Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4635856 | Applied Mathematics and Computation | 2007 | 12 Pages |
Abstract
Using the above result and combining it with the theory of random dynamics of complex polynomials, we consider the following: Let Ï be a Borel probability measure in the space {gâC[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 Ï. Let Tâ(z) be the probability of tending to ââC¯ starting from the initial value zâC¯ and let GÏ be the polynomial semigroup generated by the support of Ï. Suppose that the support of Ï is compact, the postcritical set of GÏ is bounded in the complex plane and the Julia set of GÏ is disconnected. Then, we show that (1) in each component U of the complement of the Julia set of GÏ, Tââ£U equals a constant CU, (2) Tâ:C¯â[0,1] is a continuous function on the whole C¯, and (3) if J1, J2 are two components of the Julia set of GÏ with J1 ⩽ J2, then maxzâJ1Tâ(z)⩽minzâJ2Tâ(z). Hence Tâ is similar to the devil's-staircase function.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Hiroki Sumi,