Some Metric Properties For Continued ?-Fraction Expansions

Combining continued fractions and ?-expansions,the thesis studys a new kind of expansion of real numbers:the continued ?-fraction expansions.We focus on the metric properties of the continued ?-fraction expansions and obtain the results as followings:First,we study some basic properties of ?-integers and continued ?-fractions.Let?>1 be a root of the polynomial t2=at-1+1 with a ? N,a?1 or a root of the polynomial t2=at-1 with a?N,a?3.Let an(x)be the n-th partial quotients in the continued?-fraction expansions of x and ? be a postive function defined on N.We show that the Lebesgue measure of the following set E(?)={x?[0,1):an(x)??(n)for infinitely many n?N}obeys a dichotomy law according to the convergence or divergence of a series.More precisely,the Lebesgue measure of E(?)is null or full according to the convergence or divergence of the series?n=1? 1/?(n).As a result,the set of numbers in the interval[0,1)with bounded partial quotients in their continued ?-fraction expansions is of zero Lebesgue measure.Second,we study the measure preserving transformation with ergodic root systems,and show that if a measure preserving transformation admits an ergodic ?-th root system,then such a system has at most ? ergodic components,and it can be represented as the convex combinations of these ergodic components.