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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!

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Do you want $A,B$ to be in $GL(d,Z)$? If they are only in $GL(d,R)$ then they do not necessarily induce an action on the torus. If two non-degenerate matrices commute then they must have the same invariant subspaces, but the converse is not true in general (consider two three-dimensional upper-triangular matrices). To obtain $h(AB)=h(A)+h(B)$ for $A,B \in GL(2,Z)$ we are asking that the spectral radii of $A$ and $B$ should multiply, which under mild conditions implies that they are simultaneously diagonalisable. – Ian Morris Oct 9 '13 at 16:03
It would be nice to have the conditions in the question itself, so as to make it self contained. – j.c. Dec 17 '13 at 18:21

For a start, I guess that one should be very familiar of the proofs of Furstenberg's diophantine result, as this paper generalizes this theorem. Secondly, it might be of interest that Zhiren Wang in his PhD thesis generalized Berend's results to an effective proof (in a method which is some-what similar to the method which Bourgain-Lindenstrauss-Michel-Venkatesh used to generalize Hillel results).

$(3)$ is easy, just like in Furstenberg's theorem, otherwise, the whole semi-group would be contained in a semigroup spanned by one element, and it is well-known that such semigroups (in the case of diagonal actions, in contrary to Ratner's theorem) do not enjoy measure rigidity. This is somewhat equivalent to the assumption of higher-rank in the Einsiedler-Katok-Lindenstruass theorem about rigidity of higher-rank diagonalizable actions.

$(2)$ roughly corresponds to the fact that the system is expansive system. Otherwise (as shown by Berend in his paper about necessity of the conditions) - you can find some small set "which won't grow", due to the fact that every matrix will contract the set and won't spread it out.

$(1)$ is more mysterious, and as Dani sates in his paper, it implies that the action is basically a diagonal action (actually conjugated to a diagonal one, but it won't effect the rigidity).

The other question should be answered in another topic I guess, but Ian said the condition more or less, if you are familiar with Sinai/Pesin's formula for entropy (or in this case, the geometry of Markov partitions from the Adler-Weiss theorem), you can massage the proof a bit to get a proper condition (assuming some conditions on the matrices, say commutation relations). I believe that the formal proofs appear in Walters or in the new Einsedler-Ward volume about entropy.

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