The 3x C 1 map T is defined on the 2-adic integers Z 2 by T .x/ D x=2 for even x and T .x/ D .3x C 1/=2 for odd x. Under iteration of T , the sequence .T k .x/ mod 2/ 1 kD0 , called the parity vector of x 2 Z 2 , can be interpreted as an infinite word over the alphabet ¹0; 1º or as the digits of the 2-adic integerˆ 1 .x/ D P 1 kD0 .T k .x/ mod 2/ 2 k . For any v 2 Z 2 (or equivalently for the infinite word v), the inverse mapˆ(called the 3x C 1 conjugacy map) yields the unique x 2 Z 2 with parity vector v. It is unknown if there exists any aperiodic v with an eventually periodicˆ.v/. In this paper we computeˆ.v/ for a class of aperiodic infinite words v of minimal complexity, the mechanical words with irrational slope and intercept 0. Our main result is a generalized continued fraction expansion of 1=ˆ.x/, convergent under the 2-adic metric of Z 2 . The given examples suggest thatˆalways maps Sturmian words to infinite words of full complexity.

An infinite word x over an alphabet with b letters has full complexity if for each m 2 N all the b m words of length m are factors of x. We prove that the periods of˙1=3 k in the 2-adic expansion approach full complexity for increasing k: For any m 2 N, the periods for k > d.m C 1/ ln.2/= ln.3/e have complexity 2 m . Amazingly, these 2 m words occur in the period almost the same number of times. On the way, first we prove the same for the binary period. We get a similar result for the decimal period of 1=3 k .