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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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