a brief history of quasicrystals and a short introduction to classical lattice-gas models of interacting particles and show how ergodic theory and topology enter mathematical physics.

We will discuss uniquely ergodic measures of non-periodic tilings and one-dimensional non-periodic sequences which form symbolic dynamical systems. Symbols can be interpreted as particles and we are concerned with configurations which minimize interaction energy of particles, the so-called ground-state configurations. One of the main questions is whether ground-state configurations are stable with respect to small perturbations of interactions. Such stability may be interpreted as a sort of rigidity of non-periodic structures such as tilings, substitution systems, and other uniquely ergodic non-periodic systems. We introduce a property of bounded fluctuations of finite patterns of non-periodic configurations, the so-called Strict Boundary Property. We discuss how relevant this property is for stability of non-periodic structures. Examples of Robinson's tilings, Thue-Morse, Fibonacci, and general Sturmian systems are presented. Some open problems are formulated.,