Higher-dimensional Shifts and Ergodic Convergence

University Of Memphis, Memphis TN

Investigators

Abstract

Proposal Number: DMS-0200703 PI: Anthony Quas ABSTRACT The work will focus on the areas of higher-dimensional shifts of finite type and almost everywhere convergence in ergodic theory. Higher-dimensional shifts of finite type are symbolic dynamical systems whose points are arrays of symbols from a finite alphabet satisfying certain adjacency rules. While there is a well-developed theory for one-dimensional shifts of finite type, the general situation for higher-dimensional shifts has been very unclear. Recently a number of organizing concepts have started to emerge and it is hoped that these will lead to a satisfactory theory for at least some reasonable collection of higher-dimensional systems. Birkhoff's theorem is concerned with the effects of averaging a sequence of measurements of a system taken at equal time intervals. The result then states that in the case of a measure-preserving transformation, for almost every initial condition, the time averages are convergent. Further, the limit is identified if the transformation is ergodic. The work will focus on variations to this scenario where the measurements are taken at varying intervals. The higher-dimensional shifts of finite type that will be studied are of relevance in designing efficient data storage schemes. Due to physical restrictions on storage media, in many cases certain patterns of binary digits cannot be reliably written and read back from digital storage media. In these cases, it is necessary to encode the data so as to ensure that the encoded data does not contain any of the forbidden patterns of binary digits. This applies to magnetic tapes, but also to conventional hard disks, where data is stored in a number of tracks, all separated from one another. It is likely that in order to continue the growth in the amount of data that can be stored on a single device, it will be desirable or maybe even physically necessary to consider two-dimensional storage schemes. Efficient methods for doing this will depend on a good understanding of the properties of higher-dimensional shifts of finite type. The almost everywhere convergence results lie at the intersection of a number of diverse areas of mathematics: probability, dynamical systems, ergodic theory, number theory and harmonic analysis. It is hoped that progress made in this area will have knock-on effects in the other disciplines.