Let $\Sigma=\{A,B,C\}$ be an alphabet, and let $\Sigma^{\mathbb{N}}$ be the set of infinite sequences on $\Sigma$ (ie $ABCBCCCBABC...$). By outside conditions, I have several subsequences that are disallowed, namely $AA, BB, CC, ABAB, ACAC, BABA, BCBC, CACA, CBCB, ACB, BAC, CBA$, and I wish to show that all $\sigma\in\Sigma^{\mathbb{N}}$ are eventually periodic. I'd like to know if there are any results on this subject or any suggestions of ways to proceed on a proof.
Previous attempts: My original thought was that I could create a counterexample by making blocks of length 3 to represent 0 and 1 and then to create some sort of irrational decimal with these (say $\pi$ in binary, for example). I went through the allowable combinations and found that this was impossible. Doing this again for larger blocks seems possible, but I'd like to find some general results on this problem.
(Apologies for the tags, I can't quite figure out what good ones are for this question.)