We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$-maps $T_{\unicode[STIX]{x1D6FD}}$, for arbitrary $\unicode[STIX]{x1D6FD}>1$.

中文翻译：

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图导向马尔可夫系统上的 Smale 自同态

我们研究 Smale 偏斜积自同态（在 Mihailescu 和 Urbański 中介绍 [Skew product Smale 自同态在有限类型的可数移位上。埃尔戈德。钍。&动态。系统。doi: 10.1017/etds.2019.31。2019 年 6 月在线发表]）现在超过了可数的图有向马尔可夫系统，我们证明了纤维中条件测量的精确维数，然后证明了平衡测度本身的全局精确维数。我们的结果适用于大类系统并有许多应用。例如，它们适用于图导向马尔可夫系统的自然扩展。另一个应用是在抛物线系统上倾斜产品。我们还给出了遍历数论中的应用，例如连分数展开式和后向分数展开式。最后，我们获得了关于自然延伸的诱导图的平衡测量的 Hausdorff（和逐点）维数的通用公式${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$的$\unicode[STIX]{x1D6FD}$-地图$T_{\unicode[STIX]{x1D6FD}}$, 对于任意$\unicode[STIX]{x1D6FD}>1$.