On mean-equicontinuity

• Applicant: Dr. Till Hauser
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 530703788

The aim of the project is to extend the theory of group actions. For this the important concept of mean-equicontinuity will be studied in the context of actions of discrete countable groups. The main goal is to extend the theory in two directions. In the first part of this project, the concept is to be generalised from actions to factormaps. For this purpose, the correct definition of this generalisation must first be developed. In this sub-project, various known results are to be lifted to this setting. According to Downarowicz, Glasner, Fuhrmann, Gröger and Lenz, mean-equicontinuous actions are precisely the topo-isomorphic extensions of equicontinuous systems. If one considers the latter as an equicontinuous extension of a point, the following questions arise: Is it possible to develop a notion of mean-equicontinuity for factor mappings such that they are exactly the compositions of a topo-isomorphism and an equicontinuous factormap? Furthermore, it will be asked how the notions of proximal and distal factormap are related to the notions of mean-equicontinuous, topo-isomorphic and equicontinuous factormap. An important part of the project is the search for further examples. Special attention will be paid to the class of Toeplitz systems. Mean-equicontinuity has so far only been studied in the context of amenable groups. However, recent research on Toeplitz systems in the non-amenable setting suggests developing the notion for arbitrary discrete countable groups. In the second part of the project, a joint project with Gabriel Fuhrmann and Maik Gröger, actions of non-amenable groups will be considered. The starting point will be the observation that, in contrast to the definition, various characterisations of mean-equicontinuity can also be formulated for non-amenable groups. An important goal of this project is to check whether these properties are still equivalent in the non-amenable setting. Furthermore, pseudometrics are to be explored that might generalise the Weyl-pseudometrics from the amenable setting. An important part of this project is also the search for examples.

• DFG Programme: WBP Fellowship
• International Connection: Chile
• Host: Professorin Dr. María Isabel Cortez