Exploring Quasi-Similarity and Kolmogorov Entropy in Dynamical Systems
Dynamical systems are a rich field of study in mathematics, bridging various areas such as topology, geometry, and probability. An intriguing question within this realm concerns the connections between quasi-similarity of dynamical systems and Kolmogorov entropy. In recent research, we delve into this topic and uncover significant findings. Notably, we prove that all Bernoulli actions of a given countably infinite group are quasi-similar to each other. This discovery paves the way for further exploration into the nature of dynamical systems and their behaviors. Interestingly, the existence of non-Bernoulli actions within the same quasi-similarity class remains an open problem, inviting further investigation. In contrast to quasi-similarity, the concept of disjointness, or independence of actions, offers a different perspective. Pinsker’s theorem provides a foundational result, demonstrating that a deterministic action is independent of an action with completely positive entropy. ...