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.

中文翻译：

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论乘法级联对措施的作用

我们考虑 Mandelbrot 乘法级联对符号空间支持的概率测度的作用。对于一般概率测度，我们几乎可以得到一个限制测度非退化的尖锐标准；它依赖于度量的上下 Hausdorff 维度和随机权重的熵。我们还获得了限制措施的下 Hausdorff 和上包装尺寸的清晰界限。当原始度量是与具有一定连续性模量（比 Hölder 弱）的势相关的 Gibbs 度量时，我们所有的结果都是尖锐的。这改进了 Kahane 和 Peyrière、Ben Nasr 和 Fan 之前获得的结果。我们利用我们的结果来推导出一些随机分形测量的维数估计和绝对连续性。