Extensions and restrictions of Wythoffʹs game preserving its positions

We consider extensions and restrictions of Wythoffʹs game having exactly the same set of P positions as the original game. No strict subset of rules gives the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P positions. Testing if a move belongs to such an extended set of rules is shown to be doable in polynomial time. Many arguments rely on the infinite Fibonacci word, automatic sequences and the corresponding numeration system. With these tools, we provide new two-dimensional morphisms generating an infinite picture encoding respectively P positions of Wythoffʹs game and moves that can be adjoined.