The atomic cross-chain swapping means I can have assets or a coin on one chain and transfer it to another. In this case maybe transfer coins from bitcoin to the xaya blockchain for chi. It wouldn't really get rid of the transaction fees for bitcoin, but what it would do is allow you to transfer some amount of bitcoin out if you know you are going to be doing high frequency transactions. You could just do it on a lower fee blockchain and then put it back in your bitcoin account when you have done what you wanted to do.

There is another possibility on Xaya and I believe Ethereum has implemented as well now. On Xaya they are called game channels. The idea is that you allow a side branching chain to develop between an arbitrary number of people, then when a game is over you inject the end of the side chain back into the main chain and verify it. They invented it for near realtime verification between consenting parties, then you only pay the main transaction fee when you get back on the main block instead of very every action. But that type of system doesn't have to be only done with games obviously, that could be applied to anything.

If you are interested more in it, you can check this video out. They have lots of documentation and tutorials for the network as well.

https://www.youtube.com/watch?v=wMd9mvc8pqA