Invariant games and non-homogeneous Beatty sequences

The aim of this talk is to introduce some notions arising in combinatorial game theory and make the connection with combinatorics on words.
We characterize all pairs of complementary non-homogenous Beatty sequences (A_n)n≥0 and (B_n)n≥0 for which there exists an invariant game having exactly {(A_n,B_n)∣n≥0}∪{(B_n,A_n)∣n≥0} as set of P-positions. Using the notion of Sturmian word and tools arising in symbolic dynamics and combinatorics on words, this characterization can be translated to a decision procedure relying only on a few algebraic tests about algebraicity or rational independence. Given any four real numbers defining the two sequences, up to these tests, we can therefore decide whether or not such an invariant game exists.