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Dynamical systems arising from elliptic curves. (English) Zbl 0965.37020
The organization of the paper under review is as follows: 1. Introduction. 2. The solenoid. 3. Elliptic curves. 4. The \(\beta\)-transformation and a \(p\)-adic analogue. 5. Dynamics on the elliptic adeles. 6. Putative elliptic dynamics.
The main results: Let \(q \in {\mathbb Q}_p\) and \(\log^{+}x\) denote \(\max\{\log x,0\}. \) For a generic element \(x\) of \({\mathbb Z}_p\) the authors define the \(q\)-transformation \(T_{q}(x)\) (a \(p\)-adic analogue of the \(\beta\)-transformation). Then the topological entropy of the \(p\)-adic \(\beta\)-transformation is given by \(h(T_{q}) = \log^{+}|q|_p\) (Theorem 4.1). If \(|q|_p \geq 1\) then the map \(T_{q}\) is ergodic with respect to Haar measure for \(|q|_p > 1\) and is not ergodic for \(|q|_p = 1\) (Theorem 4.2).
Let \(\text{Per}_{n}(T_{q})\) denote the subgroup of \({\mathbb Z}_p\) consisting of elements of period \(n\) under \(T_{q}\). Let \(U\) be the set of unit roots of \({\mathbb Q}_p\) and \(q \in {\mathbb Q}_p \setminus U\). Then \[ \log |Per_{n}(T_{q})|= n \log^{+} |q|_{p}. \] (Theorem 4.3). The authors use the topological entropy and measure theoretical arguments based on volume growth rate and arithmetic of \({\mathbb Z}_p. \)
Let \(Q\) be a rational point of an elliptic curve over \({\mathbb Q}\) and let \(\widehat{h}(Q)\) be the global canonical height on rational points of the elliptic curve. Then with the definitions and assumptions of the paper under review and \(q = a/b = x(Q)\), (i) the entropy of \(T_Q\) is given by \(h(T_{Q}) = 2\widehat{h}(Q),\) and (ii) the asymptotic growth rate of the periodic points is given by the division polynomial \(\nu_{n}(x) \): \( \log |Per_{n}(T_{Q}|\sim \log |b^{n} \nu_{n}(q)|\) as \(n \rightarrow \infty. \) (Theorem 5.2). In the latter case the authors also use the elliptic analogue of Baker’s theorem, which is described in the paper of S. David [Lower bounds for linear forms in elliptic logarithms (French), Mem. Soc. Math. Fr., Nouv. Sér. 62 (1995; Zbl 0859.11048)] and the paper of G. Everest and T. Ward [Exp. Math. 7, 305–316 (1998; Zbl 0927.11009)].
In the last section the authors set out a precise conjecture about the existence of elliptic dynamical systems and discuss a possible connection with mathematical physics. It seems that the conjecture is proved in a forthcoming paper of G. Everest, M. Einsiedler and T. Ward.

MSC:
37C35 Orbit growth in dynamical systems
37B40 Topological entropy
11G07 Elliptic curves over local fields
11G50 Heights
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