Anomalous transport: results, conjectures and applications to quasicrystals

We review the main rigorous results accumulated during the last 5 years concerning the theory of transport in aperiodic media. Motivated by the transport properties of quasicrystals, we introduce the mathematical background needed to describe transport in aperiodic media. We then give the definitions and the properties of spectral and diffusion exponents with an emphasis on ‘anomalous transport’. The last section is devoted to discussing the relevance of this theory for quasicrystals.