Discretized Rotation with fixed initial points

Shigeki Akiyama, Bill Mance

We prove that if $\max \{|a_0|,|a_1|\}\le 10$ and $\hat{I}»\in ]-2,2[$, then the sequence defined by $$0\le a_{n+2}+\hat{I}»a_{n+1}+a_n<1$$ is periodic.