Everest, Graham; Ward, Thomas A dynamical interpretation of the global canonical height on an elliptic curve. (English) Zbl 0927.11009 Exp. Math. 7, No. 4, 305-316 (1998). This paper can be regarded as an introduction to the recent text by the same authors [Heights of polynomials and entropy in algebraic dynamics. Springer, London (1999; Zbl 0919.11064)]. They begin with a leisurely and enlightening review of properties of the (logarithmic) Mahler measure \(m(F(u_1,\dots,u_d))\) and its meaning as the topological entropy of an algebraic dynamical system defined by the polynomial \(F(u_1,\dots,u_d)\). They then turn to the definition of the elliptic Mahler measure of G. R. Everest and B. ni Fhlathúin [Math. Proc. Camb. Philos. Soc. 120, 13-25 (1996; Zbl 0865.11068)] and conjecture that this arises in a similar but more complicated way as the entropy of a suitable algebraic dynamical system. They deduce some necessary properties of this conjectured dynamical system but its existence remains an open question. Table 1 on p. 315 gives a summary of analogous properties of the classical Mahler measure and the elliptic Mahler measure indicating clearly some of the interesting open questions concerning the latter. Reviewer: D.W.Boyd (Vancouver) Cited in 1 ReviewCited in 1 Document MSC: 11C08 Polynomials in number theory 11G50 Heights 22D40 Ergodic theory on groups 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 28D99 Measure-theoretic ergodic theory 37B40 Topological entropy Keywords:Mahler measure; entropy; algebraic dynamics; elliptic curve; Lehmer’s problem; heights; elliptic Mahler measure Citations:Zbl 0865.11068; Zbl 0919.11064 PDFBibTeX XMLCite \textit{G. Everest} and \textit{T. Ward}, Exp. Math. 7, No. 4, 305--316 (1998; Zbl 0927.11009) Full Text: DOI Euclid EuDML EMIS References: [1] Abramov L. M., Teor. Veroyatnost. i Primenen. 4 pp 249– (1959) [2] DOI: 10.1090/S0002-9947-1965-0175106-9 [3] DOI: 10.1090/S0002-9947-1971-0274707-X [4] Boyd D. W., J. Number Theory 13 (1) pp 116– (1981) · Zbl 0447.12003 [5] Boyd D. W., Canad. Math. Bull. 24 (4) pp 453– (1981) · Zbl 0474.12005 [6] Boyd D. W., Experiment. Math. 7 (1) pp 37– (1998) [7] Chothi V., J. Reine Angew. Math. 489 pp 99– (1997) [8] DOI: 10.1007/BF00533332 · Zbl 0261.28015 [9] Deninger C., J. Amer. Math. Soc. 10 (2) pp 259– (1997) · Zbl 0913.11027 [10] Einsiedler M., ”A generalisation of Mahler measure and its application in algebraic dynamical systems” (1997) · Zbl 0931.11043 [11] Einsiedler M., ”Fitting ideals for finitely presented algebraic dynamical systems” (1997) · Zbl 0972.22005 [12] Everest G. R., J. London Math. Soc. (1999) [13] Everest G. R., Math. Proc. Cambridge Philos. Soc. 120 (1) pp 13– (1996) · Zbl 0865.11068 [14] Kitchens B., Ergodic Theory Dynamical Systems 9 (4) pp 691– (1989) [15] Lawton W., Recent advances in topological dynamics pp 182– (1973) [16] Lawton W., J. Sci. Fac. Chiangmai Univ. 4 pp 15– (1977) [17] Lawton W. M., J. Number Theory 16 (3) pp 356– (1983) · Zbl 0516.12018 [18] DOI: 10.2307/1968172 · Zbl 0007.19904 [19] Lind D. A., Ergodic Theory Dynamical Systems 2 (1) pp 49– (1982) · Zbl 0507.58034 [20] Lind D. A., Ergodic Theory Dynamical Systems 8 (3) pp 411– (1988) [21] Lind D., Invent. Math. 101 (3) pp 593– (1990) · Zbl 0774.22002 [22] DOI: 10.1112/S0025579300001637 · Zbl 0099.25003 [23] Mahler K., J. London Math. Soc. 37 pp 341– (1962) · Zbl 0105.06301 [24] Rudolph D. J., Invent. Math. 120 (3) pp 455– (1995) · Zbl 0835.28007 [25] Schmidt K., Dynamical systems of algebraic origin (1995) · Zbl 0833.28001 [26] Silverman J. H., The arithmetic of elliptic curves (1986) · Zbl 0585.14026 [27] DOI: 10.1007/978-1-4612-0851-8 [28] Smyth C. J., Bull. London Math. Soc. 3 pp 169– (1971) · Zbl 0235.12003 [29] Smyth C. J., Canad. Math. Bull. 24 (4) pp 447– (1981) · Zbl 0475.12002 [30] Weil A., Basic number theory,, 3. ed. (1974) · Zbl 0326.12001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.