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Fractal behaviour, i.e. scale invariance in spatio-temporal dynamics, has been found to describe and model many systems in nature, in particular fluid mechanics and geophysical related geometrical objects, like the convective boundary layer of cumulus cloud fields, topographic landscapes, solar granulation patterns, and observational astrophysical time series, like light curves of pulsating stars. The main interest in the study of fractal properties in such physical phenomena lies in the close relationships they have with chaotic and turbulent dynamic. In this work we introduce some statistical tools for fractal analysis of light curves: Rescaled Range Analysis (R/S), Multifractal Spectra Analysis, and Coarse Graining Spectral Analysis (CGSA), an FFT based algorithm, which can discriminate in a time series the stochastic fractal power spectra from the harmonic one. An interesting application of fractal analysis in asteroseismology concerns the joint use of all these tools in order to develop classification criteria and algorithms for ¿ Scuti pulsating stars. In fact from the fractal and multifractal fingerprints in the background noise of the light curves we could infer properties of different mechanisms of stellar dynamic, among them rotation, modes excitation and magnetic activity. © 2018 The Author(s). ; SdF, JPG and RG acknowledge funding support from Spanish public funds for research under projects ESP2015-65712-C5-3-R. JCS acknowledges funding support from Spanish public funds for research under projects ESP2017-87676-2-2 and ESP2015-65712-C5-5-R, and from project RYC-2012-09913 under the 'Ramon y Cajal' program of the Spanish MINECO. SdF, JPG and RG acknowledge support from the 'Junta de Andalucia' regional government under project 2012-P12-TIC-2469. ; Peer Reviewed

Open Access. Attribution 4.0 International (CC BY 4.0) ; This paper aims at being a provocative guide to the future of asteroseismology from the perspective of the analysis of time series, where the fundamentals of harmonic analysis are subjected to stress tests. In this context, we give an annotated summary of our research over the last decades on harmonic analysis of A-F stars. We discuss and explore the consequences of our findings, which may extend to any kind of pulsators. As well, we analyse the impact of this reconsideration on future asteroseismic studies, which would entail a paradigm shift. This includes a discussion on the presence of fractal behavior in delta Sct stars, and how this can be used to develop a stopping criterion of the pre-whitening process, as an alternative to SNR (or significance) criterion. Drilling a scientific paradigm has its natural resilience, hence the path described here is being arduous, although fruitful at the same time. ; SF, JP-G, JR, ML-M, and RG acknowledges financial support from the State Agency for Research of the Spanish MCIU through the Center of Excellence Severo Ochoa award for the Instituto de Astrofisica de Andalucia (SEV-2017-0709) and Spanish public funds for research under projects ESP2015-65712C5-5-R. JS and AG acknowledge funding support from Spanish public funds for research under projects ESP2017-87676-C5-2R. JS also acknowledges funding support from project RYC2012-09913 under the Ramon y Cajal program of the Spanish MINECO. AG acknowledges support from Universidad de Granada under project E-FQM-041-UGR18 from the Programa Operativo FEDER 2014-2020 programme by Junta de Andalucia regional Government. ; Peer reviewed