From Fractal Groups to Fractal Sets

Abstract

The idea of self-similarity is one of the most fundamental in the modern mathematics. The notion of “renormalization group”, which plays an essential role in quantum field theory, statistical physics and dynamical systems, is related to it. The notions of fractal and multi-fractal, playing an important role in singular geometry, measure theory and holomorphic dynamics, are also related. Self-similarity also appears in the theory of C*-algebras (for example in the representation theory of the Cuntz algebras) and in many other branches of mathematics. Starting from 1980 the idea of self-similarity entered algebra and began to exert great influence on asymptotic and geometric group theory.KeywordsInverse SemigroupFractal GroupIterate Function SystemHyperbolic GroupPenrose TilingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.