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.

Uncovers new and interesting connections between number theory and dynamics - The book covers many results of Mahler's measure of the height of a polynomial. A book has never before been published covering this topic - The early parts of the book are based on lectures given to post- graduates (at several levels) at UEA during semester 2 of the year 1997/98

Inhalt

1. Lehmer, Mahler and Jensen.- 2. Dynamical Systems.- 3. Mahler's Measure in Many Variables.- 4. Higher-Dimensional Dynamical Systems.- 5. Elliptic Heights.- 6. The Elliptic Mahler Measure.- A. Algebra.- A.1 Algebraic Integers.- A.2 Integer Matrices.- A.3 Hilbert's Nullstellensatz.- B. Analysis.- B.1 Stone-Weierstrass Theorem.- B.2 The Gelfand Transform.- C. Division Polynomials.- E.1 Lehmer Primes.- E.2 Elliptic Primes.- F. Exercises and Questions.- F.1 Hints for the Exercises.- F.2 List of Questions.- G. List of Notation.