In the supervisory control framework of discrete-event systems (DES) with infinite behavior initiated by Thistle and Wonham, a supervisor satisfying the minimal acceptable specification and the maximal legal specification is synthesized. However, this supervisor may incur livelocks as it cannot ensure that the infinite behavior under supervision will always visit some marker states. To tackle this problem, we propose the definition of markability by requiring that all infinite cycles include at least one marker state. Then we formulate the problem of $\omega-$nonblocking supervisory control of DES with infinite behavior to synthesize an $\omega-$nonblocking (i.e. nonblocking, deadlock-free and livelock-free) supervisor. An algorithm is proposed to achieve $\omega-$nonblockingness by computing the supremal $*-$controllable, $*-$closed, $\omega-$controllable and markable sublanguage. We utilize the example of a robot as a running example.