Measures, Dimensions and Dynamics

该项目涉及确定性和随机共形动力系统。这包括作者建立的共形迭代函数系统理论、黎曼球的有理函数、全函数和亚纯函数。我们提出研究Hilbert空间中的共形IFS，特别是研究极限集的超维。将a . Zdunik的工作推广到大于2的维数，我们提出了处理IFS的极限集（其闭包是拓扑Cantor集）调和测度的Hausdorff维数。发展刚性调查我们打算证明高维几何刚度的极限集的闭包是拓扑磁盘。本文拟给出正则IFS的保形测度满足倍性的充分必要条件，并给出产生具有保形测度满足倍性的连分数系统的所有自然数子集的刻画。传递给有理函数，我们建议处理非循环映射（更准确地说，Julia集合中包含的临界点被假设为非循环）。我们想要证明这些图的豪斯多夫维数和盒维数相等，研究压力和逃逸率的各种定义。利用仿射层合的概念，我们提议与柳比奇一起发展半双曲有理函数的热力学形式。与J. Kotus和A. Zdunik一起，我们提议探索指数族成员的不变和几何（Hausdorff, packing）测度，以及适合沃尔特斯热力学形式的亚纯函数。我们打算处理最近出现的量化维数的问题，将其理论与多重分形形式化联系起来。我们的最后一个子项目涉及随机IFS的几何度量，特别是关于精确包装度量函数的存在性。我们建议继续和发展我们正在进行的研究，涉及各种度量- hausdorff， packing，共形，Gibbs状态，不变测度，绝对连续的勒贝格测度等，以及相关的函数，如容量和压力，最后是相关的维度。这些概念是从两个相互交织的观点来研究的。一个是各种动力系统的问题，另一个是应用于递归生成对象的几何测度理论问题。这些方法自然地相互融合，产生了一些有趣的想法。我们发展了保形迭代函数系统的广泛理论，无限多个保形映射的迭代，这些保形映射可以是双曲的（一致收缩）或抛物线的（至少一个映射有一个不动点），并将其应用于一些众所周知的问题，例如连分式和阿波罗填充。在应用的背景下，还存在一些基本的突出问题和其他几个问题。我们希望扩展和发展这一理论，以涵盖更结构化的映射迭代-其由有向图或替换和随机迭代以及这些系统的适当的多重分形形式化控制。这不仅适用于有理函数、全函数和亚纯函数的研究，而且适用于统计学和工程学中新出现的量化维数问题理论。我们还打算将我们的理论应用于由各种元胞自动机生成的极限集。这对于压缩视觉图像非常重要。

The project concerns deterministic and random conformal dynamical systems. This includes the theory of conformal iterated function systems (IFS) founded by the proposers , rational functions of the Riemann sphere and entire and meromorphic functions. We propose to study conformal IFS in a Hilbert space, in particular to examine the transfinite dimensions of the limit sets . Extending the work with A. Zdunik to the dimensions greater than 2, we propose to deal with the Hausdorff dimension of the harmonic measure of the limit set (whose closure is a topological Cantor set) of an IFS. Developing our rigidity investigations we intend to prove the geometrical rigidity of higher dimensional limits sets whose closures are topological disks. We intend to provide necessary and sufficient conditions for conformal measures of regular IFS to satisfy the doubling property and as an application we would present a characterization of all subsets of natural numbers which generate the continued fraction systems with conformal measures satisfying the doubling property. Passing to rational functions we propose to deal with no-recurrent maps (more precisely, the critical points contained in the Julia sets are assumed to be non-recurrent). We would like to work on the proof of equality of the Hausdorff dimension and box dimension of these maps, to study various definitions of pressure and the escape rates. Employing the concept of affine laminations, we propose to develop together with M. Lyubich the thermodynamic formalism of semi-hyperbolic rational functions. Together with J. Kotus and A. Zdunik we propose to explore invariant and geometric (Hausdorff, packing) measures for the members of the exponential family, as well as meromorphic functions fitting into Walters' thermodynamic formalism. We intend to deal with recently emerged topic of quantization dimension relating its theory to the multifractal formalism. Our last sub-project concerns geometric measures of random IFS, in particular concerning the existence of an exactpacking measure function.We propose to continue and develop our ongoing research involving measures of various kinds-Hausdorff, packing, conformal, Gibbs states, invariant measures absolutely continuous with respect to Lebesgue measure, etc., and associated functions such as capacities and pressureand finally associated dimensions. These notions are studied from two interlaced viewpoints. One is that of various dynamical systems and the other is that of geometric measure theory as applied to recursively generated objects. These approaches naturally meld with one another and lead to some interesting mixtures of ideas. We have developed an extensive theory of conformal iterated function systems, the iteration of infinitely many conformal maps, which may be hyperbolic (uniformly contracting) or parabolic (at least one map has an indifferent fixed point) and also have made some applications of it to some well known problems, e.g., continued fractions and Apollonian packings. There are yet some fundamental outstanding problems and several others which arise within the context of applications. We want to extend and develop this theory to cover a more structured iteration of maps-whose governed by directed graphs or substitutions and random iterations and an appropriate multifractal formalisn for these systems. This has many applications not only to the study of rational, entire and meromorphic functions, but also to newly emerging theory of quantization dimension problems arising from statistics and engineering. We also intend to apply our theory to limit sets generated by various cellular automata. This could be very important for compressing visual images.