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Thermodynamic formalism for invariant measures in iterated function systems with overlaps
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2021-05-21 , DOI: 10.1142/s0219199721500413
Eugen Mihailescu 1
Affiliation  

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 ν=πμ is a self-conformal measure, then HD(ν)<h(μ)|χ(μ)| if and only if the overlap number o(𝒮,μ)>1. Examples are also discussed.



中文翻译:

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ν=π*μ是自适形测度,则高清(ν)<H(μ)|χ(μ)|当且仅当重叠数(𝒮,μ)>1. 还讨论了示例。

更新日期:2021-05-21
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