Abstract
Let K be a number field, let \({\varphi \in K(t)}\) be a rational map of degree at least 2, and let \({\alpha, \beta \in K}\) . We show that if α is not in the forward orbit of β, then there is a positive proportion of primes \({\mathfrak{p}}\) of K such that \({\alpha {\rm mod} \mathfrak{p}}\) is not in the forward orbit of \({\beta {\rm mod} \mathfrak{p}}\) . Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace α by a hypersurface, such as the ramification locus of a morphism \({\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}}\) .
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Benedetto, R.L., Ghioca, D., Hutz, B. et al. Periods of rational maps modulo primes. Math. Ann. 355, 637–660 (2013). https://doi.org/10.1007/s00208-012-0799-8
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DOI: https://doi.org/10.1007/s00208-012-0799-8