@article{GuetschowNesmeWerner2012,
  author    = {Guetschow, Johannes and Nesme, Vincent and Werner, Reinhard F.},
  title     = {Self-similarity of cellular automata on abelian groups},
  series = {Journal of cellular automata},
  volume    = {7},
  journal   = {Journal of cellular automata},
  number    = {2},
  publisher = {Old City Publishing Science},
  address   = {Philadelphia},
  issn      = {1557-5969},
  pages     = {83 -- 113},
  year      = {2012},
  abstract  = {It is well known that the spacetime diagrams of some cellular automata have a self-similar fractal structure: for instance Wolfram's rule 90 generates a Sierpinski triangle. Explaining the self-similarity of the spacetime diagrams of cellular automata is a well-explored topic, but virtually all of the results revolve around a special class of automata, whose typical features include irreversibility, an alphabet with a ring structure, a global evolution that is a ring homomorphism, and a property known as (weakly) p-Fermat. The class of automata that we study in this article has none of these properties. Their cell structure is weaker, as it does not come with a multiplication, and they are far from being p-Fermat, even weakly. However, they do produce self-similar spacetime diagrams, and we explain why and how.},
  language  = {en}
}
@phdthesis{Braun2023,
  author    = {Braun, Tobias},
  title     = {Recurrences in past climates},
  doi       = {10.25932/publishup-58690},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-586900},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xxviii, 251},
  year      = {2023},
  abstract  = {Our ability to predict the state of a system relies on its tendency to recur to states it has visited before. Recurrence also pervades common intuitions about the systems we are most familiar with: daily routines, social rituals and the return of the seasons are just a few relatable examples. To this end, recurrence plots (RP) provide a systematic framework to quantify the recurrence of states. Despite their conceptual simplicity, they are a versatile tool in the study of observational data. The global climate is a complex system for which an understanding based on observational data is not only of academical relevance, but vital for the predurance of human societies within the planetary boundaries. Contextualizing current global climate change, however, requires observational data far beyond the instrumental period. The palaeoclimate record offers a valuable archive of proxy data but demands methodological approaches that adequately address its complexities. In this regard, the following dissertation aims at devising novel and further developing existing methods in the framework of recurrence analysis (RA). The proposed research questions focus on using RA to capture scale-dependent properties in nonlinear time series and tailoring recurrence quantification analysis (RQA) to characterize seasonal variability in palaeoclimate records ('Palaeoseasonality'). In the first part of this thesis, we focus on the methodological development of novel approaches in RA. The predictability of nonlinear (palaeo)climate time series is limited by abrupt transitions between regimes that exhibit entirely different dynamical complexity (e.g. crossing of 'tipping points'). These possibly depend on characteristic time scales. RPs are well-established for detecting transitions and capture scale-dependencies, yet few approaches have combined both aspects. We apply existing concepts from the study of self-similar textures to RPs to detect abrupt transitions, considering the most relevant time scales. This combination of methods further results in the definition of a novel recurrence based nonlinear dependence measure. Quantifying lagged interactions between multiple variables is a common problem, especially in the characterization of high-dimensional complex systems. The proposed 'recurrence flow' measure of nonlinear dependence offers an elegant way to characterize such couplings. For spatially extended complex systems, the coupled dynamics of local variables result in the emergence of spatial patterns. These patterns tend to recur in time. Based on this observation, we propose a novel method that entails dynamically distinct regimes of atmospheric circulation based on their recurrent spatial patterns. Bridging the two parts of this dissertation, we next turn to methodological advances of RA for the study of Palaeoseasonality. Observational series of palaeoclimate 'proxy' records involve inherent limitations, such as irregular temporal sampling. We reveal biases in the RQA of time series with a non-stationary sampling rate and propose a correction scheme. In the second part of this thesis, we proceed with applications in Palaeoseasonality. A review of common and promising time series analysis methods shows that numerous valuable tools exist, but their sound application requires adaptions to archive-specific limitations and consolidating transdisciplinary knowledge. Next, we study stalagmite proxy records from the Central Pacific as sensitive recorders of mid-Holocene El Ni{\~n}o-Southern Oscillation (ENSO) dynamics. The records' remarkably high temporal resolution allows to draw links between ENSO and seasonal dynamics, quantified by RA. The final study presented here examines how seasonal predictability could play a role for the stability of agricultural societies. The Classic Maya underwent a period of sociopolitical disintegration that has been linked to drought events. Based on seasonally resolved stable isotope records from Yok Balum cave in Belize, we propose a measure of seasonal predictability. It unveils the potential role declining seasonal predictability could have played in destabilizing agricultural and sociopolitical systems of Classic Maya populations. The methodological approaches and applications presented in this work reveal multiple exciting future research avenues, both for RA and the study of Palaeoseasonality.},
  language  = {en}
}