Contemporary Mathematics- Advances in Dimension Theory, Fractal Functions and Measures

Discussions from virtual symposiums delve into fractal geometry and dynamical systems, illuminating Mandelbrot’s legacy and contemporary research. The proceedings explore fractal dimensions, patterns of irregular shapes, and diverse topics like continued fractions, inhomogeneous attractors, and neural network uncertainty. This volume contains the proceedings of the AMS Special Session on Fractal Geometry and Dynamical Systems, held at the Spring Eastern Virtual Sectional Meeting on April 1-2, 2023, and the virtual Conference on Functional Analysis and Fractals organized by the Indian Institute of Information Technology Allahabad (IIIT-A), India, on February 16-18, 2024. Fifty years ago, Mandelbrot created a new type of geometry called fractal. One of the novelties of this new mathematics is a systematic qualitative and quantitative approach to the concepts of irregular shapes and roughness. Galileo said that the universe is written in mathematical language and its characters are triangles, circles, and ""other"" geometric figures. Mandelbrot masterly defined ""other"" geometric objects whose main property is the self-similarity and coined the term ""fractal"" for them. Such models fit better complex patterns such as the circulatory system, the coastline of a littoral country or a stock market chart. One way of quantifying the complexity of such structures is the computation of their fractal dimension. This book presents modern advances in the concept of dimension and its related notion of fractal measure. The text is oriented to give insight into the current research in the area, and it contains novel contributions of important scientists in the field. The book deals with very diverse topics such as the Hausdorff dimension of a set of continued fractions, dimension theory of inhomogeneous attractors, ergodic conjecture of falling balls systems, or Hausdorff measures to represent uncertainty in neural networks.