Y by Jupiterballs
Let
be an irrational number and
be a integer such that
. A sequence
satisfies that
and for all positive integers
,
Prove that
(i)
is eventually periodic.
(ii) The eventual fundamental period of
is an odd integer which doesn't depend on the choice of
.






![\[ x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \textup{if} \; x_n \leqslant L \\\left \lfloor \frac{x_n}{\alpha} \right \rfloor & \textup{if} \; x_n > L \end{cases}. \]](http://latex.artofproblemsolving.com/0/f/6/0f69e8256213a0b14768cc29ad7db84b7d3880df.png)
(i)

(ii) The eventual fundamental period of


This post has been edited 2 times. Last edited by Photaesthesia, Nov 27, 2024, 7:43 AM