PERIODS OF DISCRETIZED LINEAR ANOSOV MAPS

Integer m x m matrices A with determinant 1 define diffeomorphisms of the m-dimensional torus T-m = (R/Z)(m) into itself. Likewise, they def ine bijective self-maps of the discretized tori (Z/nZ)(m) = (Z(n))(m). We present estimates of the surprisingly low order (or period) Per(A) (n) of the iteration A(r), r = 1,2, 3,..., on the discretized torus (Z (n))(m). We obtain Per(A)(n) less than or equal to 3n for dimension m = 2. In the special case of the Anosov map [GRAPHICS] this result is d ue to Dyson and Talk [DT92]. For arbitrary dimensions m > 2 we obtain Per(A)(n) less than or equal to constant . n(m-1) provided n is a powe r of a prime number. For general n, number theoretic problems arise.