Measures, Dimensions and Dynamics

主要研究者建议进一步推进和发展共形动力系统和全纯动力系统的研究理论和方法。这包括超越整函数和亚纯函数的确定性迭代和随机迭代，Hilbert空间上的图有向马尔可夫系统和有限维迭代函数系统的维谱。结合代数几何和微分几何的方法，建立了复射影空间上有理函数的热力学形式。此外，还提出了有理半群理论的发展方向。文中还利用所给出的方法研究了沿平方和高次方的散度遍历平均，并分析了Cantor空间上同胚Bernoulli试测度的结构。除了遍历理论和动力系统外，该研究还涉及复分析、概率论、几何测度论、拓扑学、维度论、数论、代数几何和微分几何等数学领域。

The principal investigators propose to further advance and develop the theory and methods for investigations of conformal and holomorphic dynamical systems. This includes deterministic and random iterations of transcendental entire and meromorphic functions, graph directed Markov systems on Hilbert spaces and dimension spectra for finite-dimensional iterated function systems. In conjunction with the methods of algebraic geometry and differential geometry, the thermodynamic formalism for rational functions on complex projective spaces is proposed to be built. Furthermore, the theory of rational semigroups is proposed to be advanced. The methods laid out are also used to investigate divergent ergodic averages along squares and higher powers, and the structure of homeomorphic Bernoulli trial measures on the Cantor space is analyzed. Apart from ergodic theory and dynamical systems, the proposed research also involves such areas of mathematics as complex analysis, probability theory, geometric measure theory, topology, dimension theory, number theory, algebraic geometry and differential geometry.