We had a neighbor like that in our old house, however he only butchered our bushes and trapped our cat.
We also have one up here which will shoot deer when they're clearly on our property. If we catch him in the act then he says, "I wounded it on my property and it ran over to yours.". If you're such a bad shot then why are you even hunting?
Here's what you do to fix the problem wrote:
Dig a pit (about 5ft deep should do) right under the gate, then add 4ft of 100% pure mud (the thicker the better! No solid dirt), then build a trap door over it. Rig the trap door to open when the gate is opened. Then you put a sign 5ft inside your yard which says, "I wouldn't open that if I were you.".
If he disobeys the sign and opens the gate then the trapdoor opens and he takes a mud bath (like in Ooglog!

). Depending on how heavy and strong he is determines whether he can get out by himself or not. If the mud is thick enough, and since there is 4 feet of it, he should have quite a time trying to get out.
If he can't get out and you finally have to help him out then write up a paper on the computer which says he promises he'll never come into your yard without your permission first. Then print it and make him sign it. If he doesn't sign then he stays in the mud bath until someone else comes. If he does sign then you go into your house with the signed document and scan it to your computer and then save it as a weird filename and hide the original in an inconspicuous place (just in case he comes in your house to destroy the paper). After the paper is safe you can help him out, but be prepared to run.
If that doesn't work, or you just don't want to go through all that, then just tell him that he's trespassing and if he does it again you
will call the police.
If he's on your property without your permission then you can call the police, don't be shy.