We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems (IFSs) with overlaps ${𝒮}$. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for IFSs with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If $\nu ={\pi }_{\ast }\mu$ is a self-conformal measure, then $\mathrm{\text{HD}}(\nu )<\frac{h(\mu )}{|\chi (\mu )|}$ if and only if the overlap number $o({𝒮},\mu )>1$. Examples are also discussed.

中文翻译：

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具有重叠的迭代函数系统中不变测度的热力学形式

我们研究了与具有重叠的共形迭代函数系统 (IFS) 相关的一类不可逆变换的平衡 (Gibbs) 状态图像${𝒮}$. 我们证明了这些图像度量的精确维度，并使用它们的重叠数找到了维度公式。特别是，我们根据重叠数获得了具有重叠的 IFS 的自适形测量维度的几何公式​​。这暗示了降维的充要条件。如果$\nu ={\pi }_{*}\mu$是自适形测度，则$\mathrm{\text{高清}}(\nu )<\frac{H(\mu )}{|\chi (\mu )|}$当且仅当重叠数$○({𝒮},\mu )>1$. 还讨论了示例。