For $\alpha \in (0,1)$, write $\alpha$ as a continued fraction like $\alpha=[a_1, a_2, \ldots]$ (note that the implicit $0$th coefficient $a_0=0$ has been omitted), and let $\frac{p_n}{q_n}$ be the $n$th convergent to $\alpha$. If you have $T$ the Gauss map, $T(x) = \frac{1}{x}-\left\lfloor\frac{1}{x}\right\rfloor$ (which acts on a continued fraction by $T([a_1,a_2,\ldots]) = [a_2, a_3, \ldots])$ then how can the value of $$\sum_{T^n x=x}\frac{1}{q_n^t},$$ be estimated for $t$ positive and fixed $n \in \mathbb{N}$?
Thanks.