Real analyticity of Hausdorff dimension for expanding rational semigroups
Real analyticity of Hausdorff dimension for expanding rational semigroups
复制标题
扩展有理半群的Hausdorff维数的实解析性
DOI：
10.1017/s0143385709000297
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
H. Sumi and M. Urbanski
中科院分区：
文献类型：
--
作者：
Shimomura;Shun;熊谷 隆;H.Shiga;H. Sumi and M. Urbanski
我们考虑黎曼球面上有限多个有理映射生成的扩张半群的动力学问题。我们证明了对于这样的半群的解析族，Bowen参数函数是实解析的，并且是多重次调和的。结合第一作者的一个结果，我们证明了：如果这样一个扩展半群的解析族的每个半群都满足开集条件，则Julia集的Hausdorff维是参数的实解析多重次调和函数。此外，我们还给出了满足上述所有条件的半群解析族的大量例子，并详细分析了相应的Bowen参数和Hausdorff维函数。
We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the Hausdorff dimension of the Julia set is a real-analytic and plurisubharmonic function of the parameter. Moreover, we provide an extensive collection of examples of analytic families of semigroups satisfying all of the above conditions and we analyze in detail the corresponding Bowen’s parameters and Hausdorff dimension function.