This volume provides readers with an overview of the most recent developments in the mathematical fields related to fractals. It includes both original research contributions, as well as surveys from many of the leading experts on modern fractal geometry theory and applications. The contributions contained in the book stem from the conference Fractals and Related Fields IV", that was held in 2022 on the Island of Porquerolles, France. Various aspects of fractal geometry in connection with harmonic analysis, geometric measure theory, ergodic theory and dynamical systems, probability theory, number theory, functional analysis, additive combinatorics, embedding theory, and signal and image processing are addressed within its pages.

We hope that the book will be interesting for pure and applied mathematicians in these areas, as well as for other researchers curious to discover more about fractals.

Provides an overview of recent developments in mathematical fields related to fractals Includes original research contributions as well as surveys written by experts in their respective fields Readers will find interesting and motivating results as well as new avenues for further research

Inhalt

Amir Algom and Meng Wu: Large slices through self affine carpets.- Demi Allen and Edouard Daviaud: A survey of recent extensions and generalisations of the Mass Transference Principle.- Peter Arzt and Uta Freiberg: Spectral Exponents of Gap Diffusions on Random Homogeneous Cantor-Sets.- Athanasios Batakis, Nga Nguyen and Michel Zinsmeister: A new Model of City Growth and its Application to a middle sized French City.- Wejdene Ben Nasr, Véronique Billat, Stéphane Jaffard,, Florent Palacin and Guillaume Saës: The weak scaling multifractal spectrum: Mathematical setting and applications to marathon runners physiological data.- Argyrios Christodoulou and Natalia Jurga: Self-projective sets.- Hervé Queffélec and Martine Queffélec: Old and new results on the Furstenberg sets.- Michael Hochman: On the analytic self-maps of Cantor sets in the line.- Xiong Jin: A Chung-Fuchs type theorem for skew product dynamical systems.- Sabrina Kombrink and Lucas Schmidt: Eigenvalue counting functions and parallel volumes for examples of fractal sprays generated by the Koch snowflake.- Eugen Mihailescu: Exact dimensional measures in deterministic and random systems.- Tuomas Sahlsten: Fourier transforms and iterated function systems.- Nicolas de Saxcé: A presentation of the discretized sum-product in division algebras.- Xavier Tolsa: Carleson's 𝜀2 conjecture in higher dimensions and Faber-Krahn inequalities.