Bounded Remainder Sets

Abstract

The paper considers the category (\( \mathcal{T} \), S, X) consisting of mappings S :\( \mathcal{T} \) −→\( \mathcal{T} \) of spaces \( \mathcal{T} \) with distinguished subsets X ⊂ \( \mathcal{T} \). Let r_X (i, x₀) be the distribution function of points of an S-orbit x₀, x₁ = S(x₀), . . . , x_i−1 = Sⁱ⁻¹(x₀) getting into X, and let δ_X (i, x₀) be the deviation defined by the equation r_X (i, x₀) = aX i + δ_X (i, x₀), where aX ⁱ is the average value. If δ_X (i, x₀) = 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 \( \mathcal{T} \) 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 ⊂ \( \mathcal{T} \) or a sequence of subsets depending on the iteration number i = 0, 1, 2, . . .. Bibliography: 27 titles.