Let us consider a continued fraction of the form $$\frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}}, $$ which we will denote by $[a_1, a_2, a_3, \cdots ]$, where $(a_1, a_2, \cdots)$ is a sequence of positive integers (and all continued fractions will consists of sequences of positive integers).
For a continued fraction, we may consider its "left shifts" $$G([a_1, a_2, a_3, \cdots ]) := [a_2, a_3, a_4, \cdots, ], $$ (that is, $G$ denotes the Gauss map).
I would like to construct a continued fraction $x := [a_1, a_2, a_3, \cdots ]$ for which $$\lim_{n \to \infty} G^{2n}(x) = \frac{1}{3}. $$ (I believe the question could be formulated for any number in $(0, 1)$, not only for $1/3$).
I am not exactly sure how to proceed with the above problem. I am not exactly sure what the relationship is between the above convergence and the even iterates of the Gauss map.
I thought there is some relationship between the even iterates of the Gauss map and the convergents of a continued fraction $[a_1, a_2, \cdots]$, but I am not sure about it. According to All Even-Numbered Convergents of a Finite Continued Fraction Are Less Than the Value , if $(p_n)_{n \in \mathbb{N}}$ and $(q_n)_{n \in \mathbb{N}}$ are the convergents of $\{a_1, a_2, a_3, \cdots \}$, defined as $$p_0 := 0, p_1 := 1, p_n := a_n p_{n - 1} + p_{n - 2} \text{ for } n \geq 2 $$ and $$q_0 := 1, q_1 := a_1, q_n := a_n q_{n - 1} + q_{n - 2} \text{ for } n \geq 2,$$ we then have $$\frac{p_n}{q_n} = [a_1, a_2, \cdots, a_n] $$ for every $n \geq 1$ and the sequence $$ \left( \frac{p_{2n}}{q_{2n}} \right)_{n \in \mathbb{N}} $$ is increasing. Does this imply that the sequence $(G^{2n}(x))_{n \in \mathbb{N}}$ would be increasing or decreasing?