In 1970s Bowen related hyperbolic dynamics with specification property and used this to show existence of a unique measure of maximal entropy. Almost the same time Sigmund used specification property as a tool in characterization of simplex of invariant measures. These results have several consequences. First, it became clear that (broadly understood) tracing of well well-chosen trajectories can provide good insight into the simplex of invariant measures. Second, tracing of trajectories can lead to emerging structures and properties in dynamics (e.g. uniform spread of some trajectories necessary for measure of maximal entropy; forming of some patterns; irregular motions, etc.). Finally, well defined tracing may be stable under perturbations, leading to better understanding of features of typical dynamics. Over the years, these results were inspiration for numerous mathematicians in various studies of dynamical systems.

In this talk we will present selected questions and recent results fitting into the above framework of research.