Recently, I am reading a paper titled "multi-invariant sets on tori" by D.Berend. I am puzzled by the three necessary and sufficient conditions given there. Could you provide me with some concrete examples satisfying the three conditions, say endomorphisms of T^2 or T^3? In addition, does there exist an convenient procedure for deciding whether two given non-degenerate matrices are communicate or not (e.g. for A,B in GL(2,R) or GL(3,R))?

Another question may be off the point. It is about entropy of the composition. Take A,B in GL(2,R)(or GL(3,R)) for example, can we decide when h(AB)=h(A)+h(B) or when the inequality holds?

Looking forward to your answers or suggestions.I will appreciate it very much!