ATS stands for Alternative Trading System.
Regulatory change in recent years has brought the ATS type of securities exchange into existence.
An ATS is an after-market exchange where people who own securities can buy and sell.
Because all Reg A+ offerings are Public Offerings, listing Reg A+ securities during the offering and after the Reg A+ completes on an ATS is a useful way for a company to provide liquidy to its investors.
Rule 144A securities can be bought and sold on ATS exchanges.
One big advantage of ATS exchanges is that shorting of stocks is not possible and naked shorting is not possible. (stockbrokers are allowed to put shorts on stocks on the OTC, NASDAQ, and NYSE, without being required to borrow the securities - so some put naked short positions on stocks - which can be very damaging to the companies they choose). the SEC regulation only requires stockbrokers to have reasonable access to borrow. That is a very ambiguous term that in practice allows naked shorting.
The fees to list are approximately $10k to $20k depending on the specific ATS, plus the cost of filing with the States for secondary market Blue Sky exemption.