On the Objectivity and Quasi-Objectivity of TSE and TRA

We analyze two recently-introduced flow measured that are based on a single trajectory only: trajectory stretching exponent (TSE) to detect hyperbolic (stretching) behavior, and trajectory angular velocity (TRA) to detect elliptic (rotation) behavior. Haller et al. [$\href{https://doi.org/10.1063/5.0044151}{2021}$] and Haller et al. [$\href{https://doi.org/10.1063/5.0090124}{2022}$] introduced TSE, TRA as well as the concept of quasi-objectivity, and formulated theorems about the objectivity and quasi-objectivity of TSE and TRA. In this paper, we present two counter-examples showing that all theorems in Haller et al. [$\href{https://doi.org/10.1063/5.0044151}{2021}$] and Haller et al. [$\href{https://doi.org/10.1063/5.0090124}{2022}$] are incorrect