Geodesic
$p$-Adic Entropies of Logistic Maps
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 257-263 
G. B. Shabat. $p$-Adic Entropies of Logistic Maps. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 257-263. http://geodesic.mathdoc.fr/item/TM_2004_245_a25/
@article{TM_2004_245_a25,
     author = {G. B. Shabat},
     title = {$p${-Adic} {Entropies} of {Logistic} {Maps}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {257--263},
     year = {2004},
     volume = {245},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_245_a25/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We study the dynamic properties of the logistic maps $x\to\lambda x(1-x)$ over the fields of $p$-adic numbers. We are interested in the chaotic behavior of trajectories; it turns out that, for a fixed rational $\lambda$, such behavior occurs only for finitely many $p$'s. This fact is consistent with the main result of the paper: the calculation of topological entropies of these maps. The possibility of the adelic interpretation of this result is discussed.

[1] Adler R. L., Konheim A. G., McAndrew M. H., “Topological entropy”, Trans. Amer. Math. Soc., 114:2 (1965), 309–319 | DOI | MR | Zbl

[2] Dremov V. A., “Ob odnom $p$-adicheskom mnozhestve Zhyulia”, UMN, 58:6 (2003), 151–152 | MR | Zbl

[3] Nevins M., Rogers Th. D., Quadratic maps as dynamical systems on the $p$-adic numbers, , 2000 http://citeseer.nj.nec.com/451088.html

[4] Thiran E., Verstegen D., Weyers J., “$p$-Adic dynamics”, J. Stat. Phys., 54:3/4 (1989), 893–913 | DOI | MR | Zbl