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

Abstract

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 K 1, K 2 in C ¯ , K 1 ⩽ K 2 indicates that K 1 = K 2, or K 1 is included in a bounded component of C ¯ ⧹ K 2 . 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 G τ be the polynomial semigroup generated by the support of τ. Suppose that the support of τ is compact, the postcritical set of G τ is bounded in the complex plane and the Julia set of G τ is disconnected. Then, we show that (1) in each component U of the complement of the Julia set of G τ , T ∞∣ U equals a constant C U , (2) T ∞ : C ¯ → [ 0 , 1 ] is a continuous function on the whole C ¯ , and (3) if J 1, J 2 are two components of the Julia set of G τ with J 1 ⩽ J 2, then max z ∈ J 1 T ∞ ( z ) ⩽ min z ∈ J 2 T ∞ ( z ) . Hence T ∞ is similar to the devil’s-staircase function.