Zoysia is a species of warm-season grass known for its dense clumping growth habit and hard-wearing nature. It is notably slow growing and hence easier to control near hardscape surfaces than a Bermuda grass would be. While slow growing and clumping by nature, zoysia does produce creeping stolons and rhizomes to help it spread in the landscape. Occasional maintenance will easily keep zoysia from invading sidewalk space.
Mow your zoysia lawn regularly to maintain it at a blade height of between one and three inches. Regular mowing will help to keep the spreading-overground stolons in check.
Engage the edging function on your lawn mower every other or every third time you mow the lawn. Run the edger side of the mower along the seam between the zoysia grass and the sidewalk to cut a gully and sever any creeping rhizomes crossing between the two.
Use a separate mechanical or hand edger tool between the zoysia grass and any adjacent sidewalks or hard surfaces, to cut the a line between the two surfaces and sever the creeping plant material.
Don gloves and hand pull any errant creeping zoysia rhizomes or stolons that you see on or near the sidewalk between mowings and edging maintenance. Ripping the stolons and rhizomes up and out of the soil will stunt regrowth considerably, and the grass will likely not regrow in that location before your next session of maintenance edging.