Using only one button in a platformer is an unusual way of controlling things, but I think I’ve come up with an easy way to do it.
The “Jumper” as the protagonist is called, will keep going in one direction. You can make him jump with the button, and holding down the button will cause him to stop and turn around. You can hold down the button for any length of time and the jumper with rock back and forth, indicating which direction he’ll go in once you release the button.
And the first screenshot of the game (with horribly ugly placeholder art):