Title: Typical dimension and absolute continuity for classes of dynamically defined measures, Part II : exposition and extensions Show Chinese title

Title: 典型维数和绝对连续性对于动态定义测度的类，第二部分：阐述和扩展

Abstract: This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of ergodic measures for parametrized families of smooth iterated function systems (IFS) on the line. Under some assumptions, most crucially, a transversality condition, we obtain formulas for the Hausdorff dimension of the measure and absolute continuity for almost every parameter in the appropriate parameter region. The main novelty of [1] and the present paper is that not only the IFS, but also the ergodic measure in the symbolic space, whose push-forward we consider, depends on the parameter. This includes many interesting families of measures, in particular, invariant measures for IFS's with place-dependent probabilities and natural (equilibrium) measures for smooth IFS's. One of the goals of this paper is to present an exposition of [1] in a more reader-friendly way, emphasizing the ideas and proof strategies, but omitting the more technical parts. This exposition/survey is based in part on the series of lectures by K\'aroly Simon at the Summer School "Dynamics and Fractals" in 2023 at the Banach Center, Warsaw. The main new feature, compared to [1], is that we consider multi-parameter families; in other words, the set of parameters is allowed to be multi-dimensional. This broadens the scope of applications. A new application considered here is to a class of Furstenberg-like measures. [1] B. B\'ar\'any, K. Simon, B. Solomyak and A. \'Spiewak: Typical absolute continuity for classes of dynamically defined measures. Advances in Mathematics, Volume 399, 2022, 108258, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2022.108258. Show Chinese abstract

Abstract: 本文部分是阐述，部分是对我们工作[1]在多参数情况下的扩展。 我们考虑某些参数化动态定义测度的类。 这些是在自然投影下的遍历测度的前推，针对线上的参数化光滑迭代函数系统（IFS）族。 在一些假设下，最关键的是一个横截性条件，我们得到了测度的豪斯多夫维数公式以及在适当参数区域中几乎所有参数的绝对连续性。 [1]和本文的主要新意在于，不仅IFS依赖于参数，而且我们所考虑其前推的符号空间中的遍历测度也依赖于参数。 这包括了许多有趣的测度族，特别是具有位置依赖概率的IFS的不变测度以及光滑IFS的自然（平衡）测度。 本文的目标之一是以更易理解的方式阐述[1]，强调思想和证明策略，但省略了更技术性的部分。 这一阐述/综述部分基于Károly Simon在2023年华沙巴拿赫中心夏季学校“动力学与分形”系列讲座的内容。 与[1]相比，本论文的主要新特点是我们考虑多参数族；换句话说，参数集被允许是多维的。 这扩大了应用范围。 这里考虑的一个新应用是类似于Furstenberg的测度类。 [1] B. Bárány, K. Simon, B. Solomyak and A. Śpiewak: 动态定义测度类的典型绝对连续性。 《数学进展》，第399卷，2022年，108258，ISSN 0001-8708，https://doi.org/10.1016/j.aim.2022.108258。