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A dynamical interpretation of the global canonical height on an elliptic curve. (English) Zbl 0927.11009
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.

##### 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
##### Citations:
Zbl 0865.11068; Zbl 0919.11064
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##### 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
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