A horse that won't go into an arena is referred to as an "arena sour horse". This refusal to enter an arena is common with rodeo horses- particularly speed event horses. The horse learns after a few experiences that going into the arena means they will have to work very hard, and that they may be spurred, whipped, or have their mouths handled harshly when they do enter- therefore they begin to fight going in the arena.
If your horse won't go into the arena, consider this behavior a redflag to step back and examine your training and riding methods. This article covers the steps to retrain an arena sour horse, but unless you change the factors that caused the horse to become arena sour, the horse will revert back to not wanting to enter the arena again.
The best way to break an arena sour horse is to work in an arena several times a week- but only practice speed on the patterns about 25% of the time. So if you ride 4 days a week, only ask for speed once a week. In your first week of retraining, you may simply need to win the argument to get your horse through the in gate, then halt and dismount immediately when you go into the arena, and remove your horse's tack right where you stop.
This helps teach your horse to think- dismounting and untacking is the ultimate reward in horse training, if you reward going through in in gate so strongly your horse will begin to look forward to going through the in gate, in hopes he'll be rewarded.
After several sessions, you'll find your horse entering the arena increasingly willingly. When your horse starts entering without argument, begin mixing up your routine. Keep workouts easy, work frequently on the rail, then begin practicing turns and rollbacks on the rail. Frequently walk and trot your pattern- insisting your horse stay calm and in the gait you choose.
Speed horses WILL not loose time off their competition speed time through this type of training- on the contrary, it teaches them to think and listen to their rider instead of run a pattern mindlessly- and this gives a good rider the ability to have better turns and more control- which shaves more time in the long run.
So if your horse won't enter the arena 1 analyze why he won't enter the arena, and 2. mix up your training routine so your horse learns the arena doesn't mean unpleasant things will happen.