Papers › A local-global principle in the dynamics of quadratic polynomials
A local-global principle in the dynamics of quadratic polynomials
David Krumm
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Let K be a number field, f∈K[x] a quadratic polynomial, and n∈{1,2,3}. We show that if f has a point of period n in every non-archimedean completion of K, then f has a point of period n in K. For n∈{4,5} we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over K for which this local-global principle fails. By considering a stronger form of this principle, we strengthen global results obtained by Morton and Flynn-Poonen-Schaefer in the case K=𝐐. More precisely, we show that for every quadratic polynomial f∈𝐐[x] there exist infinitely many primes p such that f does not have a point of period 4 in the p-adic field 𝐐ₚ. Conditional on knowing all rational points on a particular curve of genus 11, the same result is proved for points of period 5.
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