Arithmetic geometry and algebraic dynamical systems are flourishing areas of mathematics. Both subjects have highly technical aspects, yet both of� fer a rich supply of down-to-earth examples. Both have much to gain from each other in techniques and, more importantly, as a means for posing (and sometimes solving) outstanding problems. It is unlikely that new graduate students will have the time or the energy to master both. This book is in� tended as a starting point for either topic, but is in content no more than an invitation. We hope to show that a rich common vein of ideas permeates both areas, and hope that further exploration of this commonality will result. Central to both topics is a notion of complexity. In arithmetic geome� try 'height' measures arithmetical complexity of points on varieties, while in dynamical systems 'entropy' measures the orbit complexity of maps. The con� nections between these two notions in explicit examples lie at the heart of the book. The fundamental objects which appear in both settings are polynomi� als, so we are concerned principally with heights of polynomials. By working with polynomials rather than algebraic numbers we avoid local heights and p-adic valuations.

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The main theme of the book is the theory of heights as they appear in various guises. This includes a large body of results on Mahler's measure of the height of a polynomial of which topic there is no book available. The genesis of the measure in a paper by Lehmer is looked at, which is extremely well-timed due to the revival of interest following the work of Boyd and Deninger on special values of Mahler's measure. The authors'approach is very down to earth as they cover the rationals, assuming no prior knowledge of elliptic curves. The chapters include examples and particular computations. A large chunk of the book has been devoted to the elliptic Mahler's measure. Special calculation have been included and will be self-contained. One of the most important results about Mahler's measure is that it is the entropy associated to a dynamical system. The authors devote space to discussing this and to giving some convincing and original examples to explain this phenomenon.

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