Combinatorial constructions in Smooth Ergodic Theory

历史上由统计力学问题引起的遍历理论研究动力系统的统计性质。特别是，人们对系统的长期行为及其时间和空间平均值之间的关系感兴趣。遍历理论中的一个主要问题是，是否存在具有特定遍历性质的光滑映射。这也是本研究项目的核心问题。构造具有规定遍历或拓扑性质的光滑微分同态的最有力工具之一是由D. Anosov和a . Katok提出的所谓共轭逼近法，该方法适用于具有非平凡圆作用的至少2维的任意光滑紧连流形。这些微分同胚被构造为属于圆作用的映射的共轭极限。在这个研究项目中，我们的目标是顺利实现进一步的遍历和光谱性质。此外，我们希望将共轭逼近法继续推广到实解析范畴。

Historically motivated by problems in statistical mechanics Ergodic Theory examines statistical properties of dynamical systems. In particular, one is interested in the long-term behaviour of the system as well as the relationship between its time and space averages. One of the main questions in Ergodic Theory asks if there are smooth maps with specific ergodic properties. This is also the central question of this research project.One of the most powerful tools of constructing smooth diffeomorphisms with prescribed ergodic or topological properties is the so-called approximation by conjugation-method developed by D. Anosov and A. Katok which works on arbitrary smooth compact connected manifolds of dimension at least 2 admitting a non-trivial circle action. These diffeomorphisms are constructed as limits of conjugates of maps belonging to the circle action. In this research project we aim at the smooth realization of further ergodic as well as spectral properties. Moreover, we want to continue extending the approximation by conjugation-method to the real-analytic category.