Foggy Bottom,
You would have to look principally in the ICAO Convention on International Civil Aviation - it is unlikely that you will find the answers you seek in any of the Annexes. It will also be necessary to look in the Primary Legislation of both States.
in General (as I am not a lawyer):
You need to provide more information; what exactly are the operations you are conducting - are they General Aviation (GA) - i.e. private, or commercial? It is generally accepted that you are operating Internationally if conducting operations outside the boundaries of the State of Registration (not just when crossing the border).
If the aircraft is N registered and flying in Peru for GA; you should be flying: in compliance with the relevant Parts of FARs; the more stringent of the Rules of The Air as defined by the FAA, ICAO or Peru; or the more stringent of the operational requirements provided by the FAA or Peru.
A number of States limit the amount of time the aircraft resides - if it remains in their airspace (this is difficult to establish if records of flights are not retained by the State of Operation); this can be done by requiring you to vacate the airspace or re-register the aircraft in the State of Operation. Even if you remain in a foreign State, responsibility for oversight still remains with the State of Registration.
Conducting commercial operations in another State requires either, some protocol agreement (as exists between the US and Canada), or a transfer of Operational Oversight using ICAO Bis 83 (this is an agreement between the two States that responsibility for oversight has been formally transferred) - this permits the aircraft to remain registered to the original State. Other local agreements on (wet or dry) leasing can also be concluded.
ICAO requires CAT Operators to hold an AOC; this has to be issued by the State of Operations (which is usually the State of the Operator). The issuance of an AOC by the State of Operations to a foreign operator, without a Bis 83 transfer, should be seen as a rare occurrence - it places both the State of Registration and the State of Operations in difficult positions.
More information please.
Jim