3 ms·
The page doesn't say why matrices need to have whole-number (really just integer) entries, but I'd suspect it's because bad approximations to non-integer ration
by JadeNB 2y ago
The page doesn't say why matrices need to have whole-number (really just integer) entries, but I'd suspect it's because bad approximations to non-integer rationals accumulate sufficiently to make the recurrence unrecognizable.
- pepinator 2y agoIt says that the underlying action is on the torus (R/Z)^2. If the entries of the matrix are not integers, do we have a well defined action on the torus? It seems to me that the answer is no because Z^2 would not be invariant by the action.
- JadeNB 2y ago> It says that the underlying action is on the torus (R/Z)^2. If the entries of the matrix are not integers, do we have a well defined action on the torus? It seems to me that the answer is no because Z^2 would not be invariant by the action. Ah, good point.
- kkylin 2y agoOne can still define a map by taking the fractional part of the matrix-vector product, but the resulting map won't be continuous (with respect to the topology of the torus). In addition, if one wants the map be a homeomorphism (continuous with continuous inverse) then the determinant must have absolute value 1.