Let f be a holomorphic map containing an irrationally indifferent fixed point z0. If f is locally linearizable at z0, then the maximal linearizable domain containing z0 is called the Siegel disk of f centered at z0. The topology of the boundaries of Siegel disks has been studied extensively in past 3 decades. This was motivated by the prediction of Douady and Sullivan that the Siegel disk of every non-linear rational map is a Jordan domain.

For the topology of whole Julia sets of holomorphic maps with Siegel disks, the results appear less. Petersen proved that the quadratic Julia sets with bounded type Siegel disks are locally connected. Later Yampolsky proved the same result by an alternative method based on the existence of complex a prior bound of unicritical circle maps. A big progress was made by Petersen and Zakeri in 2004. They proved that for almost all rotation numbers, the quadratic Julia sets with Siegel disks are locally connected. Recently J. Yang proved a striking result that the Julia set of any polynomial (assumed to be connected) is locally connected at the boundary points of their bounded type Siegel disks.

As a generalization of Petersen's result, we prove that the Julia sets of a number of rational maps and transcendental entire functions with bounded type Siegel disks are locally connected. This is based on establishing an expanding property of a long iteration of a class of quasi-Blaschke models near the unit circle.

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