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.
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.
One 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.
> 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.
Managed to time-sync the frame transitions to an old Juno Reactor album and i'm right back in some weird MTV esque vibe like I'm watching Beavis and Butthead or Duckman or something.
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.
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.
One 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.
> 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.
Managed to time-sync the frame transitions to an old Juno Reactor album and i'm right back in some weird MTV esque vibe like I'm watching Beavis and Butthead or Duckman or something.
Cool, though I didn't understand it at all
I clicked the link wondering if it would be about this "Arnold the Cat" or perhaps related to Simon's Cat:
https://www.publishersweekly.com/9780316638111