Simultaneously preperiodic integers for quadratic polynomials

摘要

In this article, we study the set of parameters c ∈ C for which two given complex numbers a and b are simultaneously preperiodic for the quadratic polynomial fc(z) = z2 + c. Combining complex-analytic and arithmetic arguments, Baker and DeMarco showed that this set of parameters is infinite if and only if a2 = b2. Recently, Buff answered a question of theirs, proving that the set of parameters c ∈ C for which both 0 and 1 are preperiodic for fc is equal to {-2, -1, 0}. Following his approach, we complete the description of these sets when a and b are two given integers with |a| not equal to |b|.