论文标题
关于乘法级联对措施的作用
On the action of multiplicative cascades on measures
论文作者
论文摘要
我们考虑Mandelbrot乘法级联反应对符号空间支持的概率度量的作用。对于一般概率措施，我们获得了限制度量的非分类的急剧标准。它依赖于度量的下部和上部豪斯多夫尺寸和随机重量的熵。我们还获得了下部豪斯多夫的锋利边界和限制度量的上部堆积尺寸。当原始度量是与某些连续性模量（比Hölder弱）相关的Gibbs量度时，我们所有的结果都很清晰。这改善了Kahane和Peyrière，Ben Nasr和Fan的结果。我们利用结果来得出某些随机分形测量的尺寸估计和绝对连续性。
We consider the action of Mandelbrot multiplicative cascades on probability measures supported on a symbolic space. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on the lower and upper Hausdorff dimensions of the measure and the entropy of the random weights. We also obtain sharp bounds for the lower Hausdorff and upper packing dimensions of the limiting measure. When the original measure is a Gibbs measure associated with a potential of certain modulus of continuity (weaker than Hölder), all our results are sharp. This improves results previously obtained by Kahane and Peyrière, Ben Nasr, and Fan. We exploit our results to derive dimension estimates and absolute continuity for some random fractal measures.
