LR60 Captain on extended left downwind at MVY reports conflict with aircraft doing aerobatics the Tower reported the traffic was not in MVY airspace.

Narrative

Weather was clear with good visibility. We were cleared for visual approach to Runway 24 MVY. Checked in with MVY Tower and cleared for left downwind to Runway 24. I descended to pattern altitude (1500msl) and entered the left downwind. Tower instructed us to extend our downwind for landing traffic/departure and told us the tower would call our base turn. As we continued on the downwind I observed an aircraft slightly higher than our altitude and between us and the extended centerline of MVY Runway 24 just past the shoreline (about 4 to 5 miles from MVY). The aircraft was flying erratically and there was no TCAS target. A white biplane was observed spinning down to our altitude and was near to abeam of our position when tower called our base turn. By that time the biplane was beginning a loop. I refused the turn from tower and continued downwind until I was sure to be clear of the aerobatic traffic. We then turned base to final with no incident. When I advised tower of the incident the reply was (he; the biplane is not in my airspace).Tower should not extend traffic into airspace they are not going to be responsible for. Aerobatic activities should be NOTAM'ed. I asked a few questions after my Passenger told me several biplanes operated commercially in the area. I was told there are two biplanes operating out of a field near MVY; one of which matched the aircraft I observed. The biplane operators should not operate near active runway approach paths. If commercial aerobatic operations are going on; how about designating a box for aerobatic operations like found anywhere else there are regular aerobatic ops like St. Augustine; FL. If the aircraft had his transponder on we might have had more warning of his proximity and I am sure if his transponder was on we would have got an RA. That might be why his transponder was off. I don't know if it is common practice to kill your squawk while doing aerobatics but I would recommend leaving it on. Tower did not seem to be concerned; MVY FBO personnel were familiar with the aerobatic operations.