Potential theory and a characterization of polynomials in complex dynamics

Abstract

Abstract. We obtain a characterization of polynomials among rational functions on P 1 from the point of view of complex dynamics and potential theory. This characterization generalizes a theorem of Lopes. Our proof applies both classical and (dynamically) weighted potential theory.

Effective divisors on the projective line having small diagonals and small heights and their application to adelic dynamics

“…was first established by DeMarco [18] for archimedean K and was generalized to rational functions defined over a number field by Baker-Rumely [4] (for a simple proof of (8.3) which also works for general K, see Baker-Rumely [2, Appendix A] or Stawiska and the author [37,Appendix]). The dynamical Green function g f of f on P 1 is the unique normalized weight on P 1 such that µ g f = µ f , i.e., for any lift F of f , g f ≡ g F + V g F /2 on P 1 .…”

A generalization of the converse of Brolin’s theorem

“…The implication (ii)⇒(i) in Theorem 1 was first claimed by Oba and Pitcher [7], also assuming f (∞) = ∞ (so f (D ∞ ) = D ∞ ) and f ′ (∞) = 0. It was established by Lopes [4] (see also Lalley [3] and Mañé-da Rocha [5]) under a relaxed additional assumption f (∞) = ∞, and by the present authors [8,Theorem 1]…”

“…Theorem 1 is not much stronger than [8,Theorem 1]. Indeed, with a little extra effort, it can be obtained by combining [8, Theorem 1] with Sullivan's no wandering domain theorem [12] and the Riemann-Hurwitz formula.…”

Limit functions of discrete dynamical systems

“…For each rational function f of degree at least 2 there exists a probability measure μ whose support is J so that f is μ-invariant and strong-mixing ( [6], [11]). In the case of a polynomial f , this measure coincides with the equilibrium measure ν of the Julia set (see [4] see also [15]). From Theorem 1.7 we get…”

“…Again, let f be a rational function of degree at least 2. Even if ∞ is a fixed point belonging to the Fatou set, which implies that J is a compact subset of the plane with positive capacity and having the property that f is invariant with respect to the equilibrium measure ν of J (see [9,Theorem 1]), the maximal entropy measure μ is different from the equilibrium measure ν unless f is a polynomial (see [10], and [15] for an extension). In particular, if ∞ is a neutral fixed point, then the system (J, f, ν) is a factor of an irrational rotation of the circle, hence ergodic with respect to ν, and ν and μ are mutually singular (again, see [9,Theorem 1]).…”