Result: Random dynamics of polynomials and devil's-staircase-like functions in the complex plane

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 T. Suppose that the support of τ is compact, the postcritical set of Gτ is bounded in the complex plane and the Julia set of Gτis 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 Gτwith J1≤ J2, then maxz∈j1 T∞(z) < minz∈J2 T∞(z). Hence T∞ is similar to the devil's-staircase function.