Random dynamics of polynomials and devil’s-staircase-like functions in the complex plane

摘要

We consider the dynamics of polynomial semigroups with bounded postcritical set and random dynamics of complex polynomials in the complex plane.A polynomial semigroup G is a semigroup generated by polynomials in one variable with the semigroup operation being functional composition. We show that if the postcritical set of G, that is the closure of the G-orbit of the union of any critical values of any generators of G, is bounded in the complex plane, then the space of components of the Julia set of G (Julia set is the set of points in the Riemann sphere C¯ in which G is not normal) has a total order “⩽”, where for two compact connected sets K1, K2 in C¯, K1 ⩽ K2 indicates that K1 = K2, or K1 is included in a bounded component of ⧹C¯⧹K2.Using the above result and combining it with the theory of random dynamics of complex polynomials, we consider the following: Let τ be a Borel probability measure in the space {g∈C[z]|deg(g)⩾2} with topology induced by the uniform convergence on the Riemann sphere C¯. We consider the i.i.d. random dynamics in C¯ such that at every step we choose a polynomial according to the distribution τ. Let T∞(z) be the probability of tending to ∞∈C¯ starting from the initial value z∈C¯ and let Gτ be the polynomial semigroup generated by the support of τ. Suppose that the support of τ is compact, the postcritical set of Gτ is bounded in the complex plane and the Julia set of Gτ is disconnected. Then, we show that (1) in each component U of the complement of the Julia set of Gτ, T∞∣U equals a constant CU, (2) T∞:C¯→[0,1] is a continuous function on the whole C¯, and (3) if J1, J2 are two components of the Julia set of Gτ with J1 ⩽ J2, then maxz∈J1T∞(z)⩽minz∈J2T∞(z). Hence T∞ is similar to the devil’s-staircase function.