We study the Hausdorff dimension of Gibbs measures with infinite entropy with respect to maps of the interval with countably many branches. We show that under simple conditions, such measures are symbolic-exact dimensional, and provide an almost sure value for the symbolic dimension. We also show that the lower local dimension dimension is almost surely equal to zero, while the upper local dimension is almost surely equal to the symbolic dimension. In particular, we prove that a large class of Gibbs measures with infinite entropy for the Gauss map have Hausdorff dimension zero and packing dimension equal to $1/2$, and so such measures are not exact dimensional.

中文翻译：

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具有无穷熵的 Gibbs 测度的维度

我们研究了具有无限熵的 Gibbs 测度的 Hausdorff 维数，该维度相对于具有可数许多分支的区间图。我们表明，在简单条件下，此类度量是符号精确维度，并为符号维度提供几乎确定的值。我们还表明，较低的局部维数几乎肯定等于零，而较高的局部维数几乎肯定等于符号维数。特别是，我们证明了高斯图的一大类具有无限熵的吉布斯测度的豪斯多夫维数为零，包装维数等于$1/2$，因此这些测度不是精确维数。