Entropy of nonautonomous dynamical systems

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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.