Bounded Remainder Sets

摘要

The paper considers the category (T, S, X) consisting of mappings S :T −→T of spaces T with distinguished subsets X ⊂ T. Let r_X (i, x_0) be the distribution function of points of an S-orbit x_0, x_1 = S(x_0),.. , x_(i−1) = S~(i−1)(x_0) getting into X, and let δ_X (i, x_0) be the deviation defined by the equation r_X (i, x_0) = aX i + δ_X (i, x_0), where aX ~i is the average value. If δ_X (i, x_0) = O(1), then such sets X are called bounded remainder sets. In the paper, bounded remainder sets X are constructed in the following cases: (1) the space T is the circle, torus, or the Klein bottle; (2) the map S is a rotation of the circle, a shift or an exchange mapping of the torus; (3) X is a fixed subset X ⊂ T or a sequence of subsets depending on the iteration number i = 0, 1, 2,.. Bibliography: 27 titles.