# Aperiodic Order: Dynamical Systems, Combinatorics, and Operators (04w5001)

Arriving in Banff, Alberta Saturday, May 29 and departing Thursday June 3, 2004

## Organizers

Michael Baake (Institut fuer Mathematik)

Michael Baake (University of Bielefeld)

David Damanik (California Institute of Technology)

Ian Putnam (University of Victoria)

Boris Solomyak (University of Washington)

## Objectives

While the meeting in Waterloo was of a rather general and broad nature to get the mathematical research in this area going, the first meeting in Oberwolfach was firmly rooted in discrete geometry and tiling theory. The second meeting concentrated more on topological aspects and recent developments in the packing approach to tilings.

It is our plan to counterbalance this with a meeting where the spectral theory is the central topic, with emphasis on its connections to ergodic theory, combinatorics, and operator theory. Let us explain this in more detail.

One of the main features of quasicystalline structures is that they are highly ordered, and hence quite different from random structures, yet they lack global translation invariance which sets them apart from crystalline structures. This type of intermediate behaviour can be studied from different viewpoints, for example, ergodic theory, combinatorics, and operator theory.

During the last 10 years, it has become obvious that methods from ergodic theory are very suitable to explore and prove properties of perfectly ordered structures such as tilings and point sets obtained from either the projection method or an inflation procedure. Questions ranging from the general concept of symmetry to the spectral nature of the structures and their associated operators in applications find a universal frame in an ergodic theory setting. We expect that the forthcoming activities in this direction will have an important impact on the classification and taxonomy of ordered structures between (fully) periodic and amorphous, and there is also a very natural link to combinatorial methods.

From a combinatorial perspective, it is natural to assign a complexity function to a given structure. This function assigns to a given radius $r$ the number of different local patterns of radius $r$ in this structure. Growth behaviour of the complexity function corresponds to order properties of the structure. For example, crystalline structures are characterized by having a bounded complexity function, random structures have fast growing complexity functions, and complexity functions of structures with intermediate order properties have intermediate growth behaviour. It is therefore natural to investigate this correspondence between order properties and growth behaviour in more detail, and many recent works have dealt with this issue. On the other hand, research on the combinatorics side is very active on its own account, and it is producing results that have applications to problems motivated by physics. One of the objectives of the workshop is therefore to bring together researchers from both the quasicrystal community and the combinatorics community to exchange ideas, open problems, and promising approaches.

Transport properties of a structure are studied by assigning a self-adjoint operator to it and investigating spectral properties of this operator. The spectral type, too, seems to reflect order properties of the given structure. For example, crystalline structures always lead to an absolutely continuous spectrum, random structures tend to have a pure point part in their spectrum, and for structures with intermediate order properties one expects, and can prove for some of them, a singular continuous component. This correspondence has been firmly established in one dimension and the extension to higher dimensions is under study. This problem motivates questions on the operator side that will be of independent interest and an exchange of ideas between the quasicrystal community and researchers working on operator theory, another objective of the workshop, will likely be very fruitful.

On the diffraction side of the coin, recent progress has resulted in a complete characterization of translation bounded measures with pure point diffraction. In general, the diffraction measure will be of mixed type, and the logical next step will consist in a more detailed analysis of the continuous part of it. The new aspect here, due to a combination of ergodic theory and the theory of almost periodic measures, is that certain key properties can be described directly in terms of the original measure or its autocorrelation rather than in terms of the Fourier properties, and this once again gives way to the application of combinatorial methods. We expect some important developments in this direction and want to use the meeting to discuss and advance them. In particular, through the link to the combinatorial and complexity issues, we hope to establish a closer connection between the spectral theories of measures and operators.

The diffraction side is closely related to dynamics. In fact, it was recently shown that, under quite general assumptions, pure point diffraction is equivalent to pure point spectrum in the sense of ergodic theory. On the other hand, as was also shown recently, a complete characterization of translation bounded measures with pure point spectrum is possible via strong almost periodicity of the corresponding autocorrelation. It remains a challenging problem to link these results up and to extend them to a characterization of those systems with singular continuous spectrum. In dynamical systems and ergodic theory, the study of several commuting transformations (corresponding to systems in higher dimensions) has begun in recent years. There is no unified theory at this point, but several classes of particular interest have emerged, some of which are related to quasicrystals in general, and to spectral properties in particular. Extending classical notions and results from symbolic dynamics to higher dimensions involves rather subtle properties which are geometric in nature, and possibly linked to phenomena in quasicrystals. We would hope that the mix of people for the proposed meeting would form a perfect discussion forum to substantiate these connections.

Also, topological methods have provided new invariants for quasicrystals, which show up in spectral and transport properties. Here, further progress is to be expected towards systems in more than one dimension. In this respect, quasicrystals provide interesting links to C*-algebras and K-theory. There are even connections with Elliott's classification program for amenable C*-algebras which deserves to be further exploited.

