The fibers will not naturally breakdown so their lifespan is determined by the overall wear-and-tear on the arena as well as how well the arena is maintained.
The better you keep the fibers integrated into your sand, the longer they will last for. If you allow the arena to get very dry for long periods of time, your fibers will be more likely to rise to the surface where they are more susceptible to displacement by the wind or being tracked out of the arena for example.
You will of course gradually lose some footing in the horses’ feet and on your maintenance equipment, but this will only become noticeable after a long period of time.
The average arena finds it beneficial to refresh a portion of their sand and fibers after about 5 years. If your sand was more worn to begin with or you live in a particularly harsh climate, your arena may need to be refreshed sooner.
Arenas utilizing a very durable silica sand can expect a 10+ year lifespan out of their footing.