Rigidity properties for commuting automorphisms on tori and solenoids

In this paper, the authors investigate measure rigidity theorems for higher rank actions on tori and solenoids by commuting automorphisms. They assume positive entropy and utilize this assumption to establish the measure rigidity theorem. Furthermore, Einsiedler and Lindenstrauss apply their results to achieve a comprehensive classification of disjointness and measurable factors for these actions. By doing so, they provide valuable insights into the behavior of commuting automorphisms on tori and solenoids. The paper spans 43 pages in memory of Anatole Katok. These categories pertain to topics such as smooth dynamical systems, ergodicity, and entropy. Overall, this research contributes to our understanding of measure rigidity in higher rank actions on tori and solenoids by commuting automorphisms. It also offers a complete classification of disjointness and measurable factors associated with these actions that can help us gain further insight into the dynamics of such systems.

Measure Rigidity Theorems for Higher Rank Actions on Tori and Solenoids by Commuting Automorphisms

In this paper, Einsiedler and Lindenstrauss investigate measure rigidity theorems for higher rank actions on tori and solenoids by commuting automorphisms. This research is dedicated to Anatole Katok, a renowned mathematician who made significant contributions to the fields of smooth dynamical systems, ergodicity, and entropy. Spanning 43 pages in his memory, this paper offers valuable insights into the behavior of such systems.

Positive Entropy Assumption

The authors assume positive entropy in order to establish their measure rigidity theorem. Positive entropy is a concept that states that if two different points are chosen at random from a given set of points within a system, then it is highly likely that they will be far apart from each other. In other words, there is an increased probability that these two points will not be close together or have any sort of relationship with one another. This assumption allows them to better understand how higher rank actions on tori and solenoids behave when acted upon by commuting automorphisms.

Classification of Disjointness & Measurable Factors

Einsiedler and Lindenstrauss apply their results to achieve a comprehensive classification of disjointness and measurable factors for these actions. By doing so, they provide further insight into the dynamics of such systems by showing how certain measurable factors can affect the behavior of higher rank actions on tori and solenoids when acted upon by commuting automorphisms. For example, they show how certain measurable factors can lead to disjointness between two sets within the system or cause them to overlap with one another depending on their respective properties.

Conclusion

Overall, this research contributes to our understanding of measure rigidity in higher rank actions on tori and solenoids by commuting automorphisms as well as provides us with a complete classification of disjointness and measurable factors associated with these actions that can help us gain further insight into the dynamics of such systems.