Even though we have selected spectral theory as the central theme of this proposal, we hope that it has become apparent how widely linked the field of aperiodic order is with the mathematical disciplines. We sincerely believe that this is a unique opportunity to bring people with rather different background together to solve new and fascinating problems. We believe that BIRS would be a perfect environment, both scientifically and from its general atmosphere.

It is our plan to counterbalance this with a meeting where the spectral theory is the central topic, with emphasis on its connections to ergodic theory, combinatorics, and operator theory. Let us explain this in more detail.

One of the main features of quasicystalline structures is that they are highly ordered, and hence quite different from random structures, yet they lack global translation invariance which sets them apart from crystalline structures. This type of intermediate behaviour can be studied from different viewpoints, for example, ergodic theory, combinatorics, and operator theory.

During the last 10 years, it has become obvious that methods from ergodic theory are very suitable to explore and prove properties of perfectly ordered structures such as tilings and point sets obtained from either the projection method or an inflation procedure. Questions ranging from the general concept of symmetry to the spectral nature of the structures and their associated operators in applications find a universal frame in an ergodic theory setting. We expect that the forthcoming activities in this direction will have an important impact on the classification and taxonomy of ordered structures between (fully) periodic and amorphous, and there is also a very natural link to combinatorial methods.

From a combinatorial perspective, it is natural to assign a complexity function to a given structure. This function assigns to a given radius $r$ the number of different local patterns of radius $r$ in this structure. Growth behaviour of the complexity function corresponds to order properties of the structure. For example, crystalline structures are characterized by having a bounded complexity function, random structures have fast growing complexity functions, and complexity functions of structures with intermediate order properties have intermediate growth behaviour. It is therefore natural to investigate this correspondence between order properties and growth behaviour in more detail, and many recent works have dealt with this issue. On the other hand, research on the combinatorics side is very active on its own account, and it is producing results that have applications to problems motivated by physics. One of the objectives of the workshop is therefore to bring together researchers from both the quasicrystal community and the combinatorics community to exchange ideas, open problems, and promising approaches.

Transport properties of a structure are studied by assigning a self-adjoint operator to it and investigating spectral properties of this operator. The spectral type, too, seems to reflect order properties of the given structure. For example, crystalline structures always lead to an absolutely continuous spectrum, random structures tend to have a pure point part in their spectrum, and for structures with intermediate order properties one expects, and can prove for some of them, a singular continuous component. This correspondence has been firmly established in one dimension and the extension to higher dimensions is under study. This problem motivates questions on the operator side that will be of independent interest and an exchange of ideas between the quasicrystal community and researchers working on operator theory, another objective of the workshop, will likely be very fruitful.

On the diffraction side of the coin, recent progress has resulted in a complete characterization of translation bounded measures with pure point diffraction. In general, the diffraction measure will be of mixed type, and the logical next step will consist in a more detailed analysis of the continuous part of it. The new aspect here, due to a combination of ergodic theory and the theory of almost periodic measures, is that certain key properties can be described directly in terms of the original measure or its autocorrelation rather than in terms of the Fourier properties, and this once again gives way to the application of combinatorial methods. We expect some important developments in this direction and want to use the meeting to discuss and advance them. In particular, through the link to the combinatorial and complexity issues, we hope to establish a closer connection between the spectral theories of measures and operators.

The diffraction side is closely related to dynamics. In fact, it was recently shown that, under quite general assumptions, pure point diffraction is equivalent to pure point spectrum in the sense of ergodic theory. On the other hand, as was also shown recently, a complete characterization of translation bounded measures with pure point spectrum is possible via strong almost periodicity of the corresponding autocorrelation. It remains a challenging problem to link these results up and to extend them to a characterization of those systems with singular continuous spectrum. In dynamical systems and ergodic theory, the study of several commuting transformations (corresponding to systems in higher dimensions) has begun in recent years. There is no unified theory at this point, but several classes of particular interest have emerged, some of which are related to quasicrystals in general, and to spectral properties in particular. Extending classical notions and results from symbolic dynamics to higher dimensions involves rather subtle properties which are geometric in nature, and possibly linked to phenomena in quasicrystals. We would hope that the mix of people for the proposed meeting would form a perfect discussion forum to substantiate these connections.

Also, topological methods have provided new invariants for quasicrystals, which show up in spectral and transport properties. Here, further progress is to be expected towards systems in more than one dimension. In this respect, quasicrystals provide interesting links to C*-algebras and K-theory. There are even connections with Elliott's classification program for amenable C*-algebras which deserves to be further exploited.

Even though we have selected spectral theory as the central theme of this proposal, we hope that it has become apparent how widely linked the field of aperiodic order is with the mathematical disciplines. We sincerely believe that this is a unique opportunity to bring people with rather different background together to solve new and fascinating problems. We believe that BIRS would be a perfect environment, both scientifically and from its general atmosphere.