Reduction, Dynamics, and Julia Sets of Rational Functions☆

By Robert L. Benedetto. 2001

62 citations recorded by the source corpus.

Abstract We consider a rational function φ ( z )∈ K ( z ) in one variable defined over an algebraically closed field K which is complete with respect to a valuation v . We study how the reduction (modulo v ) of such functions behaves under composition, and in particular under iteration. We also investigate the relationship between bad reduction and the Julia set of φ . In particular, we prove that under certain conditions, bad reduction is equivalent to having a nonempty Julia set. We also give several examples of maps not satisfying those conditions and having both bad reduction and an empty Julia set.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0fa8e5b625c50b1bc70474ab2afa7d9b33576809

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

A finiteness theorem for canonical heights attached to rational maps over function fields (2006, background): “we present new proofs of some theorems of R. Benedetto and J. Rivera-Letelier concerning reduction of rational maps. For example, we easily deduce from Theorem 3.14 the following result of Benedetto ([7], Theorem B, see also [17], §7): Corollary 3.24. ϕ has potentially good reduction iﬀ ϕ(n) has pot”
Preperiodic points of polynomials over global fields (2005, background): “duction. However, after our coordinate changes, φis a monic polynomial without constant term. The assumption that rv >1 means that |ai|v >1 for some coeﬃcient ai of φ. Thus, by Corollary 4.6 of [5], φcannot have good reduction even after a change of coordinates. Remark 2.6. The fact that rv ≥ ”
Current trends and open problems in arithmetic dynamics (2018, background): “ this conjecture, although there are weaker bounds in which the constant C(N,d,D) is replaced by a constant that depends in various weakwaysonthe number ofprimes ofbad reduction of f; see for example [27, 31], as wellasthe discussionin Section12. Indeed, evenfor the family ofpolynomialsx2+c, we have”
ON MINIMAL DECOMPOSITION OF p-ADIC POLYNOMIAL DYNAMICAL SYSTEMS (2010, background): “sentpaper, only the sets Bi in the abovedecomposition are called minimal components. Recently the theory of Non-Archimedean, in particular of p-adic, dynamical systems has been intensively developed ([6, 7, 8], [13], [14], [16], [17], [18], [19, 20], [24], [26], [30], [33], [37], [39]). See also the”