On 14 Nov 2004 06:31:37 GMT, email@example.com (Robert Israel) wrote:
>Let F be the mapping (a,b,c,d) -> (|a-b|,|b-c|,|c-d|,|d-a|) on R^4. >Note that F is positively homogeneous (i.e. F(tx) = t F(x) if >t >= 0), so multiplying a vector by a positive scalar doesn't >change the number of iterations needed to reach (0,0,0,0). >[...] this mapping has arbitrarily long orbits (on Z^4 as >well as R^4).
If k is the unique real solution of k^3 + 2k^2 - 2 = 0, and u is the vector (1, k + 1, (k + 1)^2, k^2 + 3k + 3), then F(u) = (k, k(k + 1), k + 2, k^2 + 3k + 2) = ku (because k(k + 1)^2 = k + 2), therefore F^n(u) = (k^n)u \neq (0, 0, 0, 0) for all n >= 0.