In general, the police need a warrant to come into your home. There are some exceptions to the warrant requirement, though. One of them is "hot pursuit."
It sounds like - from your posts - that you have a different version of what happened than the police. The judge or jury will listen to both sides, and decide who's telling the truth.
Without knowing what the police are saying, it's hard to give you advice.
Talk to a lawyer, and find out exactly what the police are saying you did.
If it's a fine-only offense - one for which the maximum penalty is just a fine - you'll have to decide whether it's worth it to you to fight the case, or whether you'd prefer to just pay the fine.
It sounds like - from your posts - that you have a different version of what happened than the police. The judge or jury will listen to both sides, and decide who's telling the truth.
Without knowing what the police are saying, it's hard to give you advice.
Talk to a lawyer, and find out exactly what the police are saying you did.
If it's a fine-only offense - one for which the maximum penalty is just a fine - you'll have to decide whether it's worth it to you to fight the case, or whether you'd prefer to just pay the fine.