Entropy of nonautonomous dynamical systems

本研究项目的目的是促进非自治确定性动力系统的熵理论的发展。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，在随后的几年里，几位作者对它进行了研究。这一概念推广了自治动力系统理论中经典的拓扑熵概念，作为这类系统中最重要的不变量，自20世纪60年代以来一直被深入研究，现在被认为是众所周知的。系统的拓扑熵是一个实数，它可以用来衡量轨道结构的全局指数复杂度。轨迹依赖于其初始值的越复杂或越混乱，熵的值就越高。Kolyada和Snoha引入的量对于具有时变动力学的系统是类似的度量，例如，它是由具有显式时变右端的微分方程产生的。此外，它还推广了其他几个熵的概念，特别是拓扑序列熵、非紧状态空间系统的拓扑熵和随机动力系统的拓扑熵。最后，还有一个与控制论的熵概念的关系，它是完成控制任务所需的最低比特率的度量。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论对应关系，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实现的。这一原理断言拓扑熵等于关于系统的所有不变度量的度量熵的上确界，这是关于拓扑熵的大多数非平凡结果的基础。进一步要讨论的主题是：通过非游荡集刻画具有零熵的系统，双曲系统的熵，以及逃逸率和熵之间的关系。所有这些话题也都与控制论问题有关。

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.