In general, and I'm not a lawyer, you buy a product and it has a warranty. I think one year is short for a $1000 monitor - Dell and HP offer 3 year warranties on their upscale IPS hi-res displays, but, regardless, it's one year with the Cinema Display.
If the warranty runs out, then, it's on you to fix it.
Now, if you buy a new product, and I'm only generalizing here and again, I am not a lawyer, but when you buy a new product and it breaks you take it in for warranty covered repairs. Then say it breaks again, and you get it fixed under warranty, and then it breaks again, while under warranty. Now you can start to press for a lemon law. You bought a new product, and even though it is covered under warranty, it's not acting like a new product and you are spending a lot of time getting it fixed and are without the use of the product while it is in for repairs.
So, if you are out of warranty, and this is the first issue, then, it's pretty much on you to get it fixed, again, disclaiming myself from anything resembling legal advice here.