Finite-to-one topomorphic extensions of equicontinuous dynamical systems

• Applicant: Professor Dr. Tobias Henrik Oertel-Jäger
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552775134

Understanding and quantifying the dynamical complexity of time-dependent systems is one of the major goals of Dynamical Systems Theory. Thereby, a central role is played by the notion of entropy, which allows to measure the ‘chaoticity’ inherent in a system. However, even when entropy is zero, one may find a remarkable variety of dynamical behaviours and many weaker forms of chaoticity. Hence, the study of zero-entropy systems is a subject of independent interest and has seen considerable activity over the past decades. A particular motivation for these endeavours comes from the theory of aperiodic order, where mathematical models of quasicrystals and other aperiodic structures with long-range order, such as the Penrose tiling, induce associated dynamical systems. Different degrees of long-range order then correspond to different levels of dynamical complexity. An alternative viewpoint is to study dynamical systems via extension structures. This approach has led to seminal results such as the Furstenberg Structure Theorem for distal flows, the Furstenberg-Zimmer Structure Theorem for measure preserving flows or the Veech Structure Theorem for point-distal flows. In the context of low-complexity systems, one may hope to characterise important system classes by basic extension structures. For instance, there has recently been considerable progress in the understanding of mean equicontinuous minimal systems. Glasner and Downarowicz showed that these are isomorphic to their maximal equicontinuous factor (MEF), an underlying compact group rotation which encodes the `regular' (non-chaotic) part of the dynamics. Subsequent results established a hierarchy of subclasses, characterised by different invertibility properties of the factor map to the MEF. The aim of the project is to build on these recent advances and extend the theory to a more general class of systems – referred to as finite-to-one topomorphic extensions. This provides a natural next step in the understanding of low-complexity dynamics from the viewpoint of extension structures. Moreover, this class comprises a broad scope of systems of both theoretical and practical interest, including paradigmatic examples in the field of aperiodic order, like the Thue-Morse or Rudin-Shapiro subshifts. The first objective of the project is to provide dynamical characterisations of finite-to-one topomorphic extensions and important subclasses, defined again in terms of structural properties of the MEF factor map. Further dynamical properties will then be related to the classification. In order to measure the complexity of finite-to-one topomorphic extensions, we intend to introduce a sequence of taylor-made topological invariants, generalising the recently introduced notion of amorphic complexity. Moreover, the construction and investigation of novel examples will play a prominent role and go hand in hand with the development of the general theory.

• DFG Programme: Research Grants