Illegal Dribble or Travel
 

A1 is dribbling up the court, drives to the basket for a layup, takes his permitted step, while airborne decides not to shoot (or pass) and lets the ball fall to the floor directly in front of him. Ball bounces on the floor, A1 returns to the floor (not really sure which happened first), and A1 then gathers the ball without it having been touched by any other player. Is this an illegal dribble violation or a travel violation?

Thanks.

Illegal dribble.

Both, by deliberately dropping the ball to the floor the player started a second dribble. The player also began a dribble after lifting his pivot foot from the floor, which violates the traveling rule.
It doesn't matter which violation is called. The result is the same.

It's both. Put another ball in play and let B inbound them both.

Thanks. This explains why every time I thought it was one I thought of reasons why it was the other.

From The List ...
 

The shooter can retrieve his or her own airball, if the referee considers it to be a shot attempt. The release ends team control. It is not a violation for that player to start another dribble at that point. When an airborne player keeps control of an attempted shot that is blocked, is unable to release the ball, and returns to the floor with it, that player has not traveled; it is a held ball. If, in this situation, the shooter loses control of the ball because of the block, then this is simply a blocked shot, and play continues. If, in this situation, the defender simply touches the ball, and the airborne shooter returns to the floor holding the ball, it’s a traveling violation. When an airborne player tries for goal, sees that the try will be blocked, purposely drops the ball, and touches the ball after it hits the floor, that player has traveled by starting a dribble with the pivot foot off the floor.