Mathison and the Dynamo shuffle

Mathison has bought a new deck of cards that contains 2N cards, numbered and ordered from 0 to 2N-1. Mathison is getting bored and decides to learn the Dynamo shuffle (or Sybil cut) - a flashy manner to shuffle cards. Unfortunately, the Dynamo shuffle is a little too complicated so Mathison decides to create his own shuffle. This newly invented shuffle is done in N steps. At step k (0 ≤ k N) the