The Eventually Periodic Continued ?-Fractions And Their Lévy Constants

In this paper,we study the arithmetic and dynamic properties of continued-fraction.(?),we obtain some properties of the eventually periodic continued-fractions and the invariant measures of corresponding continued-fraction transforma-tion.Specifically,it can be divided into several parts:Firstly,the symbolic space corresponding to the continued-fraction expansion is a finite sub-shift composed of countable symbols(denoted as (?)(?)~N is not admissible and We give the necessary and sufficient conditions for the admissible sequence.Furthermore,similar to Lagrange's theorem in continuous frac-tions,we prove that the nonnegative real number whose continued-fraction expansion is eventually periodic is the root of a quadratic irreducible polynomial with the coefficients in Z[?].Secondly,the L?evy constant reflects the exponential growth rate of the denomina-tor of convergence of continued-fraction expansion.We prove that the set of L?evy (?)(ntually periodic continued-fractions is dense in[c,+?),where)Thirdly,measure-preserving transformation is an important research object of dy-namic system.Then we obtain the existence of an invariant measure equivalent to the Lebesgue measure for continued?-fraction transformation and give the relation the rela-tion of density function of invariant measure relative to the Lebesgue measure.