Self-similar measures on the Julia sets (Q1593042)

In the first part of the present paper, the authors prove the following result. For any rational function \(R\) if there is a simply connected domain \(U\) satisfying the condition that \(R^{-1}\subset U\) and \(\overline U\) does not contain any critical value \(R\), then there exists a unique self-similar measure on the Julia set \(J(R)\). In the second part of the paper, the authors discuss random iteration of a finite set of rational functions and construct a self-similar measure such that the measure is non-atomic and its support is equal to the Julia set. Finally, some applications to condensed matter physics are given.