Vanishing of the entropy pseudonorm for certain integrable systems Boris S. Kruglikov and Vladimir S. Matveev Abstract We introduce the notion of entropy pseudonorm for an action of R n and prove that it vanishes for the group actions associated with a big class of integrable Hamiltonian systems. 1. Entropy pseudonorm Let W be a smooth manifold and Φ : (R n , +) → Diff(W ) a smooth action on it. Assume there exists a compact Φ-invariant exhaustion of W . Define the following function on R n (where h top is the topological entropy): ρ Φ (v)= h top (Φ(v)), v ∈ R n . This function is a pseudonorm on R n (ρ Φ (v) is well-defined because with our hypothesis the entropy h d of [Bo] does not depend on the distance function d, homogeneity is standard and the triangle inequality follows from the Hu formula [H]). We call ρ Φ the entropy pseudonorm. We will investigate it in the case of the Poisson action corresponding to an integrable Hamil- tonian system on a symplectic manifold (W 2n ,ω). Namely, let (W 2n ,ω) possess pair-wise Poisson commuting functions I 1 ,I 2 ,...,I n , which are functionally independent almost everywhere. Denote by ϕ τ i the time τ shift along the Hamiltonian vector field of the function I i . The maps ϕ τ i commute and therefore generate the Poisson action of the group (R n , +), Φ(τ 1 ,...,τ n ) def = ϕ τ1 1 ◦···◦ ϕ τ n n : W 2n → W 2n , with the corresponding momentum map Ψ = (I 1 ,...,I n ): W 2n → R n , see [A]. The entropy pseudonorm ρ Φ vanishes in the following important cases: − Williamson-Vey-Eliasson-Ito non-degenerate singularities [E, I]; − Taimanov non-degeneracy condition [T]. In the first case vanishing of topological entropy of the Hamiltonian flow was proved in [P2], in the second case in [T]. Since there is nothing special about the Hamiltonian in these situations, it can be changed to any of the integrals and ρ Φ ≡ 0 follows. Also in [P1, BP]) vanishing of h top was proven for the cases: − Systems integrable with periodic integrals; − Collectively integrable systems (the definition is in [GS]). It is not difficult to see that in both cases the entropy pseudonorm ρ Φ vanishes as well. Note that Liouville integrability does not imply vanishing of topological entropy, see [BT] (more examples in [Bu]). For these examples the entropy pseudonorm is degenerate, but it is possible to construct integrable examples [K] such that ρ Φ is a norm. In the present paper we prove vanishing of the entropy pseudonorm for another class of in- tegrable systems. These systems were recently actively studied in mathematical physics in the 1