## Final Report Summary - TOPDSC (Topological aspects of dynamical independence and chaos)

The main aim of the project was an extensive study of systems with complicated dynamics acting on compact metric spaces. When we want to detect if a system is chaotic, especially in applications, we usually examine its local properties and try to construct a factor map from invariant subset to a system with known properties. We can also try to deduce information on dynamical behavior of the system form some global properties, like proximality, mixing, specification property, the structure of minimal sets, etc. The main emphasis in our research was put on various kinds of mixing in various contexts.

We were studying structure of minimal subsystems, alpha and omega-limit sets under reasonably mild conditions. We proved that on the unit interval every alpha-limit set is an omega-limit set, while the converse is not true in general. We also proved that notions of internal chain transitivity and weak incompressibility coincide in compact metric spaces, and while both are properties of omega-limits sets, they are more general notions. We also provided a few sufficient conditions when these notions coincide (i.e. internally chain-transitive set becomes omega-limit set of some point), generalizing and extending a few previously known results.

In the project we put lots of effort in characterization of mixing and its local aspects expressed in terms of weakly mixing sets (of order n). For every n we were able to provide an example of minimal dynamical system containing many perfect weakly mixing sets of order n but without nontrivial weakly mixing sets of order n+1 (i.e. other than singletons). We were also able to relate weakly mixing sets with other notions such as topological sequence entropy, equicontinuity, proximality, etc. and various types of chaos. We also found an interesting connection, that in invertible minimal systems with non-trivial weakly mixing sets of order three topological sequence entropy is always positive. Additionally, we related weakly mixing sets with product recurrence, which leaded us to an alternative characterization of distality in terms of very weak product recurrence.

In our studies we were also focusing on definitions of chaos in terms of pairs and tuples. Among others, we were studying chaos in the sense of Li and Yorke and distributional chaos. We proved that while syndetically proximal relation is an obstruction for distributional chaos, it is still possible to construct syndetically scrambled sets for the dynamics of continuous self-maps of compact metric spaces. We prove that maps with specification property have invariant distributionally scrambled sets and that this kind of chaos can be transferred through finite-to-one factor maps. We also show that in case of (various) definitions of chaos n-tuples do not force (n+1)-tuples to exist.

Finally, we proved that our results can work well in practice, for example when we are given a first return map on a cross section in a non-autonomous time-periodic differential equation (of special type of stationary points), or a sequence of covering relations with properly arranged inter-relations (so called coupled-expanding maps) .

We were studying structure of minimal subsystems, alpha and omega-limit sets under reasonably mild conditions. We proved that on the unit interval every alpha-limit set is an omega-limit set, while the converse is not true in general. We also proved that notions of internal chain transitivity and weak incompressibility coincide in compact metric spaces, and while both are properties of omega-limits sets, they are more general notions. We also provided a few sufficient conditions when these notions coincide (i.e. internally chain-transitive set becomes omega-limit set of some point), generalizing and extending a few previously known results.

In the project we put lots of effort in characterization of mixing and its local aspects expressed in terms of weakly mixing sets (of order n). For every n we were able to provide an example of minimal dynamical system containing many perfect weakly mixing sets of order n but without nontrivial weakly mixing sets of order n+1 (i.e. other than singletons). We were also able to relate weakly mixing sets with other notions such as topological sequence entropy, equicontinuity, proximality, etc. and various types of chaos. We also found an interesting connection, that in invertible minimal systems with non-trivial weakly mixing sets of order three topological sequence entropy is always positive. Additionally, we related weakly mixing sets with product recurrence, which leaded us to an alternative characterization of distality in terms of very weak product recurrence.

In our studies we were also focusing on definitions of chaos in terms of pairs and tuples. Among others, we were studying chaos in the sense of Li and Yorke and distributional chaos. We proved that while syndetically proximal relation is an obstruction for distributional chaos, it is still possible to construct syndetically scrambled sets for the dynamics of continuous self-maps of compact metric spaces. We prove that maps with specification property have invariant distributionally scrambled sets and that this kind of chaos can be transferred through finite-to-one factor maps. We also show that in case of (various) definitions of chaos n-tuples do not force (n+1)-tuples to exist.

Finally, we proved that our results can work well in practice, for example when we are given a first return map on a cross section in a non-autonomous time-periodic differential equation (of special type of stationary points), or a sequence of covering relations with properly arranged inter-relations (so called coupled-expanding maps) .