alt.hn

1/11/2025 at 10:25:37 PM

Arnold's Cat (2018)

http://gerdbreitenbach.de/arnold_cat/cat.html

by bradly

1/15/2025 at 8:21:25 PM

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.

by JadeNB

1/15/2025 at 8:34:18 PM

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.

by pepinator

1/16/2025 at 3:15:10 AM

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.

by kkylin

1/15/2025 at 8:45:37 PM

> 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.

by JadeNB

1/16/2025 at 3:03:59 AM

Cool, though I didn't understand it at all

by 01HNNWZ0MV43FF

1/16/2025 at 4:54:06 AM

[dead]

by bnetd