Lectures on fractal geometry and dynamical systems /

Pesin and Climenhaga (both, Penn State) measure the enormous popularity of fractals by the millions of entries retrieved from a Google query, but more than 1,000 WorldCat book entries tell the story just as well. All these monographs presumably must distinguish themselves, whether by level of rigor, application domain, intended audience, connections to other subjects, treatment of computation, or special inclusion of pet topics. Many books likewise treat dynamics, the study of how functions behave in the long term when iterated. The strength in this work lies in the precise handling of the implications of formal dynamics for the geometry of fractals, particularly the computation of their fractional dimensions. Rigorous but elementary, this book might enliven topology and analysis (particularly measure theory) for undergraduates. The authors present mostly familiar examples--e.g., logistic maps, Smale horseshoes, and Lorenz equations, though a chapter on the FitzHugh-Nagumo model of impulse propagation along an axon seems fresher. Considering the nonexpert audience, the spare illustrations seem merely serviceable, and the silence concerning computational issues, both practical and theoretical, odd. Pesin's celebrated expertise notwithstanding, the total package does not stand out when measured against such a crowded field. Summing Up: Recommended. Upper-division undergraduates, graduate students, and professionals. D. V. Feldman University of New Hampshire