Generalized Gauss transformations

摘要

For a fixed real number p⩾1, let Tp be the transformation on the unit interval defined by Tp(x)={1/xp}, where {x} is the fractional part of x. Let ρp(x)dx be the Tp-invariant absolutely continuous ergodic measure. In [Appl. Math. Comput. 109 (2000) 287], it was asked whether ρp(x) converges to 1 or not in the supremum norm as p→∞. We give a positive answer for this question. Let [a1,a2,…]Tp be a symbolic representation of x∈[0,1) induced by Tp. It is also shown that the distribution of relative frequency of k∈N in [a1,a2,…]Tp satisfies the central limit theorem.