标题： 共轭测度在共形迭代函数系统中的重叠函数 显示英文标题

标题： Overlap functions for measures in conformal iterated function systems

摘要： 我们研究具有任意重叠的共形迭代函数系统（IFS）$\mathcal S = \{\phi_i\}_{i \in I}$，以及极限集$\Lambda$上的测度$\mu$，这些测度是关于某个提升映射$\Phi$在$\Sigma_I^+ \times \Lambda$上的平衡测度$\hat \mu$的投影。不假设任何类型的开放集条件。 我们引入了关于测度$\hat \mu$与$\mathcal S$的重叠函数和重叠数的概念；特别是引入了（拓扑）重叠数$o(\mathcal S)$的概念。 这些概念考虑了极限集中的点之间的$n$链。 我们证明$o(\mathcal S, \hat \mu)$与相对于提升$\Phi$的条件熵$\hat \mu$相关。 各种类型的投影到不变测度的$\Lambda$被研究。 我们通过使用压强函数和$o(\mathcal S, \hat \mu)$对$\mu$在$\Lambda$上的 Hausdorff 维数$HD(\mu)$得到了上界估计。 特别是，这适用于 Bernoulli 测度在$\Sigma_I^+$上的投影。 接下来，我们将结果应用于伯努利卷积$\nu_\lambda$对于$\lambda \in (\frac 12, 1)$，这对应于由具有重叠的IFS的收缩组合（以相等概率）确定的自相似测度$\mathcal S_\lambda$。 我们证明对于所有$\lambda \in (\frac 12, 1)$，存在$HD(\nu_\lambda)$与重叠数$o(\mathcal S_\lambda)$之间的关系。 数字$o(\mathcal S_\lambda)$用积分在$\Sigma_2^+$上近似，相对于均匀伯努利测度$\nu_{(\frac 12, \frac 12)}$。 我们还估计了$o(\mathcal S_\lambda)$对于某些值的$\lambda$。 显示英文摘要

摘要： We study conformal iterated function systems (IFS) $\mathcal S = \{\phi_i\}_{i \in I}$ with arbitrary overlaps, and measures $\mu$ on limit sets $\Lambda$, which are projections of equilibrium measures $\hat \mu$ with respect to a certain lift map $\Phi$ on $\Sigma_I^+ \times \Lambda$. No type of Open Set Condition is assumed. We introduce a notion of overlap function and overlap number for such a measure $\hat \mu$ with respect to $\mathcal S$; and, in particular a notion of (topological) overlap number $o(\mathcal S)$. These notions take in consideration the $n$-chains between points in the limit set. We prove that $o(\mathcal S, \hat \mu)$ is related to a conditional entropy of $\hat \mu$ with respect to the lift $\Phi$. Various types of projections to $\Lambda$ of invariant measures are studied. We obtain upper estimates for the Hausdorff dimension $HD(\mu)$ of $\mu$ on $\Lambda$, by using pressure functions and $o(\mathcal S, \hat \mu)$. In particular, this applies to projections of Bernoulli measures on $\Sigma_I^+$. Next, we apply the results to Bernoulli convolutions $\nu_\lambda$ for $\lambda \in (\frac 12, 1)$, which correspond to self-similar measures determined by composing, with equal probabilities, the contractions of an IFS with overlaps $\mathcal S_\lambda$. We prove that for all $\lambda \in (\frac 12, 1)$, there exists a relation between $HD(\nu_\lambda)$ and the overlap number $o(\mathcal S_\lambda)$. The number $o(\mathcal S_\lambda)$ is approximated with integrals on $\Sigma_2^+$ with respect to the uniform Bernoulli measure $\nu_{(\frac 12, \frac 12)}$. We also estimate $o(\mathcal S_\lambda)$ for certain values of $\lambda$.