Omega-Language to Büchi automaton
Intuitively, the language is exactly the set of words that have $a$ in all the odd places. Thus, the corresponding NBW (in fact, $DBW$) has three states, $q,s,r$, where $q$ is initial, $s$ is accepting, and $r$ is a rejecting sink. The transitions are $\delta(q,a)=s, \delta(q,b)=r$, and $\delta(s,a)=\delta(s,b)=q$.