It is quite possible to cycle a 1 gallon with no filtration. The majority of your beneficial bacteria will live in your gravel and on your ornaments. The tank will remain cycled even with 50-75% weekly/twice a week water changes. But it takes a looong time, like 12 weeks,twice as long as a large body of water.
Do you have a tank that is up and running already? If so, you can use gravel and water from it to boost the cycle along.
If you use a betta, you will have to extremely diligent with your water changes for the next few weeks. Fishless is good too, I know of one member who fishless cycled their small betta tanks and it took 8 weeks I think.