Measures, Dimensions and Dynamics

该项目涉及确定性和随机共形动力系统。这包括共形迭代函数系统（IFS）的理论建立的提案，合理的功能，黎曼球面和整个亚纯函数。我们建议在Hilbert空间中研究共形IFS，特别是研究极限集的超限维数。扩展了A. Zdunik推广到大于2的维数，我们提出了IFS的极限集（其闭包是拓扑Cantor集）的调和测度的Hausdorff维数.发展我们的刚性调查，我们打算证明几何刚性的高维极限集的闭包是拓扑磁盘。 给出了正则IFS的共形测度满足加倍性质的充要条件，并作为应用给出了生成具有满足加倍性质的共形测度的连分式系统的自然数子集的一个刻画.传递到有理函数，我们建议处理非递归映射（更确切地说，假设Julia集中包含的临界点是非递归的）。我们想证明这些映射的Hausdorff维数和盒维数相等，研究压力和逃逸率的各种定义。采用仿射叠层的概念，我们建议与M。半双曲有理函数的热力学形式。Kotus和A. Zdunik我们建议探索不变和几何（豪斯多夫，包装）措施的成员指数家庭，以及亚纯函数拟合到沃尔特斯的热力学形式主义。我们打算处理最近出现的话题的量子化维数有关其理论的多重分形形式主义。我们的最后一个子项目是关于随机IFS的几何测度，特别是关于精确填充测度函数的存在性，我们建议继续和发展我们正在进行的研究，涉及各种测度--Hausdorff测度、填充测度、共形测度、Gibbs态测度、关于Lebesgue测度绝对连续的不变测度等，以及相关的功能，如容量和压力，最后是相关的尺寸。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，另一个是几何测量理论，适用于递归生成的对象。这些方法自然地相互融合，并导致一些有趣的想法混合。我们发展了一个广泛的共形迭代函数系统理论，即无穷多个共形映射的迭代，这些映射可以是双曲的（一致收缩的）或抛物的（至少一个映射有一个无关不动点），并且还将它应用于一些众所周知的问题，例如，连分数和阿波罗填充。但在应用方面仍存在一些根本性的未决问题和其他一些问题。我们希望扩展和发展这一理论，以涵盖一个更结构化的迭代映射，其由有向图或替换和随机迭代和适当的多重分形formalisn这些系统。这不仅有许多应用研究的理性，整个和亚纯函数，而且还新兴的理论所产生的统计和工程的量化维数问题。我们还打算将我们的理论应用于各种细胞自动机产生的极限集。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.