A truncated Gauss-Kuzmin law Academic Article

The transformations T n {T_n} which map x [ 0 , 1 ) x in [0,,1) onto 0 0 (if x 1 / ( n + 1 ) x leqslant 1/(n + 1) ), and to { 1 / x } { 1/x} otherwise, are truncated versions of the continued fraction transformation T : x { 1 / x } T:x o { 1/x} (but 0 0 0 o 0 ). An analog to the Gauss-Kuzmin result is obtained for these T n {T_n} , and is used to show that the Lebesgue measure of T n k { 0 } T_n^{ - k}{ 0} approaches 1 1 exponentially. From this fact is obtained a new proof that the ratios / k
u /k , where