# MATH 845, Arithmetic of Dynamical Systems -- Questions and errata

• Theorem 2.21: (question by everybody) The theorem on behaviour of periodic points under good reduction is only stated for rational functions of degree at least 2. However, the proof does not seem to depend on the degree. Is there a reason the the statement excludes maps of degree 1?
• Excercises 2.8, 2.11 (erratum by Alex Molnar) In 2.8, the v-adic chordal metric is probably meant and in 2.11 definitely the archimedean chordal metric
• Conjecture 3.15: (question by Jason Bell) We can also consider dynamical systems over finitely generated extensions. Via a specialization argument one can show that the uniform boundedness conjecture for preperiodic points over number fields implies that such a system has only finitely many preperiodic points too. Is anything proven in that case?
• Theorem 3.43: (question by Nils Bruin) The proof of this theorem definitely uses that d>1. However, one can still ask if a fractional linear transformation for which infinity is a wandering point (i.e., no power is a polynomial map) can have any infinite orbits with infinitely many integral points.
• Page 268, line 2 "Proposition 2.29(a)" should be "Proposition 2.29(b)"
• Page 268, line 18 "P0 in ...:" Point P is excluded, but P1 is meant