This talk is based on a work with Cantat and Deserti. According to the degree sequence, there are 4 types (elliptic, Jonquieres, Halphen and Loxodromic) of elements f in Bir().  For a fixed degree , we study the set of these 4 types of elements of degree d. We show that for Halphen twists and Loxodromic transformations, such sets are constructible. This statement is not true for elliptic and Jonquieres elements.We also show that for a Jonquieres or Halphen twist f of degree d, the degree of the unique f-invariant pencil is bounded by a constant depending on d. This result may be considered as a positive answer to the Poincare problem of bounding the degree of first integrals,but for birational twists instead of algebraic foliations. As a consequence of this, we show that for two Halphen twists f and g, if they are conjugate in Bir(f), then they are conjugate by some element of degree bounded by a constant depending on deg(f)+deg(g). This statement is not true for Jonquieres twists.