Cut and project schemes, combinatorics and averaging properties for Toeplitz subshifts

点击翻译按钮获取中文摘要

The project concerns various properties of subshifts associated to Toeplitz words. Here a "word" is a map from the integers into a finite set, that is, an infinite concatenation of symbols. In Toeplitz words, this concatenation is constructed in such a way that the word is not periodic, but still exhibits a certain degree of order. This makes them important examples in the theory of aperiodic order and interesting one-dimensional models for quasicrystals.In the context of quasicrystals, the spectrum of the associated Schrödinger operator plays an important role, since it encodes the energy levels that are available to electrons. If the so-called leading sequence condition (LSC) is satisfied, then the spectrum is a Cantor set of Lebesgue measure zero (actually it even implies uniformity of locally constant cocycles). However, the LSC has so far only been shown for subshifts of so-called simple Toeplitz words. One aim of the project is to extend this proof to a wider class of Toeplitz words. In addition it is planned to find sufficient or necessary conditions for general Toeplitz words to satisfy the LSC.Parts of the LSC are related to combinatorial properties of the involved words. Here an important step is to study cut and project schemes which can be associated to the Toeplitz word. They provide a way to generate the non-periodic, one-dimensional words from periodic, higher-dimensional lattices. Understanding how combinatorial properties of the word are related to properties of the projection set is therefore a first goal of the project; analysing the combinatorial properties themselves is a second one.The remaining part of the LSC deals with averages of matrix-valued functions along Toeplitz words, but the investigation of real-valued functions is planned as well. This topic is particularly interesting, since a recent preprint reduced the much studied Sarnak conjecture to averages over Toeplitz words with "high order" (entropy zero). The conjecture concerns averages of the so-called Möbius function, which play an important role in number theory.