When operating a station in South America by remote control over the internet from the US, what regulations apply?

In amateur radio, the rules and regulations that govern a transmission are determined by the physical location of the transmitter (the RF emissions), not the location of the control operator. Because the remote station's transmitting equipment is physically located in South America, it is subject only to the laws and telecommunications regulations of that specific country. The FCC only has jurisdiction over transmitters located within the United States or its territories, or on US-registered vessels. Therefore, even if you are an American licensed operator sitting in the US and controlling the station over the internet, FCC regulations do not apply to the RF being generated in another sovereign nation.