When a lease expires the tenants become "holdover" tenants. I wouldn't say you're "bound" by the terms of the expired lease but the terms of the old lease apply to your relationship in terms of how much rent is due and when the rent is due. You can however seek to evict the tenant once the old lease has expired, which is the difference between having a lease and not having a lease. However, every time you accept rent from the holdover tenant you delay being able to evict them. Holdover proceedings require the service of a 30-day notice for the tenant to vacate. Every time you accept rent you agree to give them another 30 day period to live there and any 30-day notice that you may have sent becomes ineffective and you have to start over. Once you get to court, the judge is empowered to give them up to 6 months to stay (that doesn't mean that the judge will) but typically a deal is worked out for the tenant to leave.
So if you're only talking about the a/c fee and the tenant is otherwise paying the rent, you should consider whether its worth it to you to evict them over the a/c fee.
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