Fractals are beautiful and complex geometric objects. Their study, pioneered by Benoît Mandelbrot, is of interest in mathematics, physics and computer science. Their inherent structure, based on their self-similarity, makes the study of their geometry amenable to dynamical approaches. In this book, a theory along these lines is developed by Hillel Furstenberg, one of the foremost experts in ergodic theory, ... Read moreleading to deep results connecting fractal geometry, multiple recurrence, and Ramsey theory. In particular, the notions of fractal dimension and self-similarity are interpreted in terms of ergodic averages and periodicity of classical dynamics; moreover, the methods have deep implications in combinatorics. The exposition is well-structured and clearly written, suitable for graduate students as well as for young researchers with basic familiarity in analysis and probability theory." - Endre Szemerédi, Rényi Institute of Mathematics, Budapest Show Less