Entropy of nonautonomous dynamical systems

本研究项目的目的是推进非自治确定性动力系统的熵理论。20世纪90年代，S.Kolyada和L. Snoha提出了非自治系统的拓扑熵的概念，在随后的几年中，又有几位作者对其进行了研究。这一概念推广了自主动力系统理论中拓扑熵的经典概念，自20世纪60年代以来，拓扑熵作为自主动力系统中最重要的不变量已经得到了彻底的研究，现在被认为是很好的理解。系统的拓扑熵是一个实数，可以看作是轨道结构全局指数复杂度的度量。轨迹越复杂或混沌依赖于它们的初始值，熵的值就越高。Kolyada和Snoha引入的量是具有时间相关动力学的系统的类似度量，例如，它是由具有显式时间相关右侧的微分方程生成的。除此之外，它还推广了熵的其他几个概念，特别是拓扑序列熵、非紧致状态空间系统的拓扑熵和随机动力系统的拓扑熵。最后，还与熵的控制理论概念有关，熵是完成控制任务所需的最小比特率的度量。该研究项目的第一个目标是建立非自治系统拓扑熵的度量理论对应物，即度量熵，并将其与拓扑熵联系起来，因为这是通过变分原理为自治理论中的相应概念完成的。这个原理断言拓扑熵等于系统所有不变测度的度量熵的极值，是关于拓扑熵的大多数非平凡结果的基础。进一步的主题是：系统的特征与消失的熵通过他们的非漫游集，双曲系统的熵，和逃逸率和熵之间的关系。所有这些主题也都与控制理论问题有关。

The aim of this research project is the advancement of the entropy theory of nonautonomous deterministic dynamical systems. In the 1990s, the notion of topological entropy for nonautonomous systems has been introduced by S.Kolyada and L. Snoha, and in the subsequent years it has been investigated by several authors. This notion generalizes the classical notion of topological entropy in the theory of autonomous dynamical systems, which, as arguably the most important invariant of such systems, has been investigated thoroughly since the 1960s and nowadays is considered as well-understood. The topological entropy of a system is a real number which can be regarded as a measure for the global exponential complexity of the orbit structure. The more complicated or chaotic the trajectories depend on their initial values, the higher is the value of the entropy. The quantity introduced by Kolyada and Snoha is a similar measure for systems with time-dependent dynamics, which, for example, are generated by differential equations with explicitly time-dependent right-hand sides. Beyond that, it generalizes several other concepts of entropy, in particular topological sequence entropy, topological entropy of systems with non-compact state space and topological entropy of random dynamical systems. Finally, there is also a relation to control-theoretic notions of entropy which are measures for minimal bit rates necessary to accomplish control tasks. The first objective of the research project is to establish the measure-theoretic counterpart of topological entropy for nonautonomous systems, the metric entropy, and to relate it to the topological entropy as this is accomplished for the corresponding notions in the autonomous theory via the variational principle. This principle, which asserts that the topological entropy is equal to the supremum of the metric entropies with respect to all invariant measures of the system, is the basis for most nontrivial results about topological entropy. Further topics to be treated are: The characterization of systems with vanishing entropy via their nonwandering sets, the entropy of hyperbolic systems, and relations between escape rates and entropy. All of these topics are also related to control-theoretic problems.